derived-lemma-projective-resolutions-exist

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough projectives. \begin{enumerate} \item Any object of $\mathcal{A}$ has a projective resolution. \item If $H^n(K^\bullet) = 0$ for all $n \gg 0$ then $K^\bullet$ has a projective resolution. \item If $K^\bullet$ is a complex with $K^n = 0$ for $n > a$, then there exists a projective resolution $\alpha : P^\bullet \to K^\bullet$ with $P^n = 0$ for $n > a$ such that each $\alpha^n : P^n \to K^n$ is surjective. \end{enumerate}


derived-lemma-projective-into-acyclic-is-zero

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be an acyclic complex. Let $P^\bullet$ be bounded above and consisting of projective objects. Any morphism $P^\bullet \to K^\bullet$ is homotopic to zero.


derived-lemma-morphisms-lift-projective

Let $\mathcal{A}$ be an abelian category. Consider a solid diagram $$ \xymatrix{ K^\bullet & L^\bullet \ar[l]^\alpha \\ P^\bullet \ar[u] \ar@{-->}[ru]_\beta } $$ where $P^\bullet$ is bounded above and consists of projective objects, and $\alpha$ is a quasi-isomorphism. \begin{enumerate} \item There exists a map of complexes $\beta$ making the diagram commute up to homotopy. \item If $\alpha$ is surjective in every degree then we can find a $\beta$ which makes the diagram commute. \end{enumerate}


derived-lemma-morphisms-from-projective-complex

Let $\mathcal{A}$ be an abelian category. Let $P^\bullet$ be bounded above complex consisting of projective objects. Let $L^\bullet \in K(\mathcal{A})$. Then $$ \Mor_{K(\mathcal{A})}(P^\bullet, L^\bullet) = \Mor_{D(\mathcal{A})}(P^\bullet, L^\bullet). $$


derived-lemma-precise-vanishing

Let $\mathcal{A}$ be an abelian category. Let $P^\bullet$, $K^\bullet$ be complexes. Let $n \in \mathbf{Z}$. Assume that \begin{enumerate} \item $P^\bullet$ is a bounded complex consisting of projective objects, \item $P^i = 0$ for $i < n$, and \item $H^i(K^\bullet) = 0$ for $i \geq n$. \end{enumerate} Then $\Hom_{K(\mathcal{A})}(P^\bullet, K^\bullet) = \Hom_{D(\mathcal{A})}(P^\bullet, K^\bullet) = 0$.


derived-lemma-lift-map

Let $\mathcal{A}$ be an abelian category. Let $\beta : P^\bullet \to L^\bullet$ and $\alpha : E^\bullet \to L^\bullet$ be maps of complexes. Let $n \in \mathbf{Z}$. Assume \begin{enumerate} \item $P^\bullet$ is a bounded complex of projectives and $P^i = 0$ for $i < n$, \item $H^i(\alpha)$ is an isomorphism for $i > n$ and surjective for $i = n$. \end{enumerate} Then there exists a map of complexes $\gamma : P^\bullet \to E^\bullet$ such that $\alpha \circ \gamma$ and $\beta$ are homotopic.


derived-lemma-injective-acyclic

Let $\mathcal{A}$ be an abelian category. Let $I \in \Ob(\mathcal{A})$ be an injective object. Let $I^\bullet$ be a bounded below complex of injectives in $\mathcal{A}$. \begin{enumerate} \item $I^\bullet$ computes $RF$ relative to $\text{Qis}^{+}(\mathcal{A})$ for any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into any triangulated category $\mathcal{D}$. \item $I$ is right acyclic for any additive functor $F : \mathcal{A} \to \mathcal{B}$ into any abelian category $\mathcal{B}$. \end{enumerate}


derived-lemma-enough-injectives-right-derived

Let $\mathcal{A}$ be an abelian category with enough injectives. \begin{enumerate} \item For any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into a triangulated category $\mathcal{D}$ the right derived functor $$ RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D} $$ is everywhere defined. \item For any additive functor $F : \mathcal{A} \to \mathcal{B}$ into an abelian category $\mathcal{B}$ the right derived functor $$ RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B}) $$ is everywhere defined. \end{enumerate}


derived-lemma-right-derived-properties

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor. \begin{enumerate} \item The functor $RF$ is an exact functor $D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$. \item The functor $RF$ induces an exact functor $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$. \item The functor $RF$ induces a $\delta$-functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$. \item The functor $RF$ induces a $\delta$-functor $\mathcal{A} \to D^{+}(\mathcal{B})$. \end{enumerate}


derived-lemma-higher-derived-functors

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor. \begin{enumerate} \item For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of complexes in $\text{Comp}^{+}(\mathcal{A})$ there is an associated long exact sequence $$ \ldots \to H^i(RF(A^\bullet)) \to H^i(RF(B^\bullet)) \to H^i(RF(C^\bullet)) \to H^{i + 1}(RF(A^\bullet)) \to \ldots $$ \item The functors $R^iF : \mathcal{A} \to \mathcal{B}$ are zero for $i < 0$. Also $R^0F = F : \mathcal{A} \to \mathcal{B}$. \item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective. \item The sequence $(R^iF, \delta)$ forms a universal $\delta$-functor (see Homology, Definition \ref{homology-definition-universal-delta-functor}) from $\mathcal{A}$ to $\mathcal{B}$. \end{enumerate}


derived-lemma-cartan-eilenberg

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $K^\bullet$ be a bounded below complex. There exists a Cartan-Eilenberg resolution of $K^\bullet$.


derived-lemma-two-ss-complex-functor

Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories. Let $K^\bullet$ be a bounded below complex of $\mathcal{A}$. Let $I^{\bullet, \bullet}$ be a Cartan-Eilenberg resolution for $K^\bullet$. The spectral sequences $({}'E_r, {}'d_r)_{r \geq 0}$ and $({}''E_r, {}''d_r)_{r \geq 0}$ associated to the double complex $F(I^{\bullet, \bullet})$ satisfy the relations $$ {}'E_1^{p, q} = R^qF(K^p) \quad \text{and} \quad {}''E_2^{p, q} = R^pF(H^q(K^\bullet)) $$ Moreover, these spectral sequences are bounded, converge to $H^*(RF(K^\bullet))$, and the associated induced filtrations on $H^n(RF(K^\bullet))$ are finite.


derived-lemma-compose-derived-functors

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be left exact functors. Assume $\mathcal{A}$, $\mathcal{B}$ have enough injectives. The following are equivalent \begin{enumerate} \item $F(I)$ is right acyclic for $G$ for each injective object $I$ of $\mathcal{A}$, and \item the canonical map $$ t : R(G \circ F) \longrightarrow RG \circ RF. $$ is isomorphism of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$. \end{enumerate}


derived-lemma-grothendieck-spectral-sequence

With assumptions as in Lemma \ref{lemma-compose-derived-functors} and assuming the equivalent conditions (1) and (2) hold. Let $X$ be an object of $D^{+}(\mathcal{A})$. There exists a spectral sequence $(E_r, d_r)_{r \geq 0}$ consisting of bigraded objects $E_r$ of $\mathcal{C}$ with $d_r$ of bidegree $(r, - r + 1)$ and with $$ E_2^{p, q} = R^pG(H^q(RF(X))) $$ Moreover, this spectral sequence is bounded, converges to $H^*(R(G \circ F)(X))$, and induces a finite filtration on each $H^n(R(G \circ F)(X))$.


derived-lemma-resolution-functor

Let $\mathcal{A}$ be an abelian category with enough injectives. Given a resolution functor $(j, i)$ there is a unique way to turn $j$ into a functor and $i$ into a $2$-isomorphism producing a $2$-commutative diagram $$ \xymatrix{ K^{+}(\mathcal{A}) \ar[rd] \ar[rr]_j & & K^{+}(\mathcal{I}) \ar[ld] \\ & D^{+}(\mathcal{A}) } $$ where $\mathcal{I}$ is the full additive subcategory of $\mathcal{A}$ consisting of injective objects.


derived-lemma-into-derived-category

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives. Then a resolution functor $j$ exists and is unique up to unique isomorphism of functors.


derived-lemma-resolution-functor-quasi-inverse

Let $\mathcal{A}$ be an abelian category which has enough injectives. Let $j$ be a resolution functor. Write $Q : K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ for the natural functor. Then $j = j' \circ Q$ for a unique functor $j' : D^{+}(\mathcal{A}) \to K^{+}(\mathcal{I})$ which is quasi-inverse to the canonical functor $K^{+}(\mathcal{I}) \to D^{+}(\mathcal{A})$.


derived-lemma-functorial-injective-resolutions

Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has functorial injective embeddings, see Homology, Definition \ref{homology-definition-functorial-injective-embedding}. \begin{enumerate} \item There exists a functor $inj : \text{Comp}^{+}(\mathcal{A}) \to \text{InjRes}(\mathcal{A})$ such that $s \circ inj = \text{id}$. \item For any functor $inj : \text{Comp}^{+}(\mathcal{A}) \to \text{InjRes}(\mathcal{A})$ such that $s \circ inj = \text{id}$ we obtain a resolution functor, see Definition \ref{definition-localization-functor}. \end{enumerate}


derived-lemma-right-derived-functor

Let $\mathcal{A}$ be an abelian category with enough injectives Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor into an abelian category. Let $(i, j)$ be a resolution functor, see Definition \ref{definition-localization-functor}. The right derived functor $RF$ of $F$ fits into the following $2$-commutative diagram $$ \xymatrix{ D^{+}(\mathcal{A}) \ar[rd]_{RF} \ar[rr]^{j'} & & K^{+}(\mathcal{I}) \ar[ld]^F \\ & D^{+}(\mathcal{B}) } $$ where $j'$ is the functor from Lemma \ref{lemma-resolution-functor-quasi-inverse}.


derived-lemma-filtered-injective

Let $\mathcal{A}$ be an abelian category. An object $I$ of $\text{Fil}^f(\mathcal{A})$ is filtered injective if and only if there exist $a \leq b$, injective objects $I_n$, $a \leq n \leq b$ of $\mathcal{A}$ and an isomorphism $I \cong \bigoplus_{a \leq n \leq b} I_n$ such that $F^pI = \bigoplus_{n \geq p} I_n$.


derived-lemma-split-strict-monomorphism

Let $\mathcal{A}$ be an abelian category. Any strict monomorphism $u : I \to A$ of $\text{Fil}^f(\mathcal{A})$ where $I$ is a filtered injective object is a split injection.


derived-lemma-injective-property-filtered-injective

Let $\mathcal{A}$ be an abelian category. Let $u : A \to B$ be a strict monomorphism of $\text{Fil}^f(\mathcal{A})$ and $f : A \to I$ a morphism from $A$ into a filtered injective object in $\text{Fil}^f(\mathcal{A})$. Then there exists a morphism $g : B \to I$ such that $f = g \circ u$.


derived-lemma-strict-monomorphism-into-filtered-injective

Let $\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\text{Fil}^f(\mathcal{A})$ there exists a strict monomorphism $A \to I$ where $I$ is a filtered injective object.


derived-lemma-filtered-injective-right-resolution-single-object

Let $\mathcal{A}$ be an abelian category with enough injectives. For any object $A$ of $\text{Fil}^f(\mathcal{A})$ there exists a filtered quasi-isomorphism $A[0] \to I^\bullet$ where $I^\bullet$ is a complex of filtered injective objects with $I^n = 0$ for $n < 0$.


derived-lemma-filtered-injective-right-resolution-map

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $f : A \to B$ be a morphism of $\text{Fil}^f(\mathcal{A})$. Given filtered quasi-isomorphisms $A[0] \to I^\bullet$ and $B[0] \to J^\bullet$ where $I^\bullet, J^\bullet$ are complexes of filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then there exists a commutative diagram $$ \xymatrix{ A[0] \ar[r] \ar[d] & B[0] \ar[d] \\ I^\bullet \ar[r] & J^\bullet } $$


derived-lemma-filtered-injective-right-resolution-ses

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $0 \to A \to B \to C \to 0$ be a short exact sequence in $\text{Fil}^f(\mathcal{A})$. Given filtered quasi-isomorphisms $A[0] \to I^\bullet$ and $C[0] \to J^\bullet$ where $I^\bullet, J^\bullet$ are complexes of filtered injective objects with $I^n = J^n = 0$ for $n < 0$, then there exists a commutative diagram $$ \xymatrix{ 0 \ar[r] & A[0] \ar[r] \ar[d] & B[0] \ar[r] \ar[d] & C[0] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I^\bullet \ar[r] & M^\bullet \ar[r] & J^\bullet \ar[r] & 0 } $$ where the lower row is a termwise split sequence of complexes.


