spaces-divisors-lemma-blowing-up-gives-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties: \begin{enumerate} \item $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism, \item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$, \item there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$ \end{enumerate}


spaces-divisors-lemma-universal-property-blowing-up

\begin{slogan} Blow up a closed subset to make it Cartier. \end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. Let $\mathcal{C}$ be the full subcategory of $(\textit{Spaces}/X)$ consisting of $Y \to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$ is a final object of $\mathcal{C}$.


spaces-divisors-lemma-blow-up-effective-Cartier-divisor

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$.


spaces-divisors-lemma-blow-up-pullback-effective-Cartier

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be a blowup of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^{-1}D$ is defined (see Definition \ref{definition-pullback-effective-Cartier-divisor}).


spaces-divisors-lemma-blowing-up-two-ideals

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ and $\mathcal{J}$ be quasi-coherent sheaves of ideals. Let $b : X' \to X$ be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$ is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.


spaces-divisors-lemma-blowing-up-projective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then $b : X' \to X$ is a proper morphism.


spaces-divisors-lemma-composition-finite-type-blowups

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Let $Z \subset X$ be a closed subspace of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subspace of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subspace $Y \subset X$ of finite presentation, such that \begin{enumerate} \item $|Y| = |Z| \cup |b|(|Z'|)$, and \item the composition $X'' \to X$ is isomorphic to the blowing up of $X$ in $Y$. \end{enumerate}


spaces-divisors-lemma-strict-transform-local

In the situation of Definition \ref{definition-strict-transform}. Let $$ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B } $$ be a commutative diagram of morphisms with $U$ and $V$ schemes and \'etale horizontal arrows. Let $V' \to V$ be the blowup of $V$ in $Z \times_B V$. Then \begin{enumerate} \item $V' = V \times_B B'$ and the maps $V' \to B'$ and $U \times_V V' \to X \times_B B'$ are \'etale, \item the strict transform $U'$ of $U$ relative to $V' \to V$ is equal to $X' \times_X U$ where $X'$ is the strict transform of $X$ relative to $B' \to B$, and \item for a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the restriction of the strict transform $\mathcal{F}'$ to $U \times_V V'$ is the strict transform of $\mathcal{F}|_U$ relative to $V' \to V$. \end{enumerate}


spaces-divisors-lemma-strict-transform

In the situation of Definition \ref{definition-strict-transform}. \begin{enumerate} \item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed subspace $f^{-1}Z$ of $X$. \item For a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the strict transform $\mathcal{F}'$ is canonically isomorphic to the pushforward along $X' \to X \times_B B'$ of the strict transform of $\mathcal{F}$ relative to the blowing up $X' \to X$. \end{enumerate}


spaces-divisors-lemma-strict-transform-flat

In the situation of Definition \ref{definition-strict-transform}. \begin{enumerate} \item If $X$ is flat over $B$ at all points lying over $Z$, then the strict transform of $X$ is equal to the base change $X \times_B B'$. \item Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\mathcal{F}$ is flat over $B$ at all points lying over $Z$, then the strict transform $\mathcal{F}'$ of $\mathcal{F}$ is equal to the pullback $\text{pr}_X^*\mathcal{F}$. \end{enumerate}


spaces-divisors-lemma-strict-transform-affine

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up of $Z$ in $B$. Let $g : X \to Y$ be an affine morphism of spaces over $B$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $g' : X \times_B B' \to Y \times_B B'$ be the base change of $g$. Let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ relative to $b$. Then $g'_*\mathcal{F}'$ is the strict transform of $g_*\mathcal{F}$.