derived-lemma-right-resolution-by-filtered-injectives

Let $\mathcal{A}$ be an abelian category with enough injectives. For every $K^\bullet \in K^{+}(\text{Fil}^f(\mathcal{A}))$ there exists a filtered quasi-isomorphism $K^\bullet \to I^\bullet$ with $I^\bullet$ bounded below, each $I^n$ a filtered injective object, and each $K^n \to I^n$ a strict monomorphism.


derived-lemma-filtered-acyclic-is-zero

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet, I^\bullet \in K(\text{Fil}^f(\mathcal{A}))$. Assume $K^\bullet$ is filtered acyclic and $I^\bullet$ bounded below and consisting of filtered injective objects. Any morphism $K^\bullet \to I^\bullet$ is homotopic to zero: $\Hom_{K(\text{Fil}^f(\mathcal{A}))}(K^\bullet, I^\bullet) = 0$.


derived-lemma-morphisms-into-filtered-injective-complex

Let $\mathcal{A}$ be an abelian category. Let $I^\bullet \in K(\text{Fil}^f(\mathcal{A}))$ be a bounded below complex consisting of filtered injective objects. \begin{enumerate} \item Let $\alpha : K^\bullet \to L^\bullet$ in $K(\text{Fil}^f(\mathcal{A}))$ be a filtered quasi-isomorphism. Then the map $$ \Hom_{K(\text{Fil}^f(\mathcal{A}))}(L^\bullet, I^\bullet) \to \Hom_{K(\text{Fil}^f(\mathcal{A}))}(K^\bullet, I^\bullet) $$ is bijective. \item Let $L^\bullet \in K(\text{Fil}^f(\mathcal{A}))$. Then $$ \Hom_{K(\text{Fil}^f(\mathcal{A}))}(L^\bullet, I^\bullet) = \Hom_{DF(\mathcal{A})}(L^\bullet, I^\bullet). $$ \end{enumerate}


derived-lemma-ss-filtered-derived

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $T : \mathcal{A} \to \mathcal{B}$ be a left exact functor. Assume $\mathcal{A}$ has enough injectives. Let $(K^\bullet, F)$ be an object of $\text{Comp}^{+}(\text{Fil}^f(\mathcal{A}))$. There exists a spectral sequence $(E_r, d_r)_{r\geq 0}$ consisting of bigraded objects $E_r$ of $\mathcal{B}$ and $d_r$ of bidegree $(r, - r + 1)$ and with $$ E_1^{p, q} = R^{p + q}T(\text{gr}^p(K^\bullet)) $$ Moreover, this spectral sequence is bounded, converges to $R^*T(K^\bullet)$, and induces a finite filtration on each $R^nT(K^\bullet)$. The construction of this spectral sequence is functorial in the object $K^\bullet$ of $\text{Comp}^{+}(\text{Fil}^f(\mathcal{A}))$ and the terms $(E_r, d_r)$ for $r \geq 1$ do not depend on any choices.


derived-lemma-compute-ext-resolutions

Let $\mathcal{A}$ be an abelian category. Let $X^\bullet, Y^\bullet \in \Ob(K(\mathcal{A}))$. \begin{enumerate} \item Let $Y^\bullet \to I^\bullet$ be an injective resolution (Definition \ref{definition-injective-resolution}). Then $$ \Ext^i_\mathcal{A}(X^\bullet, Y^\bullet) = \Hom_{K(\mathcal{A})}(X^\bullet, I^\bullet[i]). $$ \item Let $P^\bullet \to X^\bullet$ be a projective resolution (Definition \ref{definition-projective-resolution}). Then $$ \Ext^i_\mathcal{A}(X^\bullet, Y^\bullet) = \Hom_{K(\mathcal{A})}(P^\bullet[-i], Y^\bullet). $$ \end{enumerate}


derived-lemma-negative-exts

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item Let $X$, $Y$ be objects of $D(\mathcal{A})$. Given $a, b \in \mathbf{Z}$ such that $H^i(X) = 0$ for $i > a$ and $H^j(Y) = 0$ for $j < b$, we have $\Ext^n_\mathcal{A}(X, Y) = 0$ for $n < b - a$ and $$ \Ext^{b - a}_\mathcal{A}(X, Y) = \Hom_\mathcal{A}(H^a(X), H^b(Y)) $$ \item Let $A, B \in \Ob(\mathcal{A})$. For $i < 0$ we have $\Ext^i_\mathcal{A}(B, A) = 0$. We have $\Ext^0_\mathcal{A}(B, A) = \Hom_\mathcal{A}(B, A)$. \end{enumerate}


derived-lemma-yoneda-extension

Let $\mathcal{A}$ be an abelian category with objects $A$, $B$. Any element in $\Ext^i_\mathcal{A}(B, A)$ is $\delta(E)$ for some degree $i$ Yoneda extension of $B$ by $A$. Given two Yoneda extensions $E$, $E'$ of the same degree then $E$ is equivalent to $E'$ if and only if $\delta(E) = \delta(E')$.


derived-lemma-ext-1

Let $\mathcal{A}$ be an abelian category. Let $A$, $B$ be objects of $\mathcal{A}$. Then $\Ext^1_\mathcal{A}(B, A)$ is the group $\Ext_\mathcal{A}(B, A)$ constructed in Homology, Definition \ref{homology-definition-ext-group}.


derived-lemma-higher-ext-zero

Let $\mathcal{A}$ be an abelian category and let $p \geq 0$. If $\Ext^p_\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\mathcal{A}$, then $\Ext^i_\mathcal{A}(B, A) = 0$ for $i \geq p$ and any pair of objects $A$, $B$ of $\mathcal{A}$.


derived-lemma-ext-2-zero

Let $\mathcal{A}$ be an abelian category. Assume $\Ext^2_\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$ of $\mathcal{A}$. Then any object $K$ of $D^b(\mathcal{A})$ is isomorphic to the direct sum of its cohomologies: $K \cong \bigoplus H^i(K)[-i]$.


derived-lemma-K-bounded-derived

Let $\mathcal{A}$ be an abelian category. Then there is a canonical identification $K_0(D^b(\mathcal{A})) = K_0(\mathcal{A})$ of zeroth $K$-groups.


derived-lemma-map-K

Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of triangulated categories. Then $F$ induces a group homomorphism $K_0(\mathcal{D}) \to K_0(\mathcal{D}')$.


derived-lemma-homological-map-K

Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor from a triangulated category to an abelian category. Assume that for any $X$ in $\mathcal{D}$ only a finite number of the objects $H(X[i])$ are nonzero in $\mathcal{A}$. Then $H$ induces a group homomorphism $K_0(\mathcal{D}) \to K_0(\mathcal{A})$ sending $[X]$ to $\sum (-1)^i[H(X[i])]$.


derived-lemma-DBA-map-K

Let $\mathcal{B}$ be a weak Serre subcategory of the abelian category $\mathcal{A}$. Then there are canonical maps $$ K_0(\mathcal{B}) \longrightarrow K_0(D^b_\mathcal{B}(\mathcal{A})) \longrightarrow K_0(\mathcal{B}) $$ whose composition is zero. The second arrow sends the class $[X]$ of the object $X$ to the element $\sum (-1)^i[H^i(X)]$ of $K_0(\mathcal{B})$.


derived-lemma-bilinear-map-K

Let $\mathcal{D}$, $\mathcal{D}'$, $\mathcal{D}''$ be triangulated categories. Let $$ \otimes : \mathcal{D} \times \mathcal{D}' \longrightarrow \mathcal{D}'' $$ be a functor such that for fixed $X$ in $\mathcal{D}$ the functor $X \otimes - : \mathcal{D}' \to \mathcal{D}''$ is an exact functor and for fixed $X'$ in $\mathcal{D}'$ the functor $- \otimes X' : \mathcal{D} \to \mathcal{D}''$ is an exact functor. Then $\otimes$ induces a bilinear map $K_0(\mathcal{D}) \times K_0(\mathcal{D}') \to K_0(\mathcal{D}'')$ which sends $([X], [X'])$ to $[X \otimes X']$.


derived-lemma-special-direct-system

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset. Assume $\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$. Let $K^\bullet$ be a complex. There exists a commutative diagram $$ \xymatrix{ P_1^\bullet \ar[d] \ar[r] & P_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}K^\bullet \ar[r] & \tau_{\leq 2}K^\bullet \ar[r] & \ldots } $$ in the category of complexes such that \begin{enumerate} \item the vertical arrows are quasi-isomorphisms and termwise surjective, \item $P_n^\bullet$ is a bounded above complex with terms in $\mathcal{P}$, \item the arrows $P_n^\bullet \to P_{n + 1}^\bullet$ are termwise split injections and each cokernel $P^i_{n + 1}/P^i_n$ is an element of $\mathcal{P}$. \end{enumerate}


derived-lemma-special-inverse-system

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{I} \subset \Ob(\mathcal{A})$ be a subset. Assume $\mathcal{I}$ contains $0$, is closed under (finite) products, and every object of $\mathcal{A}$ is a subobject of an element of $\mathcal{I}$. Let $K^\bullet$ be a complex. There exists a commutative diagram $$ \xymatrix{ \ldots \ar[r] & \tau_{\geq -2}K^\bullet \ar[r] \ar[d] & \tau_{\geq -1}K^\bullet \ar[d] \\ \ldots \ar[r] & I_2^\bullet \ar[r] & I_1^\bullet } $$ in the category of complexes such that \begin{enumerate} \item the vertical arrows are quasi-isomorphisms and termwise injective, \item $I_n^\bullet$ is a bounded below complex with terms in $\mathcal{I}$, \item the arrows $I_{n + 1}^\bullet \to I_n^\bullet$ are termwise split surjections and $\Ker(I^i_{n + 1} \to I^i_n)$ is an element of $\mathcal{I}$. \end{enumerate}


derived-lemma-pre-derived-adjoint-functors-general

In the situation above assume $F$ is right adjoint to $G$. Let $K \in \Ob(\mathcal{D})$ and $M \in \Ob(\mathcal{D}')$. If $RF$ is defined at $K$ and $LG$ is defined at $M$, then there is a canonical isomorphism $$ \Hom_{(S')^{-1}\mathcal{D}'}(M, RF(K)) = \Hom_{S^{-1}\mathcal{D}}(LG(M), K) $$ This isomorphism is functorial in both variables on the triangulated subcategories of $S^{-1}\mathcal{D}$ and $(S')^{-1}\mathcal{D}'$ where $RF$ and $LG$ are defined.


derived-lemma-pre-derived-adjoint-functors

Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{A}$ be functors of abelian categories such that $F$ is a right adjoint to $G$. Let $K^\bullet$ be a complex of $\mathcal{A}$ and let $M^\bullet$ be a complex of $\mathcal{B}$. If $RF$ is defined at $K^\bullet$ and $LG$ is defined at $M^\bullet$, then there is a canonical isomorphism $$ \Hom_{D(\mathcal{B})}(M^\bullet, RF(K^\bullet)) = \Hom_{D(\mathcal{A})}(LG(M^\bullet), K^\bullet) $$ This isomorphism is functorial in both variables on the triangulated subcategories of $D(\mathcal{A})$ and $D(\mathcal{B})$ where $RF$ and $LG$ are defined.


derived-lemma-derived-adjoint-functors

Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{A}$ be functors of abelian categories such that $F$ is a right adjoint to $G$. If the derived functors $RF : D(\mathcal{A}) \to D(\mathcal{B})$ and $LG : D(\mathcal{B}) \to D(\mathcal{A})$ exist, then $RF$ is a right adjoint to $LG$.


derived-lemma-K-injective

Let $\mathcal{A}$ be an abelian category. Let $I^\bullet$ be a complex. The following are equivalent \begin{enumerate} \item $I^\bullet$ is K-injective, \item for every quasi-isomorphism $M^\bullet \to N^\bullet$ the map $$ \Hom_{K(\mathcal{A})}(N^\bullet, I^\bullet) \to \Hom_{K(\mathcal{A})}(M^\bullet, I^\bullet) $$ is bijective, and \item for every complex $N^\bullet$ the map $$ \Hom_{K(\mathcal{A})}(N^\bullet, I^\bullet) \to \Hom_{D(\mathcal{A})}(N^\bullet, I^\bullet) $$ is an isomorphism. \end{enumerate}


derived-lemma-triangle-K-injective

Let $\mathcal{A}$ be an abelian category. Let $(K, L, M, f, g, h)$ be a distinguished triangle of $K(\mathcal{A})$. If two out of $K$, $L$, $M$ are K-injective complexes, then the third is too.