spaces-divisors-lemma-strict-transform-composition-blowups

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up with center $Z$. Let $Z' \subset B'$ be a closed subspace. Let $B'' \to B'$ be the blowing up with center $Z'$. Let $Y \subset B$ be a closed subscheme such that $|Y| = |Z| \cup |b|(|Z'|)$ and the composition $B'' \to B$ is isomorphic to the blowing up of $B$ in $Y$. In this situation, given any scheme $X$ over $B$ and $\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have \begin{enumerate} \item the strict transform of $\mathcal{F}$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $\mathcal{F}$ with respect to the blowup $B' \to B$ of $B$ in $Z$, and \item the strict transform of $X$ with respect to the blowing up of $B$ in $Y$ is equal to the strict transform with respect to the blowup $B'' \to B'$ in $Z'$ of the strict transform of $X$ with respect to the blowup $B' \to B$ of $B$ in $Z$. \end{enumerate}


spaces-divisors-lemma-composition-admissible-blowups

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open subspace. Let $b : X' \to X$ be a $U$-admissible blowup. Let $X'' \to X'$ be a $U$-admissible blowup. Then the composition $X'' \to X$ is a $U$-admissible blowup.


spaces-divisors-lemma-extend-admissible-blowups

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U, V \subset X$ be quasi-compact open subspaces. Let $b : V' \to V$ be a $U \cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \to X$ whose restriction to $V$ is $V'$.


spaces-divisors-lemma-separate-disjoint-opens-by-blowing-up

Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U, V$ be quasi-compact disjoint open subspaces of $X$. Then there exist a $U \cup V$-admissible blowup $b : X' \to X$ such that $X'$ is a disjoint union of open subspaces $X' = X'_1 \amalg X'_2$ with $b^{-1}(U) \subset X'_1$ and $b^{-1}(V) \subset X'_2$.


dga-lemma-dgm-abelian

Let $(A, d)$ be a differential graded algebra. The category $\text{Mod}_{(A, \text{d})}$ is abelian and has arbitrary limits and colimits.


dga-lemma-homotopy-direct-sums

Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ has direct sums and products.


dga-lemma-functorial-cone

Let $(A, \text{d})$ be a differential graded algebra. Suppose that $$ \xymatrix{ K_1 \ar[r]_{f_1} \ar[d]_a & L_1 \ar[d]^b \\ K_2 \ar[r]^{f_2} & L_2 } $$ is a diagram of homomorphisms of differential graded $A$-modules which is commutative up to homotopy. Then there exists a morphism $c : C(f_1) \to C(f_2)$ which gives rise to a morphism of triangles $$ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \to (K_1, L_1, C(f_1), f_2, i_2, p_2) $$ in $K(\text{Mod}_{(A, \text{d})})$.


dga-lemma-admissible-ses

Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K \to L \to M \to 0$ be an admissible short exact sequence of differential graded $A$-modules. Let $s : M \to L$ and $\pi : L \to K$ be splittings such that $\Ker(\pi) = \Im(s)$. Then we obtain a morphism $$ \delta = \pi \circ \text{d}_L \circ s : M \to K[1] $$ of $\text{Mod}_{(A, \text{d})}$ which induces the boundary maps in the long exact sequence of cohomology (\ref{equation-les}).


dga-lemma-make-commute-map

Let $(A, \text{d})$ be a differential graded algebra. Let $$ \xymatrix{ K \ar[r]_f \ar[d]_a & L \ar[d]^b \\ M \ar[r]^g & N } $$ be a diagram of homomorphisms of differential graded $A$-modules commuting up to homotopy. \begin{enumerate} \item If $f$ is an admissible monomorphism, then $b$ is homotopic to a homomorphism which makes the diagram commute. \item If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism which makes the diagram commute. \end{enumerate}


dga-lemma-make-injective

Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ be a homomorphism of differential graded $A$-modules. There exists a factorization $$ \xymatrix{ K \ar[r]^{\tilde \alpha} \ar@/_1pc/[rr]_\alpha & \tilde L \ar[r]^\pi & L } $$ in $\text{Mod}_{(A, \text{d})}$ such that \begin{enumerate} \item $\tilde \alpha$ is an admissible monomorphism (see Definition \ref{definition-admissible-ses}), \item there is a morphism $s : L \to \tilde L$ such that $\pi \circ s = \text{id}_L$ and such that $s \circ \pi$ is homotopic to $\text{id}_{\tilde L}$. \end{enumerate}


dga-lemma-sequence-maps-split

Let $(A, \text{d})$ be a differential graded algebra. Let $L_1 \to L_2 \to \ldots \to L_n$ be a sequence of composable homomorphisms of differential graded $A$-modules. There exists a commutative diagram $$ \xymatrix{ L_1 \ar[r] & L_2 \ar[r] & \ldots \ar[r] & L_n \\ M_1 \ar[r] \ar[u] & M_2 \ar[r] \ar[u] & \ldots \ar[r] & M_n \ar[u] } $$ in $\text{Mod}_{(A, \text{d})}$ such that each $M_i \to M_{i + 1}$ is an admissible monomorphism and each $M_i \to L_i$ is a homotopy equivalence.


dga-lemma-nilpotent

Let $(A, \text{d})$ be a differential graded algebra. Let $0 \to K_i \to L_i \to M_i \to 0$, $i = 1, 2, 3$ be admissible short exact sequence of differential graded $A$-modules. Let $b : L_1 \to L_2$ and $b' : L_2 \to L_3$ be homomorphisms of differential graded modules such that $$ \vcenter{ \xymatrix{ K_1 \ar[d]_0 \ar[r] & L_1 \ar[r] \ar[d]_b & M_1 \ar[d]_0 \\ K_2 \ar[r] & L_2 \ar[r] & M_2 } } \quad\text{and}\quad \vcenter{ \xymatrix{ K_2 \ar[d]^0 \ar[r] & L_2 \ar[r] \ar[d]^{b'} & M_2 \ar[d]^0 \\ K_3 \ar[r] & L_3 \ar[r] & M_3 } } $$ commute up to homotopy. Then $b' \circ b$ is homotopic to $0$.


dga-lemma-rotate-cone

Let $(A, \text{d})$ be a differential graded algebra. Let $f : K \to L$ be a homomorphism of differential graded modules. The triangle $(L, C(f), K[1], i, p, f[1])$ is the triangle associated to the admissible short exact sequence $$ 0 \to L \to C(f) \to K[1] \to 0 $$ coming from the definition of the cone of $f$.


dga-lemma-rotate-triangle

Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ and $\beta : L \to M$ define an admissible short exact sequence $$ 0 \to K \to L \to M \to 0 $$ of differential graded $A$-modules. Let $(K, L, M, \alpha, \beta, \delta)$ be the associated triangle. Then the triangles $$ (M[-1], K, L, \delta[-1], \alpha, \beta) \quad\text{and}\quad (M[-1], K, C(\delta[-1]), \delta[-1], i, p) $$ are isomorphic.


dga-lemma-third-isomorphism

Let $(A, \text{d})$ be a differential graded algebra. Let $f_1 : K_1 \to L_1$ and $f_2 : K_2 \to L_2$ be homomorphisms of differential graded $A$-modules. Let $$ (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \longrightarrow (K_1, L_1, C(f_1), f_2, i_2, p_2) $$ be any morphism of triangles of $K(\text{Mod}_{(A, \text{d})})$. If $a$ and $b$ are homotopy equivalences then so is $c$.


dga-lemma-the-same-up-to-isomorphisms

Let $(A, \text{d})$ be a differential graded algebra. \begin{enumerate} \item Given an admissible short exact sequence $0 \to K \xrightarrow{\alpha} L \to M \to 0$ of differential graded $A$-modules there exists a homotopy equivalence $C(\alpha) \to M$ such that the diagram $$ \xymatrix{ K \ar[r] \ar[d] & L \ar[d] \ar[r] & C(\alpha) \ar[r]_{-p} \ar[d] & K[1] \ar[d] \\ K \ar[r]^\alpha & L \ar[r]^\beta & M \ar[r]^\delta & K[1] } $$ defines an isomorphism of triangles in $K(\text{Mod}_{(A, \text{d})})$. \item Given a morphism of complexes $f : K \to L$ there exists an isomorphism of triangles $$ \xymatrix{ K \ar[r] \ar[d] & \tilde L \ar[d] \ar[r] & M \ar[r]_{\delta} \ar[d] & K[1] \ar[d] \\ K \ar[r] & L \ar[r] & C(f) \ar[r]^{-p} & K[1] } $$ where the upper triangle is the triangle associated to a admissible short exact sequence $K \to \tilde L \to M$. \end{enumerate}