derived-lemma-bounded-below-injectives-K-injective

Let $\mathcal{A}$ be an abelian category. A bounded below complex of injectives is K-injective.


derived-lemma-product-K-injective

Let $\mathcal{A}$ be an abelian category. Let $T$ be a set and for each $t \in T$ let $I_t^\bullet$ be a K-injective complex. If $I^n = \prod_t I_t^n$ exists for all $n$, then $I^\bullet$ is a K-injective complex. Moreover, $I^\bullet$ represents the product of the objects $I_t^\bullet$ in $D(\mathcal{A})$.


derived-lemma-K-injective-defined

Let $\mathcal{A}$ be an abelian category. Let $F : K(\mathcal{A}) \to \mathcal{D}'$ be an exact functor of triangulated categories. Then $RF$ is defined at every complex in $K(\mathcal{A})$ which is quasi-isomorphic to a K-injective complex. In fact, every K-injective complex computes $RF$.


derived-lemma-adjoint-preserve-K-injectives

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume \begin{enumerate} \item $u$ is right adjoint to $v$, and \item $v$ is exact. \end{enumerate} Then $u$ transforms K-injective complexes into K-injective complexes.


derived-lemma-replace-resolution

Let $\mathcal{A}$ be an abelian category. Let $d : \Ob(\mathcal{A}) \to \{0, 1, 2, \ldots, \infty\}$ be a function. Assume that \begin{enumerate} \item every object of $\mathcal{A}$ is a subobject of an object $A$ with $d(A) = 0$, \item $d(A \oplus B) \leq \max \{d(A), d(B)\}$ for $A, B \in \mathcal{A}$, and \item if $0 \to A \to B \to C \to 0$ is short exact, then $d(C) \leq \max\{d(A) - 1, d(B)\}$. \end{enumerate} Let $K^\bullet$ be a complex such that $n + d(K^n)$ tends to $-\infty$ as $n \to -\infty$. Then there exists a quasi-isomorphism $K^\bullet \to L^\bullet$ with $d(L^n) = 0$ for all $n \in \mathbf{Z}$.


derived-lemma-unbounded-right-derived

Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor of abelian categories. Assume \begin{enumerate} \item every object of $\mathcal{A}$ is a subobject of an object which is right acyclic for $F$, \item there exists an integer $n \geq 0$ such that $R^nF = 0$, \end{enumerate} Then \begin{enumerate} \item $RF : D(\mathcal{A}) \to D(\mathcal{B})$ exists, \item any complex consisting of right acyclic objects for $F$ computes $RF$, \item any complex is the source of a quasi-isomorphism into a complex consisting of right acyclic objects for $F$, \item for $E \in D(\mathcal{A})$ \begin{enumerate} \item $H^i(RF(\tau_{\leq a}E) \to H^i(RF(E))$ is an isomorphism for $i \leq a$, \item $H^i(RF(E)) \to H^i(RF(\tau_{\geq b - n + 1}E))$ is an isomorphism for $i \geq b$, \item if $H^i(E) = 0$ for $i \not \in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^i(RF(E)) = 0$ for $i \not \in [a, b + n - 1]$. \end{enumerate} \end{enumerate}


derived-lemma-direct-sums

Let $\mathcal{A}$ be an abelian category. If $\mathcal{A}$ has exact countable direct sums, then $D(\mathcal{A})$ has countable direct sums. In fact given a collection of complexes $K_i^\bullet$ indexed by a countable index set $I$ the termwise direct sum $\bigoplus K_i^\bullet$ is the direct sum of $K_i^\bullet$ in $D(\mathcal{A})$.


derived-lemma-compute-colimit

Let $\mathcal{A}$ be an abelian category. Assume colimits over $\mathbf{N}$ exist and are exact. Then countable direct sums exists and are exact. Moreover, if $(A_n, f_n)$ is a system over $\mathbf{N}$, then there is a short exact sequence $$ 0 \to \bigoplus A_n \to \bigoplus A_n \to \colim A_n \to 0 $$ where the first map in degree $n$ is given by $1 - f_n$.


derived-lemma-colim-hocolim

Let $\mathcal{A}$ be an abelian category. Let $L_n^\bullet$ be a system of complexes of $\mathcal{A}$. Assume colimits over $\mathbf{N}$ exist and are exact in $\mathcal{A}$. Then the termwise colimit $L^\bullet = \colim L_n^\bullet$ is a homotopy colimit of the system in $D(\mathcal{A})$.


derived-lemma-cohomology-of-hocolim

Let $\mathcal{D}$ be a triangulated category having countable direct sums. Let $\mathcal{A}$ be an abelian category with exact colimits over $\mathbf{N}$. Let $H : \mathcal{D} \to \mathcal{A}$ be a homological functor commuting with countable direct sums. Then $H(\text{hocolim} K_n) = \colim H(K_n)$ for any system of objects of $\mathcal{D}$.


derived-lemma-commutes-with-countable-sums

Let $\mathcal{D}$ be a triangulated category with countable direct sums. Let $K \in \mathcal{D}$ be an object such that for every countable set of objects $E_n \in \mathcal{D}$ the canonical map $$ \bigoplus \Hom_\mathcal{D}(K, E_n) \longrightarrow \Hom_\mathcal{D}(K, \bigoplus E_n) $$ is a bijection. Then, given any system $L_n$ of $\mathcal{D}$ over $\mathbf{N}$ whose derived colimit $L = \text{hocolim} L_n$ exists we have that $$ \colim \Hom_\mathcal{D}(K, L_n) \longrightarrow \Hom_\mathcal{D}(K, L) $$ is a bijection.


derived-lemma-products

Let $\mathcal{A}$ be an abelian category with exact countable products. Then \begin{enumerate} \item $D(\mathcal{A})$ has countable products, \item countable products $\prod K_i$ in $D(\mathcal{A})$ are obtained by taking termwise products of any complexes representing the $K_i$, and \item $H^p(\prod K_i) = \prod H^p(K_i)$. \end{enumerate}


derived-lemma-inverse-limit-bounded-below

Let $\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $(K_n)$ be an inverse system of $D^+(\mathcal{A})$. Then $R\lim K_n$ exists.


derived-lemma-difficulty-K-injectives

Let $\mathcal{A}$ be an abelian category with countable products and enough injectives. Let $K^\bullet$ be a complex. Let $I_n^\bullet$ be the inverse system of bounded below complexes of injectives produced by Lemma \ref{lemma-special-inverse-system}. Then $I^\bullet = \lim I_n^\bullet$ exists, is K-injective, and the following are equivalent \begin{enumerate} \item the map $K^\bullet \to I^\bullet$ is a quasi-isomorphism, \item the canonical map $K^\bullet \to R\lim \tau_{\geq -n}K^\bullet$ is an isomorphism in $D(\mathcal{A})$. \end{enumerate}


derived-lemma-associativity-star

Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ we have $(\mathcal{A} \star \mathcal{B}) \star \mathcal{C} = \mathcal{A} \star (\mathcal{B} \star \mathcal{C})$.


derived-lemma-smd-star

Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}$, $\mathcal{B}$ we have $smd(\mathcal{A}) \star smd(\mathcal{B}) \subset smd(\mathcal{A} \star \mathcal{B})$ and $smd(smd(\mathcal{A}) \star smd(\mathcal{B})) = smd(\mathcal{A} \star \mathcal{B})$.


derived-lemma-add-star

Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}$, $\mathcal{B}$ the full subcategories $add(\mathcal{A}) \star add(\mathcal{B})$ and $smd(add(\mathcal{A}))$ are closed under direct sums.


derived-lemma-cone-n

Let $\mathcal{T}$ be a triangulated category. Given a full subcategory $\mathcal{A}$ for $n \geq 1$ the subcategory $$ \mathcal{C}_n = smd(add(\mathcal{A})^{\star n}) = smd(add(\mathcal{A}) \star \ldots \star add(\mathcal{A})) $$ defined above is a strictly full subcategory of $\mathcal{T}$ closed under direct sums and direct summands and $\mathcal{C}_{n + m} = smd(\mathcal{C}_n \star \mathcal{C}_m)$ for all $n, m \geq 1$.


derived-lemma-in-cone-n

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D} = D(\mathcal{A})$. Let $\mathcal{E} \subset \Ob(\mathcal{A})$ be a subset which we view as a subset of $\Ob(\mathcal{D})$ also. Let $K$ be an object of $\mathcal{D}$. \begin{enumerate} \item Let $b \geq a$ and assume $H^i(K)$ is zero for $i \not \in [a, b]$ and $H^i(K) \in \mathcal{E}$ if $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$. \item Let $b \geq a$ and assume $H^i(K)$ is zero for $i \not \in [a, b]$ and $H^i(K) \in smd(add(\mathcal{E}))$ if $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$. \item Let $b \geq a$ and assume $K$ can be represented by a complex $K^\bullet$ with $K^i = 0$ for $i \not \in [a, b]$ and $K^i \in \mathcal{E}$ for $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$. \item Let $b \geq a$ and assume $K$ can be represented by a complex $K^\bullet$ with $K^i = 0$ for $i \not \in [a, b]$ and $K^i \in smd(add(\mathcal{E}))$ for $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$. \end{enumerate}


derived-lemma-forward-cone-n

Let $\mathcal{T}$ be a triangulated category. Let $H : \mathcal{T} \to \mathcal{A}$ be a homological functor to an abelian category $\mathcal{A}$. Let $a \leq b$ and $\mathcal{E} \subset \Ob(\mathcal{T})$ be a subset such that $H^i(E) = 0$ for $E \in \mathcal{E}$ and $i \not \in [a, b]$. Then for $X \in smd(add(\mathcal{E}[-m, m])^{\star n})$ we have $H^i(X) = 0$ for $i \not \in [-m + na, m + nb]$.


derived-lemma-generated-by-E-explicit

Let $\mathcal{T}$ be a triangulated category. Let $E$ be an object of $\mathcal{T}$. For $n \geq 1$ we have $$ \langle E \rangle_n = smd(\langle E \rangle_1 \star \ldots \star \langle E \rangle_1) = smd({\langle E \rangle_1}^{\star n}) = \bigcup\nolimits_{m \geq 1} smd(add(E[-m, m])^{\star n}) $$ For $n, n' \geq 1$ we have $\langle E \rangle_{n + n'} = smd(\langle E \rangle_n \star \langle E \rangle_{n'})$.


derived-lemma-find-smallest-containing-E

Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. The subcategory $$ \langle E \rangle = \bigcup\nolimits_n \langle E \rangle_n = \bigcup\nolimits_{n, m \geq 1} smd(add(E[-m, m])^{\star n}) $$ is a strictly full, saturated, triangulated subcategory of $\mathcal{D}$ and it is the smallest such subcategory of $\mathcal{D}$ containing the object $E$.


derived-lemma-right-orthogonal

Let $\mathcal{D}$ be a triangulated category. Let $E, K$ be objects of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $\Hom(E, K[i]) = 0$ for all $i \in \mathbf{Z}$, \item $\Hom(E', K) = 0$ for all $E' \in \langle E \rangle$. \end{enumerate}


derived-lemma-classical-generator-strong-generator

Let $\mathcal{D}$ be a triangulated category which has a strong generator. Let $E$ be an object of $\mathcal{D}$. If $E$ is a classical generator of $\mathcal{D}$, then $E$ is a strong generator.


derived-lemma-compact-objects-subcategory

Let $\mathcal{D}$ be a (pre-)triangulated category with direct sums. Then the compact objects of $\mathcal{D}$ form the objects of a Karoubian, saturated, strictly full, (pre-)triangulated subcategory $\mathcal{D}_c$ of $\mathcal{D}$.


derived-lemma-write-as-colimit

Let $\mathcal{D}$ be a triangulated category with direct sums. Let $E_i$, $i \in I$ be a family of compact objects of $\mathcal{D}$ such that $\bigoplus E_i$ generates $\mathcal{D}$. Then every object $X$ of $\mathcal{D}$ can be written as $$ X = \text{hocolim} X_n $$ where $X_1$ is a direct sum of shifts of the $E_i$ and each transition morphism fits into a distinguished triangle $Y_n \to X_n \to X_{n + 1} \to Y_n[1]$ where $Y_n$ is a direct sum of shifts of the $E_i$.


derived-lemma-factor-through

With assumptions and notation as in Lemma \ref{lemma-write-as-colimit}. If $C$ is a compact object and $C \to X_n$ is a morphism, then there is a factorization $C \to E \to X_n$ where $E$ is an object of $\langle E_{i_1} \oplus \ldots \oplus E_{i_t} \rangle$ for some $i_1, \ldots, i_t \in I$.


derived-lemma-brown

\begin{reference} \cite[Theorem 3.1]{Neeman-Grothendieck}. \end{reference} Let $\mathcal{D}$ be a triangulated category with direct sums which is compactly generated. Let $H : \mathcal{D} \to \textit{Ab}$ be a contravariant cohomological functor which transforms direct sums into products. Then $H$ is representable.


derived-lemma-pre-prepare-adjoint

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory invariant under all shifts. Consider a distinguished triangle $$ X \to Y \to Z \to X[1] $$ of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $Z$ is in $\mathcal{A}^\perp$, and \item $\Hom(A, X) = \Hom(A, Y)$ for all $A \in \Ob(\mathcal{A})$. \end{enumerate}


derived-lemma-orthogonal-triangulated

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full subcategory invariant under all shifts. Then both the right orthogonal $\mathcal{A}^\perp$ and the left orthogonal ${}^\perp\mathcal{A}$ of $\mathcal{A}$ are strictly full, saturated\footnote{Definition \ref{definition-saturated}.}, triangulated subcagories of $\mathcal{D}$.


derived-lemma-prepare-adjoint

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A}$ be a full triangulated subcategory of $\mathcal{D}$. For an object $X$ of $\mathcal{D}$ consider the property $P(X)$: there exists a distinguished triangle $A \to X \to B \to A[1]$ in $\mathcal{D}$ with $A$ in $\mathcal{A}$ and $B$ in $\mathcal{A}^\perp$. \begin{enumerate} \item If $X_1 \to X_2 \to X_3 \to X_1[1]$ is a distinguished triangle and $P$ holds for two out of three, then it holds for the third. \item If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \oplus X_2$. \end{enumerate}


derived-lemma-right-adjoint

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent \begin{enumerate} \item the inclusion functor $\mathcal{A} \to \mathcal{D}$ has a right adjoint, and \item for every $X$ in $\mathcal{D}$ there exists a distinguished triangle $$ A \to X \to B \to X'[1] $$ in $\mathcal{D}$ with $A \in \Ob(\mathcal{A})$ and $B \in \Ob(\mathcal{A}^\perp)$. \end{enumerate} If this holds, then $\mathcal{A}$ is saturated (Definition \ref{definition-saturated}) and if $\mathcal{A}$ is strictly full in $\mathcal{D}$, then $\mathcal{A} = {}^\perp(\mathcal{A}^\perp)$.


derived-lemma-left-adjoint

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{A} \subset \mathcal{D}$ be a full triangulated subcategory. The following are equivalent \begin{enumerate} \item the inclusion functor $\mathcal{A} \to \mathcal{D}$ has a left adjoint, and \item for every $X$ in $\mathcal{D}$ there exists a distinguished triangle $$ B \to X \to A \to K[1] $$ in $\mathcal{D}$ with $A \in \Ob(\mathcal{A})$ and $B \in \Ob({}^\perp\mathcal{A})$. \end{enumerate} If this holds, then $\mathcal{A}$ is saturated (Definition \ref{definition-saturated}) and if $\mathcal{A}$ is strictly full in $\mathcal{D}$, then $\mathcal{A} = ({}^\perp\mathcal{A})^\perp$.


derived-lemma-postnikov-system-small-cases

Let $\mathcal{D}$ be a triangulated category. Consider Postnikov systems for complexes of length $n$. \begin{enumerate} \item For $n = 0$ Postnikov systems always exist and any morphism (\ref{equation-map-complexes}) of complexes extends to a unique morphism of Postnikov systems. \item For $n = 1$ Postnikov systems always exist and any morphism (\ref{equation-map-complexes}) of complexes extends to a (nonunique) morphism of Postnikov systems. \item For $n = 2$ Postnikov systems always exist but morphisms (\ref{equation-map-complexes}) of complexes in general do not extend to morphisms of Postnikov systems. \item For $n > 2$ Postnikov systems do not always exist. \end{enumerate}


derived-lemma-maps-postnikov-systems-vanishing

Let $\mathcal{D}$ be a triangulated category. Given a map (\ref{equation-map-complexes}) consider the condition \begin{equation} \label{equation-P} \Hom(X_i[i - j - 1], X'_j) = 0 \text{ for }i > j + 1 \end{equation} Then \begin{enumerate} \item If we have a Postnikov system for $X'_n \to X'_{n - 1} \to \ldots \to X'_0$ then property (\ref{equation-P}) implies that $$ \Hom(X_i[i - j - 1], Y'_j) = 0 \text{ for }i > j + 1 $$ \item If we are given Postnikov systems for both complexes and we have (\ref{equation-P}), then the map extends to a (nonunique) map of Postnikov systems. \end{enumerate}


derived-lemma-existence-postnikov-system

Let $\mathcal{D}$ be a triangulated category. Let $X_n \to X_{n - 1} \to \ldots \to X_0$ be a complex in $\mathcal{D}$. If $$ \Hom(X_i[i - j - 2], X_j) = 0 \text{ for }i > j + 2 $$ then there exists a Postnikov system. If we have $$ \Hom(X_i[i - j - 1], X_j) = 0 \text{ for }i > j + 1 $$ then any two Postnikov systems are isomorphic.


derived-lemma-essentially-constant

Let $\mathcal{D}$ be a triangulated category. Let $(A_i)$ be an inverse system in $\mathcal{D}$. Then $(A_i)$ is essentially constant (see Categories, Definition \ref{categories-definition-essentially-constant-diagram}) if and only if there exists an $i$ and for all $j \geq i$ a direct sum decomposition $A_j = A \oplus Z_j$ such that (a) the maps $A_{j'} \to A_j$ are compatible with the direct sum decompositions and identity on $A$, (b) for all $j \geq i$ there exists some $j' \geq j$ such that $Z_{j'} \to Z_j$ is zero.


derived-lemma-essentially-constant-2-out-of-3

Let $\mathcal{D}$ be a triangulated category. Let $$ A_n \to B_n \to C_n \to A_n[1] $$ be an inverse system of distinguished triangles in $\mathcal{D}$. If $(A_n)$ and $(C_n)$ are essentially constant, then $(B_n)$ is essentially constant and their values fit into a distinguished triangle $A \to B \to C \to A[1]$ such that for some $n \geq 1$ there is a map $$ \xymatrix{ A_n \ar[d] \ar[r] & B_n \ar[d] \ar[r] & C_n \ar[d] \ar[r] & A_n[1] \ar[d] \\ A \ar[r] & B \ar[r] & C \ar[r] & A[1] } $$ of distinguished triangles which induces an isomorphism $\lim_{n' \geq n} A_{n'} \to A$ and similarly for $B$ and $C$.


derived-lemma-essentially-constant-cohomology

Let $\mathcal{A}$ be an abelian category. Let $A_n$ be an inverse system of objects of $D(\mathcal{A})$. Assume \begin{enumerate} \item there exist integers $a \leq b$ such that $H^i(A_n) = 0$ for $i \not \in [a, b]$, and \item the inverse systems $H^i(A_n)$ of $\mathcal{A}$ are essentially constant for all $i \in \mathbf{Z}$. \end{enumerate} Then $A_n$ is an essentially constant system of $D(\mathcal{A})$ whose value $A$ satisfies that $H^i(A)$ is the value of the constant system $H^i(A_n)$ for each $i \in \mathbf{Z}$.


derived-lemma-pro-isomorphism

Let $\mathcal{D}$ be a triangulated category. Let $$ A_n \to B_n \to C_n \to A_n[1] $$ be an inverse system of distinguished triangles. If the system $C_n$ is pro-zero (essentially constant with value $0$), then the maps $A_n \to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$ and the pro-object $(B_n)$.


derived-lemma-pro-isomorphism-bis

Let $\mathcal{A}$ be an abelian category. $$ A_n \to B_n $$ be an inverse system of maps of $D(\mathcal{A})$. Assume \begin{enumerate} \item there exist integers $a \leq b$ such that $H^i(A_n) = 0$ and $H^i(B_n) = 0$ for $i \not \in [a, b]$, and \item the inverse system of maps $H^i(A_n) \to H^i(B_n)$ of $\mathcal{A}$ define an isomorphism of pro-objects of $\mathcal{A}$ for all $i \in \mathbf{Z}$. \end{enumerate} Then the maps $A_n \to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$ and the pro-object $(B_n)$.


derived-proposition-9

Let $\mathcal{D}$ be a triangulated category. Any commutative diagram $$ \xymatrix{ X \ar[r] \ar[d] & Y \ar[d] \\ X' \ar[r] & Y' } $$ can be extended to a diagram $$ \xymatrix{ X \ar[r] \ar[d] & Y \ar[r] \ar[d] & Z \ar[r] \ar[d] & X[1] \ar[d] \\ X' \ar[r] \ar[d] & Y' \ar[r] \ar[d] & Z' \ar[r] \ar[d] & X'[1] \ar[d] \\ X'' \ar[r] \ar[d] & Y'' \ar[r] \ar[d] & Z'' \ar[r] \ar[d] & X''[1] \ar[d] \\ X[1] \ar[r] & Y[1] \ar[r] & Z[1] \ar[r] & X[2] } $$ where all the squares are commutative, except for the lower right square which is anticommutative. Moreover, each of the rows and columns are distinguished triangles. Finally, the morphisms on the bottom row (resp.\ right column) are obtained from the morphisms of the top row (resp.\ left column) by applying $[1]$.


derived-proposition-construct-localization

Let $\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative system compatible with the triangulated structure. Then there exists a unique structure of a pre-triangulated category on $S^{-1}\mathcal{D}$ such that the localization functor $Q : \mathcal{D} \to S^{-1}\mathcal{D}$ is exact. Moreover, if $\mathcal{D}$ is a triangulated category, so is $S^{-1}\mathcal{D}$.


derived-proposition-homotopy-category-triangulated

Let $\mathcal{A}$ be an additive category. The category $K(\mathcal{A})$ of complexes up to homotopy with its natural translation functors and distinguished triangles as defined above is a triangulated category.


derived-proposition-derived-functor

Assumptions and notation as in Situation \ref{situation-derived-functor}. \begin{enumerate} \item The full subcategory $\mathcal{E}$ of $\mathcal{D}$ consisting of objects at which $RF$ is defined is a strictly full triangulated subcategory of $\mathcal{D}$. \item We obtain an exact functor $RF : \mathcal{E} \longrightarrow \mathcal{D}'$ of triangulated categories. \item Elements of $S$ with either source or target in $\mathcal{E}$ are morphisms of $\mathcal{E}$. \item The functor $S_\mathcal{E}^{-1}\mathcal{E} \to S^{-1}\mathcal{D}$ is a fully faithful exact functor of triangulated categories. \item Any element of $S_\mathcal{E} = \text{Arrows}(\mathcal{E}) \cap S$ is mapped to an isomorphism by $RF$. \item We obtain an exact functor $$ RF : S_\mathcal{E}^{-1}\mathcal{E} \longrightarrow \mathcal{D}'. $$ \item If $\mathcal{D}'$ is Karoubian, then $\mathcal{E}$ is a saturated triangulated subcategory of $\mathcal{D}$. \end{enumerate} A similar result holds for $LF$.


derived-proposition-enough-acyclics

\begin{slogan} A functor on an Abelian categories is extended to the (bounded below or above) derived category by resolving with a complex that is acyclic for that functor. \end{slogan} Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor of abelian categories. \begin{enumerate} \item If every object of $\mathcal{A}$ injects into an object acyclic for $RF$, then $RF$ is defined on all of $K^{+}(\mathcal{A})$ and we obtain an exact functor $$ RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B}) $$ see (\ref{equation-everywhere}). Moreover, any bounded below complex $A^\bullet$ whose terms are acyclic for $RF$ computes $RF$. \item If every object of $\mathcal{A}$ is quotient of an object acyclic for $LF$, then $LF$ is defined on all of $K^{-}(\mathcal{A})$ and we obtain an exact functor $$ LF : D^{-}(\mathcal{A}) \longrightarrow D^{-}(\mathcal{B}) $$ see (\ref{equation-everywhere}). Moreover, any bounded above complex $A^\bullet$ whose terms are acyclic for $LF$ computes $LF$. \end{enumerate}


derived-proposition-left-derived-exists

Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. Let $\mathcal{P} \subset \Ob(\mathcal{A})$ be a subset. Assume \begin{enumerate} \item $\mathcal{P}$ contains $0$, is closed under (finite) direct sums, and every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$, \item for any bounded above acyclic complex $P^\bullet$ of $\mathcal{A}$ with $P^n \in \mathcal{P}$ for all $n$ the complex $F(P^\bullet)$ is exact, \item $\mathcal{A}$ and $\mathcal{B}$ have colimits of systems over $\mathbf{N}$, \item colimits over $\mathbf{N}$ are exact in both $\mathcal{A}$ and $\mathcal{B}$, and \item $F$ commutes with colimits over $\mathbf{N}$. \end{enumerate} Then $LF$ is defined on all of $D(\mathcal{A})$.


derived-proposition-generator-versus-classical-generator

Let $\mathcal{D}$ be a triangulated category with direct sums. Let $E$ be a compact object of $\mathcal{D}$. The following are equivalent \begin{enumerate} \item $E$ is a classical generator for $\mathcal{D}_c$ and $\mathcal{D}$ is compactly generated, and \item $E$ is a generator for $\mathcal{D}$. \end{enumerate}


derived-proposition-brown

\begin{reference} \cite[Theorem 4.1]{Neeman-Grothendieck}. \end{reference} Let $\mathcal{D}$ be a triangulated category with direct sums which is compactly generated. Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of triangulated categories which transforms direct sums into direct sums. Then $F$ has an exact right adjoint.


cohomology-theorem-proper-base-change

\begin{reference} \cite[Expose V bis, 4.1.1]{SGA4} \end{reference} Consider a cartesian square of topological spaces $$ \xymatrix{ X' = Y' \times_Y X \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ Assume that $f$ is proper and separated. Let $E$ be an object of $D^+(X)$. Then the base change map $$ g^{-1}Rf_*E \longrightarrow Rf'_*(g')^{-1}E $$ of Lemma \ref{lemma-base-change-map-flat-case} is an isomorphism in $D^+(Y')$.


cohomology-theorem-glueing-bbd-general

\begin{reference} Special case of \cite[Theorem 3.2.4]{BBD} without boundedness assumption. \end{reference} In Situation \ref{situation-locally-given} assume \begin{enumerate} \item $X = \bigcup_{U \in \mathcal{B}} U$, \item for $U, V \in \mathcal{B}$ we have $U \cap V = \bigcup_{W \in \mathcal{B}, W \subset U \cap V} W$, \item for any $U \in \mathcal{B}$ we have $\Ext^i(K_U, K_U) = 0$ for $i < 0$. \end{enumerate} Then there exists an object $K$ of $D(\mathcal{O}_X)$ and isomorphisms $\rho_U : K|_U \to K_U$ in $D(\mathcal{O}_U)$ for $U \in \mathcal{B}$ such that $\rho^U_V \circ \rho_U|_V = \rho_V$ for all $V \subset U$ with $U, V \in \mathcal{B}$. The pair $(K, \rho_U)$ is unique up to unique isomorphism.


cohomology-lemma-trivial-torsor

Let $X$ be a topological space. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A $\mathcal{G}$-torsor $\mathcal{F}$ is trivial if and only if $\mathcal{F}(X) \not = \emptyset$.


cohomology-lemma-torsors-h1

Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf on $X$. There is a canonical bijection between the set of isomorphism classes of $\mathcal{H}$-torsors and $H^1(X, \mathcal{H})$.


cohomology-lemma-h1-extensions

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. There is a canonical bijection $$ \Ext^1_{\textit{Mod}(\mathcal{O}_X)}(\mathcal{O}_X, \mathcal{F}) \longrightarrow H^1(X, \mathcal{F}) $$ which associates to the extension $$ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O}_X \to 0 $$ the image of $1 \in \Gamma(X, \mathcal{O}_X)$ in $H^1(X, \mathcal{F})$.


cohomology-lemma-h1-invertible

Let $(X, \mathcal{O}_X)$ be a locally ringed space. There is a canonical isomorphism $$ H^1(X, \mathcal{O}_X^*) = \Pic(X). $$ of abelian groups.


cohomology-lemma-cohomology-of-open

Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. \begin{enumerate} \item If $\mathcal{I}$ is an injective $\mathcal{O}_X$-module then $\mathcal{I}|_U$ is an injective $\mathcal{O}_U$-module. \item For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have $H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}|_U)$. \end{enumerate}


cohomology-lemma-kill-cohomology-class-on-covering

Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset X$ be an open subspace. Let $n > 0$ and let $\xi \in H^n(U, \mathcal{F})$. Then there exists an open covering $U = \bigcup_{i\in I} U_i$ such that $\xi|_{U_i} = 0$ for all $i \in I$.


cohomology-lemma-describe-higher-direct-images

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module. The sheaves $R^if_*\mathcal{F}$ are the sheaves associated to the presheaves $$ V \longmapsto H^i(f^{-1}(V), \mathcal{F}) $$ with restriction mappings as in Equation (\ref{equation-restriction-mapping}). There is a similar statement for $R^if_*$ applied to a bounded below complex $\mathcal{F}^\bullet$.


cohomology-lemma-localize-higher-direct-images

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $V \subset Y$ be an open subspace. Denote $g : f^{-1}(V) \to V$ the restriction of $f$. Then we have $$ R^pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\mathcal{F})|_V $$ There is a similar statement for the derived image $Rf_*\mathcal{F}^\bullet$ where $\mathcal{F}^\bullet$ is a bounded below complex of $\mathcal{O}_X$-modules.


cohomology-lemma-injective-restriction-surjective

\begin{slogan} Local sections in injective sheaves can be extended globally. \end{slogan} Let $X$ be a ringed space. Let $U' \subset U \subset X$ be open subspaces. For any injective $\mathcal{O}_X$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective.


cohomology-lemma-mayer-vietoris

Let $X$ be a ringed space. Suppose that $X = U \cup V$ is a union of two open subsets. For every $\mathcal{O}_X$-module $\mathcal{F}$ there exists a long exact cohomology sequence $$ 0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots $$ This long exact sequence is functorial in $\mathcal{F}$.


cohomology-lemma-relative-mayer-vietoris

Let $f : X \to Y$ be a morphism of ringed spaces. Suppose that $X = U \cup V$ is a union of two open subsets. Denote $a = f|_U : U \to Y$, $b = f|_V : V \to Y$, and $c = f|_{U \cap V} : U \cap V \to Y$. For every $\mathcal{O}_X$-module $\mathcal{F}$ there exists a long exact sequence $$ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots $$ This long exact sequence is functorial in $\mathcal{F}$.


cohomology-lemma-cech-h0

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf and \item for every open covering $\mathcal{U} : U = \bigcup_{i \in I} U_i$ the natural map $$ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) $$ is bijective. \end{enumerate}


cohomology-lemma-cech-trivial

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. If $U_i = U$ for some $i \in I$, then the extended {\v C}ech complex $$ \mathcal{F}(U) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ obtained by putting $\mathcal{F}(U)$ in degree $-1$ with differential given by the canonical map of $\mathcal{F}(U)$ into $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{F})$ is homotopy equivalent to $0$.


cohomology-lemma-cech-exact-presheaves

The functor given by Equation (\ref{equation-cech-functor}) is an exact functor (see Homology, Lemma \ref{homology-lemma-exact-functor}).


cohomology-lemma-cech-cohomology-delta-functor-presheaves

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. The functors $\mathcal{F} \mapsto \check{H}^n(\mathcal{U}, \mathcal{F})$ form a $\delta$-functor from the abelian category of presheaves of $\mathcal{O}_X$-modules to the category of $\mathcal{O}_X(U)$-modules (see Homology, Definition \ref{homology-definition-cohomological-delta-functor}).


cohomology-lemma-cech-map-into

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Denote $j_{i_0\ldots i_p} : U_{i_0 \ldots i_p} \to X$ the open immersion. Consider the chain complex $K(\mathcal{U})_\bullet$ of presheaves of $\mathcal{O}_X$-modules $$ \ldots \to \bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\mathcal{O}_{U_{i_0i_1i_2}} \to \bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\mathcal{O}_{U_{i_0i_1}} \to \bigoplus_{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}} \to 0 \to \ldots $$ where the last nonzero term is placed in degree $0$ and where the map $$ (j_{i_0\ldots i_{p + 1}})_{p!}\mathcal{O}_{U_{i_0\ldots i_{p + 1}}} \longrightarrow (j_{i_0\ldots \hat i_j \ldots i_{p + 1}})_{p!} \mathcal{O}_{U_{i_0\ldots \hat i_j \ldots i_{p + 1}}} $$ is given by $(-1)^j$ times the canonical map. Then there is an isomorphism $$ \Hom_{\mathcal{O}_X}(K(\mathcal{U})_\bullet, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ functorial in $\mathcal{F} \in \Ob(\textit{PMod}(\mathcal{O}_X))$.


cohomology-lemma-homology-complex

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{O}_\mathcal{U} \subset \mathcal{O}_X$ be the image presheaf of the map $\bigoplus j_{p!}\mathcal{O}_{U_i} \to \mathcal{O}_X$. The chain complex $K(\mathcal{U})_\bullet$ of presheaves of Lemma \ref{lemma-cech-map-into} above has homology presheaves $$ H_i(K(\mathcal{U})_\bullet) = \left\{ \begin{matrix} 0 & \text{if} & i \not = 0 \\ \mathcal{O}_\mathcal{U} & \text{if} & i = 0 \end{matrix} \right. $$


cohomology-lemma-cech-cohomology-derived-presheaves

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering of $U \subset X$. The {\v C}ech cohomology functors $\check{H}^p(\mathcal{U}, -)$ are canonically isomorphic as a $\delta$-functor to the right derived functors of the functor $$ \check{H}^0(\mathcal{U}, -) : \textit{PMod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)}. $$ Moreover, there is a functorial quasi-isomorphism $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow R\check{H}^0(\mathcal{U}, \mathcal{F}) $$ where the right hand side indicates the right derived functor $$ R\check{H}^0(\mathcal{U}, -) : D^{+}(\textit{PMod}(\mathcal{O}_X)) \longrightarrow D^{+}(\mathcal{O}_X(U)) $$ of the left exact functor $\check{H}^0(\mathcal{U}, -)$.


cohomology-lemma-injective-trivial-cech

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{I}$ be an injective $\mathcal{O}_X$-module. Then $$ \check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. $$


cohomology-lemma-cech-cohomology

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. There is a transformation $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, -) \longrightarrow R\Gamma(U, -) $$ of functors $\textit{Mod}(\mathcal{O}_X) \to D^{+}(\mathcal{O}_X(U))$. In particular this provides canonical maps $\check{H}^p(\mathcal{U}, \mathcal{F}) \to H^p(U, \mathcal{F})$ for $\mathcal{F}$ ranging over $\textit{Mod}(\mathcal{O}_X)$.


cohomology-lemma-cech-h1

Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. The map $$ \check{H}^1(\mathcal{U}, \mathcal{H}) \longrightarrow H^1(X, \mathcal{H}) $$ is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{H})$ via the bijection of Lemma \ref{lemma-torsors-h1} with the set of isomorphism classes of $\mathcal{H}$-torsors which restrict to trivial torsors over each $U_i$.


cohomology-lemma-include

Let $X$ be a ringed space. Consider the functor $i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$. It is a left exact functor with right derived functors given by $$ R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F}) $$ see discussion in Section \ref{section-locality}.


cohomology-lemma-cech-spectral-sequence

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(U, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.


cohomology-lemma-cech-spectral-sequence-application

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume that $H^i(U_{i_0 \ldots i_p}, \mathcal{F}) = 0$ for all $i > 0$, all $p \geq 0$ and all $i_0, \ldots, i_p \in I$. Then $\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})$ as $\mathcal{O}_X(U)$-modules.


cohomology-lemma-ses-cech-h1

Let $X$ be a ringed space. Let $$ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 $$ be a short exact sequence of $\mathcal{O}_X$-modules. Let $U \subset X$ be an open subset. If there exists a cofinal system of open coverings $\mathcal{U}$ of $U$ such that $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$, then the map $\mathcal{G}(U) \to \mathcal{H}(U)$ is surjective.


cohomology-lemma-cech-vanish

\begin{slogan} If higher {\v C}ech cohomology of an abelian sheaf vanishes for all open covers, then higher cohomology vanishes. \end{slogan} Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module such that $$ \check{H}^p(\mathcal{U}, \mathcal{F}) = 0 $$ for all $p > 0$ and any open covering $\mathcal{U} : U = \bigcup_{i \in I} U_i$ of an open of $X$. Then $H^p(U, \mathcal{F}) = 0$ for all $p > 0$ and any open $U \subset X$.


cohomology-lemma-cech-vanish-basis

(Variant of Lemma \ref{lemma-cech-vanish}.) Let $X$ be a ringed space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume there exists a set of open coverings $\text{Cov}$ with the following properties: \begin{enumerate} \item For every $\mathcal{U} \in \text{Cov}$ with $\mathcal{U} : U = \bigcup_{i \in I} U_i$ we have $U, U_i \in \mathcal{B}$ and every $U_{i_0 \ldots i_p} \in \mathcal{B}$. \item For every $U \in \mathcal{B}$ the open coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of open coverings of $U$. \item For every $\mathcal{U} \in \text{Cov}$ we have $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$. \end{enumerate} Then $H^p(U, \mathcal{F}) = 0$ for all $p > 0$ and any $U \in \mathcal{B}$.


cohomology-lemma-pushforward-injective

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{I}$ be an injective $\mathcal{O}_X$-module. Then \begin{enumerate} \item $\check{H}^p(\mathcal{V}, f_*\mathcal{I}) = 0$ for all $p > 0$ and any open covering $\mathcal{V} : V = \bigcup_{j \in J} V_j$ of $Y$. \item $H^p(V, f_*\mathcal{I}) = 0$ for all $p > 0$ and every open $V \subset Y$. \end{enumerate} In other words, $f_*\mathcal{I}$ is right acyclic for $\Gamma(V, -)$ (see Derived Categories, Definition \ref{derived-definition-derived-functor}) for any $V \subset Y$ open.


cohomology-lemma-pushforward-injective-flat

Let $f : X \to Y$ be a morphism of ringed spaces. Assume $f$ is flat. Then $f_*\mathcal{I}$ is an injective $\mathcal{O}_Y$-module for any injective $\mathcal{O}_X$-module $\mathcal{I}$.


cohomology-lemma-cohomology-products

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $I$ be a set. For $i \in I$ let  $\mathcal{F}_i$ be an $\mathcal{O}_X$-module. Let $U \subset X$ be open. The canonical map $$ H^p(U, \prod\nolimits_{i \in I} \mathcal{F}_i) \longrightarrow \prod\nolimits_{i \in I} H^p(U, \mathcal{F}_i) $$ is an isomorphism for $p = 0$ and injective for $p = 1$.


cohomology-lemma-injective-flasque

Let $(X, \mathcal{O}_X)$ be a ringed space. Then any injective $\mathcal{O}_X$-module is flasque.


cohomology-lemma-flasque-acyclic

Let $(X, \mathcal{O}_X)$ be a ringed space. Any flasque $\mathcal{O}_X$-module is acyclic for $R\Gamma(X, -)$ as well as $R\Gamma(U, -)$ for any open $U$ of $X$.


cohomology-lemma-flasque-acyclic-cech

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{U} : U = \bigcup U_i$ be an open covering. If $\mathcal{F}$ is flasque, then $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for $p > 0$.


cohomology-lemma-flasque-acyclic-pushforward

Let $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. If $\mathcal{F}$ is flasque, then $R^pf_*\mathcal{F} = 0$ for $p > 0$.


cohomology-lemma-vanishing-ravi

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume the restriction mappings $\mathcal{F}(U) \to \mathcal{F}(U')$ are surjective for $U'$ an arbitrary union of opens of the form $U_{i_0 \ldots i_p}$. Then $\check{H}^p(\mathcal{U}, \mathcal{F})$ vanishes for $p > 0$.


cohomology-lemma-before-Leray

Let $f : X \to Y$ be a morphism of ringed spaces. There is a commutative diagram $$ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma(X, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_X(X)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma(Y, -)} & & D^{+}(\mathcal{O}_Y(Y)) } $$ More generally for any $V \subset Y$ open and $U = f^{-1}(V)$ there is a commutative diagram $$ \xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma(U, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_X(U)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma(V, -)} & & D^{+}(\mathcal{O}_Y(V)) } $$ See also Remark \ref{remark-elucidate-lemma} for more explanation.


cohomology-lemma-modules-abelian

Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. \begin{enumerate} \item The cohomology groups $H^i(U, \mathcal{F})$ for $U \subset X$ open of $\mathcal{F}$ computed as an $\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \item Let $f : X \to Y$ be a morphism of ringed spaces. The higher direct images $R^if_*\mathcal{F}$ of $\mathcal{F}$ computed as an $\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \end{enumerate} There are similar statements in the case of bounded below complexes of $\mathcal{O}_X$-modules.


cohomology-lemma-Leray

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. There is a spectral sequence $$ E_2^{p, q} = H^p(Y, R^qf_*(\mathcal{F}^\bullet)) $$ converging to $H^{p + q}(X, \mathcal{F}^\bullet)$.


cohomology-lemma-apply-Leray

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. \begin{enumerate} \item If $R^qf_*\mathcal{F} = 0$ for $q > 0$, then $H^p(X, \mathcal{F}) = H^p(Y, f_*\mathcal{F})$ for all $p$. \item If $H^p(Y, R^qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^q(X, \mathcal{F}) = H^0(Y, R^qf_*\mathcal{F})$ for all $q$. \end{enumerate}


cohomology-lemma-higher-direct-images-compose

\begin{slogan} The total derived functor of a composition is the composition of the total derived functors. \end{slogan} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. In this case $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors from $D^{+}(X) \to D^{+}(Z)$.


cohomology-lemma-relative-Leray

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. There is a spectral sequence with $$ E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F}) $$ converging to $R^{p + q}(g \circ f)_*\mathcal{F}$. This spectral sequence is functorial in $\mathcal{F}$, and there is a version for bounded below complexes of $\mathcal{O}_X$-modules.


cohomology-lemma-functoriality

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}^\bullet$, resp.\ $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_Y$-modules, resp.\ $\mathcal{O}_X$-modules. Let $\varphi : \mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet$ be a morphism of complexes. There is a canonical morphism $$ \mathcal{G}^\bullet \longrightarrow Rf_*(\mathcal{F}^\bullet) $$ in $D^{+}(Y)$. Moreover this construction is functorial in the triple $(\mathcal{G}^\bullet, \mathcal{F}^\bullet, \varphi)$.


cohomology-lemma-functoriality-cech

Let $f : X \to Y$ be a morphism of ringed spaces. Let $\varphi : f^*\mathcal{G} \to \mathcal{F}$ be an $f$-map from an $\mathcal{O}_Y$-module $\mathcal{G}$ to an $\mathcal{O}_X$-module $\mathcal{F}$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and $\mathcal{V} : Y = \bigcup_{j \in J} V_j$ be open coverings. Assume that $\mathcal{U}$ is a refinement of $f^{-1}\mathcal{V} : X = \bigcup_{j \in J} f^{-1}(V_j)$. In this case there exists a commutative diagram $$ \xymatrix{ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] & R\Gamma(X, \mathcal{F}) \\ \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \ar[r] \ar[u]^\gamma & R\Gamma(Y, \mathcal{G}) \ar[u] } $$ in $D^{+}(\mathcal{O}_X(X))$ with horizontal arrows given by Lemma \ref{lemma-cech-cohomology} and right vertical arrow by (\ref{equation-functorial-derived}). In particular we get commutative diagrams of cohomology groups $$ \xymatrix{ \check{H}^p(\mathcal{U}, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}) \\ \check{H}^p(\mathcal{V}, \mathcal{G}) \ar[r] \ar[u]^\gamma & H^p(Y, \mathcal{G}) \ar[u] } $$ where the right vertical arrow is (\ref{equation-functorial})


cohomology-lemma-cech-always

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian sheaf. Then the map $\check{H}^1(X, \mathcal{F}) \to H^1(X, \mathcal{F})$ defined in (\ref{equation-cech-to-cohomology}) is an isomorphism.


cohomology-lemma-cech-Hausdorff-quasi-compact

Let $X$ be a Hausdorff and quasi-compact topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then the map $\check{H}^n(X, \mathcal{F}) \to H^n(X, \mathcal{F})$ defined in (\ref{equation-cech-to-cohomology}) is an isomorphism for all $n$.


cohomology-lemma-cohomology-of-closed

\begin{reference} \cite[Expose V bis, 4.1.3]{SGA4} \end{reference} Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. For every abelian sheaf $\mathcal{F}$ on $X$ the canonical map $$ \colim H^p(U, \mathcal{F}) \longrightarrow H^p(Z, \mathcal{F}|_Z) $$ where the colimit is over open neighbourhoods $U$ of $Z$ in $X$ is an isomorphism.


cohomology-lemma-base-change-map-flat-case

Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a commutative diagram of ringed spaces. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map $$ g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet $$ in $D^{+}(S')$.


cohomology-lemma-proper-base-change

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $y \in Y$. Assume that \begin{enumerate} \item $f$ is closed, \item $f$ is separated, and \item $f^{-1}(y)$ is quasi-compact. \end{enumerate} Then for $E$ in $D^+(\mathcal{O}_X)$ we have $(Rf_*E)_y = R\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$ in $D^+(\mathcal{O}_{Y, y})$.


cohomology-lemma-proper-base-change-sheaves-of-sets

Consider a cartesian square of topological spaces $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ Assume that $f$ is proper and separated. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ for any sheaf of sets $\mathcal{F}$ on $X$.


cohomology-lemma-quasi-separated-cohomology-colimit

Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties: \begin{enumerate} \item there exists a basis of quasi-compact open subsets, and \item the intersection of any two quasi-compact opens is quasi-compact. \end{enumerate} Then for any directed system $(\mathcal{F}_i, \varphi_{ii'})$ of sheaves of $\mathcal{O}_X$-modules and for any quasi-compact open $U \subset X$ the canonical map $$ \colim_i H^q(U, \mathcal{F}_i) \longrightarrow H^q(U, \colim_i \mathcal{F}_i) $$ is an isomorphism for every $q \geq 0$.


cohomology-lemma-colimit

In the situation discussed above. Let $i \in \Ob(\mathcal{I})$ and let $U_i \subset X_i$ be quasi-compact open. Then $$ \colim_{a : j \to i} H^p(f_a^{-1}(U_i), \mathcal{F}_j) = H^p(p_i^{-1}(U_i), \mathcal{F}) $$ for all $p \geq 0$. In particular we have $H^p(X, \mathcal{F}) = \colim H^p(X_i, \mathcal{F}_i)$.


cohomology-lemma-cohomology-and-closed-immersions

Let $i : Z \to X$ be a closed immersion of topological spaces. For any abelian sheaf $\mathcal{F}$ on $Z$ we have $H^p(Z, \mathcal{F}) = H^p(X, i_*\mathcal{F})$.


cohomology-lemma-irreducible-constant-cohomology-zero

Let $X$ be an irreducible topological space. Then $H^p(X, \underline{A}) = 0$ for all $p > 0$ and any abelian group $A$.


cohomology-lemma-subsheaf-of-constant-sheaf

\begin{reference} \cite[Page 168]{Tohoku}. \end{reference} Let $X$ be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let $\mathcal{F} \subset \underline{\mathbf{Z}}$ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration $$ (0) = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_n = \mathcal{F} $$ by abelian subsheaves such that for each $0 < i \leq n$ there exists a short exact sequence $$ 0 \to j'_!\underline{\mathbf{Z}}_V \to j_!\underline{\mathbf{Z}}_U \to \mathcal{F}_i/\mathcal{F}_{i - 1} \to 0 $$ with $j : U \to X$ and $j' : V \to X$ the inclusion of quasi-compact opens into $X$.


cohomology-lemma-cohomology-of-neighbourhoods-of-closed

Let $X$ be a spectral space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $E \subset X$ be a quasi-compact subset. Let $W \subset X$ be the set of points of $X$ which specialize to a point of $E$. \begin{enumerate} \item $H^p(W, \mathcal{F}|_W) = \colim H^p(U, \mathcal{F})$ where the colimit is over quasi-compact open neighbourhoods of $E$, \item $H^p(W \setminus E, \mathcal{F}|_{W \setminus E}) = \colim H^p(U \setminus E, \mathcal{F}|_{U \setminus E})$ if $E$ is a constructible subset. \end{enumerate}


cohomology-lemma-vanishing-for-profinite

Let $X$ be a profinite topological space. Then $H^q(X, \mathcal{F}) = 0$ for all $q > 0$ and all abelian sheaves $\mathcal{F}$.


cohomology-lemma-ordered-alternating

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c$ is a morphism of complexes. In fact it induces an isomorphism $$ c : \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) $$ of complexes.


cohomology-lemma-project-to-ordered

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $\pi : \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F})$ is a morphism of complexes. It induces an isomorphism $$ \pi : \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F}) $$ of complexes which is a left inverse to the morphism $c$.


cohomology-lemma-alternating-usual

Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c \circ \pi$ is homotopic to the identity on $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$. In particular the inclusion map $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is a homotopy equivalence.


cohomology-lemma-alternating-cech-trivial

Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. If $U_i = U$ for some $i \in I$, then the extended alternating {\v C}ech complex $$ \mathcal{F}(U) \to \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) $$ obtained by putting $\mathcal{F}(U)$ in degree $-1$ with differential given by the canonical map of $\mathcal{F}(U)$ into $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{F})$ is homotopy equivalent to $0$. Similarly, for any total ordering on $I$ the extended ordered {\v C}ech complex $$ \mathcal{F}(U) \to \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F}) $$ is homotopy equivalent to $0$.


cohomology-lemma-covering-resolution

Let $X$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering of $X$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Denote $\mathcal{F}_{i_0 \ldots i_p}$ the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_p}$. There exists a complex ${\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})$ of $\mathcal{O}_X$-modules with $$ {\mathfrak C}^p(\mathcal{U}, \mathcal{F}) = \prod\nolimits_{i_0 \ldots i_p} (j_{i_0 \ldots i_p})_* \mathcal{F}_{i_0 \ldots i_p} $$ and differential $d : {\mathfrak C}^p(\mathcal{U}, \mathcal{F}) \to {\mathfrak C}^{p + 1}(\mathcal{U}, \mathcal{F})$ as in Equation (\ref{equation-d-cech}). Moreover, there exists a canonical map $$ \mathcal{F} \to {\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F}) $$ which is a quasi-isomorphism, i.e., ${\mathfrak C}^\bullet(\mathcal{U}, \mathcal{F})$ is a resolution of $\mathcal{F}$.


cohomology-lemma-cech-complex-complex

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. For a bounded below complex $\mathcal{F}^\bullet$ of $\mathcal{O}_X$-modules there is a canonical map $$ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(X, \mathcal{F}^\bullet) $$ functorial in $\mathcal{F}^\bullet$ and compatible with (\ref{equation-global-sections-to-cech}) and (\ref{equation-transformation}). There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = H^p(\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \underline{H}^q(\mathcal{F}^\bullet))) $$ converging to $H^{p + q}(X, \mathcal{F}^\bullet)$.


cohomology-lemma-cech-complex-complex-computes

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. If $H^i(U_{i_0 \ldots i_p}, \mathcal{F}^q) = 0$ for all $i > 0$ and all $p, i_0, \ldots, i_p, q$, then the map $ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \to R\Gamma(X, \mathcal{F}^\bullet) $ of Lemma \ref{lemma-cech-complex-complex} is an isomorphism.


cohomology-lemma-compute-sign-cup-product-boundaries

In the situation above, assume {\v C}ech cohomology agrees with cohomology for the sheaves $\mathcal{F}_i^p$ and $\mathcal{G}_j^q$. Let $a_3 \in H^n(X, \mathcal{F}_3^\bullet)$ and $b_1 \in H^m(X, \mathcal{G}_1^\bullet)$. Then we have $$ \gamma_1( \partial a_3 \cup b_1) = (-1)^{n + 1} \gamma_3( a_3 \cup \partial b_1) $$ in $H^{n + m}(X, \mathcal{H}^\bullet)$ where $\partial$ indicates the boundary map on cohomology associated to the short exact sequences of complexes above.


cohomology-lemma-boundary-derivation-over-cup-product

Let $X$ be a topological space. Let $\mathcal{O}' \to \mathcal{O}$ be a surjection of sheaves of rings whose kernel $\mathcal{I} \subset \mathcal{O}'$ has square zero. Then $M = H^1(X, \mathcal{I})$ is a $R = H^0(X, \mathcal{O})$-module and the boundary map $\partial : R \to M$ associated to the short exact sequence $$ 0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0 $$ is a derivation (Algebra, Definition \ref{algebra-definition-derivation}).


cohomology-lemma-derived-tor-exact

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{G}^\bullet$ be a complex of $\mathcal{O}_X$-modules. The functors $$ K(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K(\textit{Mod}(\mathcal{O}_X)), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} \mathcal{F}^\bullet) $$ and $$ K(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K(\textit{Mod}(\mathcal{O}_X)), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{G}^\bullet) $$ are exact functors of triangulated categories.


cohomology-lemma-K-flat-quasi-isomorphism

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{K}^\bullet$ be a K-flat complex. Then the functor $$ K(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K(\textit{Mod}(\mathcal{O}_X)), \quad \mathcal{F}^\bullet \longmapsto \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{K}^\bullet) $$ transforms quasi-isomorphisms into quasi-isomorphisms.


cohomology-lemma-check-K-flat-stalks

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{K}^\bullet$ be a complex of $\mathcal{O}_X$-modules. Then $\mathcal{K}^\bullet$ is K-flat if and only if for all $x \in X$ the complex $\mathcal{K}_x^\bullet$ of $\mathcal{O}_{X, x}$-modules is K-flat (More on Algebra, Definition \ref{more-algebra-definition-K-flat}).


cohomology-lemma-tensor-product-K-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. If $\mathcal{K}^\bullet$, $\mathcal{L}^\bullet$ are K-flat complexes of $\mathcal{O}_X$-modules, then $\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{L}^\bullet)$ is a K-flat complex of $\mathcal{O}_X$-modules.


cohomology-lemma-K-flat-two-out-of-three

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $(\mathcal{K}_1^\bullet, \mathcal{K}_2^\bullet, \mathcal{K}_3^\bullet)$ be a distinguished triangle in $K(\textit{Mod}(\mathcal{O}_X))$. If two out of three of $\mathcal{K}_i^\bullet$ are K-flat, so is the third.


cohomology-lemma-pullback-K-flat

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback of a K-flat complex of $\mathcal{O}_Y$-modules is a K-flat complex of $\mathcal{O}_X$-modules.


cohomology-lemma-bounded-flat-K-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. A bounded above complex of flat $\mathcal{O}_X$-modules is K-flat.


cohomology-lemma-colimit-K-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots$ be a system of K-flat complexes. Then $\colim_i \mathcal{K}_i^\bullet$ is K-flat.


cohomology-lemma-resolution-by-direct-sums-extensions-by-zero

Let $(X, \mathcal{O}_X)$ be a ringed space. For any complex $\mathcal{G}^\bullet$ of $\mathcal{O}_X$-modules there exists a commutative diagram of complexes of $\mathcal{O}_X$-modules $$ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau_{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau_{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } $$ with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each $\mathcal{K}_n^\bullet$ is a bounded above complex whose terms are direct sums of $\mathcal{O}_X$-modules of the form $j_{U!}\mathcal{O}_U$, and (3) the maps $\mathcal{K}_n^\bullet \to \mathcal{K}_{n + 1}^\bullet$ are termwise split injections whose cokernels are direct sums of $\mathcal{O}_X$-modules of the form $j_{U!}\mathcal{O}_U$. Moreover, the map $\colim \mathcal{K}_n^\bullet \to \mathcal{G}^\bullet$ is a quasi-isomorphism.


cohomology-lemma-K-flat-resolution

Let $(X, \mathcal{O}_X)$ be a ringed space. For any complex $\mathcal{G}^\bullet$ there exists a $K$-flat complex $\mathcal{K}^\bullet$ whose terms are flat $\mathcal{O}_X$-modules and a quasi-isomorphism $\mathcal{K}^\bullet \to \mathcal{G}^\bullet$ which is termwise surjective.


cohomology-lemma-derived-tor-quasi-isomorphism-other-side

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\alpha : \mathcal{P}^\bullet \to \mathcal{Q}^\bullet$ be a quasi-isomorphism of K-flat complexes of $\mathcal{O}_X$-modules. For every complex $\mathcal{F}^\bullet$ of $\mathcal{O}_X$-modules the induced map $$ \text{Tot}(\text{id}_{\mathcal{F}^\bullet} \otimes \alpha) : \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{P}^\bullet) \longrightarrow \text{Tot}(\mathcal{F}^\bullet \otimes_{\mathcal{O}_X} \mathcal{Q}^\bullet) $$ is a quasi-isomorphism.


cohomology-lemma-flat-tor-zero

\begin{slogan} Tor measures the deviation of flatness. \end{slogan} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is a flat $\mathcal{O}_X$-module, and \item $\text{Tor}_1^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) = 0$ for every $\mathcal{O}_X$-module $\mathcal{G}$. \end{enumerate}


cohomology-lemma-factor-through-K-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $a : \mathcal{K}^\bullet \to \mathcal{L}^\bullet$ be a map of complexes of $\mathcal{O}_X$-modules. If $\mathcal{K}^\bullet$ is K-flat, then there exist a complex $\mathcal{N}^\bullet$ and maps of complexes $b : \mathcal{K}^\bullet \to \mathcal{N}^\bullet$ and $c : \mathcal{N}^\bullet \to \mathcal{L}^\bullet$ such that \begin{enumerate} \item $\mathcal{N}^\bullet$ is K-flat, \item $c$ is a quasi-isomorphism, \item $a$ is homotopic to $c \circ b$. \end{enumerate} If the terms of $\mathcal{K}^\bullet$ are flat, then we may choose $\mathcal{N}^\bullet$, $b$, and $c$ such that the same is true for $\mathcal{N}^\bullet$.


cohomology-lemma-derived-base-change

The construction above is independent of choices and defines an exact functor of triangulated categories $Lf^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$.


cohomology-lemma-derived-pullback-composition

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_Z) \to D(\mathcal{O}_X)$.


cohomology-lemma-pullback-tensor-product

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism $$ Lf^*( \mathcal{F}^\bullet \otimes_{\mathcal{O}_Y}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet  \otimes_{\mathcal{O}_X}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet  $$ for $\mathcal{F}^\bullet, \mathcal{G}^\bullet \in \Ob(D(X))$.


cohomology-lemma-variant-derived-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism $$ \mathcal{F}^\bullet \otimes_{\mathcal{O}_X}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet = \mathcal{F}^\bullet  \otimes_{f^{-1}\mathcal{O}_Y}^{\mathbf{L}} f^{-1}\mathcal{G}^\bullet  $$ for $\mathcal{F}^\bullet$ in $D(X)$ and $\mathcal{G}^\bullet$ in $D(Y)$.


cohomology-lemma-tensor-pull-compatibility

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{K}^\bullet$ and $\mathcal{M}^\bullet$ be complexes of $\mathcal{O}_Y$-modules. The diagram $$ \xymatrix{ Lf^*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{M}^\bullet) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y} \mathcal{M}^\bullet) \ar[d] \\ Lf^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{M}^\bullet \ar[d] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y} \mathcal{M}^\bullet) \ar[d] \\ f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} f^*\mathcal{M}^\bullet \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{M}^\bullet) } $$ commutes.


cohomology-lemma-adjoint

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma \ref{lemma-derived-base-change} are adjoint: $$ \Hom_{D(X)}(Lf^*\mathcal{G}^\bullet, \mathcal{F}^\bullet) = \Hom_{D(Y)}(\mathcal{G}^\bullet, Rf_*\mathcal{F}^\bullet) $$ bifunctorially in $\mathcal{F}^\bullet \in \Ob(D(X))$ and $\mathcal{G}^\bullet \in \Ob(D(Y))$.


cohomology-lemma-derived-pushforward-composition

Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors $D(\mathcal{O}_X) \to D(\mathcal{O}_Z)$.


cohomology-lemma-adjoints-push-pull-compatibility

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{K}^\bullet$ be a complex of $\mathcal{O}_X$-modules. The diagram $$ \xymatrix{ Lf^*f_*\mathcal{K}^\bullet \ar[r] \ar[d] & f^*f_*\mathcal{K}^\bullet \ar[d] \\ Lf^*Rf_*\mathcal{K}^\bullet \ar[r] & \mathcal{K}^\bullet } $$ coming from $Lf^* \to f^*$ on complexes, $f_* \to Rf_*$ on complexes, and adjunction $Lf^* \circ Rf_* \to \text{id}$ commutes in $D(\mathcal{O}_X)$.


cohomology-lemma-spectral-sequence-filtered-object

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a filtered complex of $\mathcal{O}_X$-modules. There exists a canonical spectral sequence $(E_r, \text{d}_r)_{r \geq 1}$ of bigraded $\Gamma(X, \mathcal{O}_X)$-modules with $d_r$ of bidegree $(r, -r + 1)$ and $$ E_1^{p, q} = H^{p + q}(X, \text{gr}^p\mathcal{F}^\bullet) $$ If for every $n$ we have $$ H^n(X, F^p\mathcal{F}^\bullet) = 0\text{ for }p \gg 0 \quad\text{and}\quad H^n(X, F^p\mathcal{F}^\bullet) = H^n(X, \mathcal{F}^\bullet)\text{ for }p \ll 0 $$ then the spectral sequence is bounded and converges to $H^*(X, \mathcal{F}^\bullet)$.


cohomology-lemma-relative-spectral-sequence-filtered-object

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet$ be a filtered complex of $\mathcal{O}_X$-modules. There exists a canonical spectral sequence $(E_r, \text{d}_r)_{r \geq 1}$ of bigraded $\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and $$ E_1^{p, q} = R^{p + q}f_*\text{gr}^p\mathcal{F}^\bullet $$ If for every $n$ we have $$ R^nf_*F^p\mathcal{F}^\bullet = 0 \text{ for }p \gg 0 \quad\text{and}\quad R^nf_*F^p\mathcal{F}^\bullet = R^nf_*\mathcal{F}^\bullet \text{ for }p \ll 0 $$ then the spectral sequence is bounded and converges to $Rf_*\mathcal{F}^\bullet$.


cohomology-lemma-godement-resolution

Let $(X, \mathcal{O}_X)$ be a ringed space. For every sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there is a resolution $$ 0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots $$ functorial in $\mathcal{F}$ such that each term $f_*f^* \ldots f_*f^*\mathcal{F}$ is a flasque $\mathcal{O}_X$-module and such that for all $x \in X$ the map $$ \mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big) $$ is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules.


cohomology-lemma-godement-resolution-bounded-below

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. There exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{G}^\bullet$ where $\mathcal{F}^\bullet$ be a bounded below complex of flasque $\mathcal{O}_X$-modules and for all $x \in X$ the map $\mathcal{F}^\bullet_x \to \mathcal{G}^\bullet_x$ is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules.


cohomology-lemma-second-cup-equals-first

This construction gives the cup product.


cohomology-lemma-cup-compatible-with-naive

In the situation above the following diagram commutes $$ \xymatrix{ f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} f_*\mathcal{M}^\bullet \ar[r] \ar[d] & Rf_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\mathcal{M}^\bullet \ar[d]^{\text{Remark \ref{remark-cup-product}}} \\ \text{Tot}( f_*\mathcal{K}^\bullet \otimes_{\mathcal{O}_Y} f_*\mathcal{M}^\bullet) \ar[d]_{\text{naive cup product}} & Rf_*(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{M}^\bullet) \ar[d] \\ f_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{M}^\bullet) \ar[r] & Rf_*\text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{M}^\bullet) } $$


cohomology-lemma-diagrams-commute

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{K}^\bullet$ and $\mathcal{M}^\bullet$ be bounded below complexes of $\mathcal{O}_X$-modules. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering Then $$ \xymatrix{ \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{K}^\bullet)) \otimes_A^\mathbf{L} \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{M}^\bullet)) \ar[d] \ar[r] & R\Gamma(X, \mathcal{K}^\bullet) \otimes_A^\mathbf{L} R\Gamma(X, \mathcal{M}^\bullet) \ar[d]^\mu \\ \text{Tot}( \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{K}^\bullet)) \otimes_A \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{M}^\bullet))) \ar[d]^{(\ref{equation-needs-signs})} & R\Gamma(X, \mathcal{K}^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{M}^\bullet) \ar[d] \\ \text{Tot}( \check{\mathcal{C}}^\bullet({\mathcal U}, \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{M}^\bullet) )) \ar[r] & R\Gamma(X, \text{Tot}(\mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{M}^\bullet)) } $$ where the horizontal arrows are the ones in Lemma \ref{lemma-cech-complex-complex} commutes in $D(A)$.


cohomology-lemma-cup-product-associative

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The relative cup product of Remark \ref{remark-cup-product} is associative in the sense that the diagram $$ \xymatrix{ Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \ar[r] \ar[d] & Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*M \ar[d] \\ Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*(L \otimes_{\mathcal{O}_X}^\mathbf{L} M) \ar[r] & Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L}  L \otimes_{\mathcal{O}_X}^\mathbf{L} M) } $$ is commutative in $D(\mathcal{O}_Y)$ for all $K, L, M$ in $D(\mathcal{O}_X)$.


cohomology-lemma-cup-product-commutative

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The relative cup product of Remark \ref{remark-cup-product} is commutative in the sense that the diagram $$ \xymatrix{ Rf_*K \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*L \ar[r] \ar[d]_\psi & Rf_*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) \ar[d]^{Rf_*\psi} \\ Rf_*L \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*K \ar[r] & Rf_*(L \otimes_{\mathcal{O}_X}^\mathbf{L} K) } $$ is commutative in $D(\mathcal{O}_Y)$ for all $K, L$ in $D(\mathcal{O}_X)$. Here $\psi$ is the commutativity constraint on the derived category (Lemma \ref{lemma-symmetric-monoidal-derived}).


cohomology-lemma-restrict-K-injective-to-open

Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. The restriction of a K-injective complex of $\mathcal{O}_X$-modules to $U$ is a K-injective complex of $\mathcal{O}_U$-modules.


cohomology-lemma-unbounded-cohomology-of-open

Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. For $K$ in $D(\mathcal{O}_X)$ we have $H^p(U, K) = H^p(U, K|_U)$.


cohomology-lemma-sheafification-cohomology

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_X)$. The sheafification of $$ U \mapsto H^q(U, K) = H^q(U, K|_U) $$ is the $q$th cohomology sheaf $H^q(K)$ of $K$.


cohomology-lemma-restrict-direct-image-open

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Given an open subspace $V \subset Y$, set $U = f^{-1}(V)$ and denote $g : U \to V$ the induced morphism. Then $(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\mathcal{O}_X)$.


cohomology-lemma-Leray-unbounded

Let $f : X \to Y$ be a morphism of ringed spaces. Then $R\Gamma(Y, -) \circ Rf_* = R\Gamma(X, -)$ as functors $D(\mathcal{O}_X) \to D(\Gamma(Y, \mathcal{O}_Y))$. More generally for $V \subset Y$ open and $U = f^{-1}(V)$ we have $R\Gamma(U, -) = R\Gamma(V, -) \circ Rf_*$.


cohomology-lemma-unbounded-describe-higher-direct-images

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $K$ be in $D(\mathcal{O}_X)$. Then $H^i(Rf_*K)$ is the sheaf associated to the presheaf $$ V \mapsto H^i(f^{-1}(V), K) = H^i(V, Rf_*K) $$


cohomology-lemma-modules-abelian-unbounded

Let $X$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_X)$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_X)$. \begin{enumerate} \item For any open $U \subset X$ there is a canonical map $R\Gamma(U, K) \to R\Gamma(U, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$. \item Let $f : X \to Y$ be a morphism of ringed spaces. There is a canonical map $Rf_*K \to Rf_*(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_Y)$. \end{enumerate}


cohomology-lemma-adjoint-lower-shriek-restrict

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $U \subset X$ be an open subset. Denote $j : (U, \mathcal{O}_U) \to (X, \mathcal{O}_X)$ the corresponding open immersion. The restriction functor $D(\mathcal{O}_X) \to D(\mathcal{O}_U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_U) \to D(\mathcal{O}_X)$.


cohomology-lemma-K-injective-flat

Let $f : X \to Y$ be a flat morphism of ringed spaces. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_X$-modules, then $f_*\mathcal{I}^\bullet$ is K-injective as a complex of $\mathcal{O}_Y$-modules.


cohomology-lemma-exact-sequence-lower-shriek

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces. For any object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ j_{U \cap V!}E|_{U \cap V} \to j_{U!}E|_U \oplus j_{V!}E|_V \to E \to  j_{U \cap V!}E|_{U \cap V}[1] $$ in $D(\mathcal{O}_X)$.


cohomology-lemma-exact-sequence-j-star

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces. For any object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ E \to  Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to Rj_{U \cap V, *}E|_{U \cap V} \to E[1] $$ in $D(\mathcal{O}_X)$.


cohomology-lemma-mayer-vietoris-hom

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces of $X$. For objects $E$, $F$ of $D(\mathcal{O}_X)$ we have a Mayer-Vietoris sequence $$ \xymatrix{ & \ldots \ar[r] & \Ext^{-1}(E_{U \cap V}, F_{U \cap V}) \ar[lld] \\ \Hom(E, F) \ar[r] & \Hom(E_U, F_U) \oplus \Hom(E_V, F_V) \ar[r] & \Hom(E_{U \cap V}, F_{U \cap V}) } $$ where the subscripts denote restrictions to the relevant opens and the $\Hom$'s and $\Ext$'s are taken in the relevant derived categories.


cohomology-lemma-unbounded-mayer-vietoris

Let $(X, \mathcal{O}_X)$ be a ringed space. Suppose that $X = U \cup V$ is a union of two open subsets. For an object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$ R\Gamma(X, E) \to R\Gamma(U, E) \oplus R\Gamma(V, E) \to R\Gamma(U \cap V, E) \to R\Gamma(X, E)[1] $$ and in particular a long exact cohomology sequence $$ \ldots \to H^n(X, E) \to H^n(U, E) \oplus H^0(V, E) \to H^n(U \cap V, E) \to H^{n + 1}(X, E) \to \ldots $$ The construction of the distinguished triangle and the long exact sequence is functorial in $E$.


cohomology-lemma-unbounded-relative-mayer-vietoris

Let $f : X \to Y$ be a morphism of ringed spaces. Suppose that $X = U \cup V$ is a union of two open subsets. Denote $a = f|_U : U \to Y$, $b = f|_V : V \to Y$, and $c = f|_{U \cap V} : U \cap V \to Y$. For every object $E$ of $D(\mathcal{O}_X)$ there exists a distinguished triangle $$ Rf_*E \to Ra_*(E|_U) \oplus Rb_*(E|_V) \to Rc_*(E|_{U \cap V}) \to Rf_*E[1] $$ This triangle is functorial in $E$.


cohomology-lemma-pushforward-restriction

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $j : U \to X$ be an open subspace. Let $T \subset X$ be a closed subset contained in $U$. \begin{enumerate} \item If $E$ is an object of $D(\mathcal{O}_X)$ whose cohomology sheaves are supported on $T$, then $E \to Rj_*(E|_U)$ is an isomorphism. \item If $F$ is an object of $D(\mathcal{O}_U)$ whose cohomology sheaves are supported on $T$, then $j_!F \to Rj_*F$ is an isomorphism. \end{enumerate}


cohomology-lemma-cohomology-with-support-sheaf-on-support

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. \begin{enumerate} \item $R\mathcal{H}_Z : D(\mathcal{O}_X) \to D(\mathcal{O}_X|_Z)$ is right adjoint to $i_* : D(\mathcal{O}_X|_Z) \to D(\mathcal{O}_X)$. \item For $K$ in $D(\mathcal{O}_X|_Z)$ we have $R\mathcal{H}_Z(i_*K) = K$. \item Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_X|_Z$-modules on $Z$. Then $\mathcal{H}^p_Z(i_*\mathcal{G}) = 0$ for $p > 0$. \end{enumerate}


cohomology-lemma-complexes-with-support-on-closed

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. \begin{enumerate} \item For $K$ in $D(\mathcal{O}_X|_Z)$ we have $i_*K$ in $D_Z(\mathcal{O}_X)$. \item The functor $i_* : D(\mathcal{O}_X|_Z) \to D_Z(\mathcal{O}_X)$ is an equivalence with quasi-inverse $i^{-1}|_{D_Z(\mathcal{O}_X)} = R\mathcal{H}_Z|_{D_Z(\mathcal{O}_X)}$. \item The functor $i_* \circ R\mathcal{H}_Z : D(\mathcal{O}_X) \to D_Z(\mathcal{O}_X)$ is right adjoint to the inclusion functor $D_Z(\mathcal{O}_X) \to D(\mathcal{O}_X)$. \end{enumerate}


cohomology-lemma-sections-with-support-K-injective

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_X$-modules, then $\mathcal{H}_Z(\mathcal{I}^\bullet)$ is K-injective complex of $\mathcal{O}_X|_Z$-modules.


cohomology-lemma-local-to-global-sections-with-support

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Then $R\Gamma(Z, - ) \circ R\mathcal{H}_Z = R\Gamma_Z(X, - )$ as functors $D(\mathcal{O}_X) \to D(\mathcal{O}_X(X))$.


cohomology-lemma-triangle-sections-with-support

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Let $U = X \setminus Z$. There is a distinguished triangle $$ R\Gamma_Z(X, K) \to R\Gamma(X, K) \to R\Gamma(U, K) \to R\Gamma_Z(X, K)[1] $$ in $D(\mathcal{O}_X(X))$ functorial for $K$ in $D(\mathcal{O}_X)$.