dga-lemma-two-split-injections

Let $(A, \text{d})$ be a differential graded algebra. Suppose that $\alpha : K \to L$ and $\beta : L \to M$ are admissible monomorphisms of differential graded $A$-modules. Then there exist distinguished triangles $(K, L, Q_1, \alpha, p_1, d_1)$, $(K, M, Q_2, \beta \circ \alpha, p_2, d_2)$ and $(L, M, Q_3, \beta, p_3, d_3)$ for which TR4 holds.


dga-lemma-left-right

Let $(A, \text{d})$ be a differential graded $R$-algebra. The functor $M \mapsto M^{opp}$ from the category of left differential graded $A$-modules to the category of right differential graded $A^{opp}$-modules is an equivalence.


dga-lemma-left-module-structure

In the situation above, let $A$ be a differential graded $R$-algebra. To give a left $A$-module structure on $M$ is the same thing as giving a homomorphism $A \to E$ of differential graded $R$-algebras.


dga-lemma-characterize-hom

Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded $R$-algebra. Let $M'$ be a right differential graded $A$-module and let $M$ be a left differential graded $A$-module. Let $N^\bullet$ be a complex of $R$-modules. Then we have $$ \Hom_{\text{Mod}_{(A, d)}}(M', \Hom(M, N^\bullet)) = \Hom_{\text{Comp}(R)}(M' \otimes_A M, N^\bullet) $$ where $M \otimes_A M$ is viewed as a complex of $R$-modules as in Section \ref{section-tensor-product}.


dga-lemma-right-module-structure

In the situation above, let $A$ be a differential graded $R$-algebra. To give a right $A$-module structure on $M$ is the same thing as giving a homomorphism $\tau : A \to E^{opp}$ of differential graded $R$-algebras.


dga-lemma-target-graded-projective

Let $(A, \text{d})$ be a differential graded algebra. Let $M \to P$ be a surjective homomorphism of differential graded $A$-modules. If $P$ is projective as a graded $A$-module, then $M \to P$ is an admissible epimorphism.


dga-lemma-hom-from-shift-free

Let $(A, d)$ be a differential graded algebra. Then we have $$ \Hom_{\text{Mod}_{(A, \text{d})}}(A[k], M) = \Ker(\text{d} : M^{-k} \to M^{-k + 1}) $$ and $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(A[k], M) = H^{-k}(M) $$ for any differential graded $A$-module $M$.


dga-lemma-source-graded-injective

Let $(A, \text{d})$ be a differential graded algebra. Let $I \to M$ be an injective homomorphism of differential graded $A$-modules. If $I$ is graded injective, then $I \to M$ is an admissible monomorphism.


dga-lemma-hom-into-shift-dual-free

Let $(A, \text{d})$ be a differential graded algebra. Then we have $$ \Hom_{\text{Mod}_{(A, \text{d})}}(M, A^\vee[k]) = \Ker(\text{d} : (M^\vee)^k \to (M^\vee)^{k + 1}) $$ and $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(M, A^\vee[k]) = H^k(M^\vee) $$ as functors in the differential graded $A$-module $M$.


dga-lemma-property-P-sequence

Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module. If $F_\bullet$ is a filtration as in property (P), then we obtain an admissible short exact sequence $$ 0 \to \bigoplus\nolimits F_iP \to \bigoplus\nolimits F_iP \to P \to 0 $$ of differential graded $A$-modules.


dga-lemma-property-P-K-projective

Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(P, N) = 0 $$ for all acyclic differential graded $A$-modules $N$.


dga-lemma-good-quotient

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $P \to M$ of differential graded $A$-modules with the following properties \begin{enumerate} \item $P \to M$ is surjective, \item $\Ker(\text{d}_P) \to \Ker(\text{d}_M)$ is surjective, and \item $P$ sits in an admissible short exact sequence $0 \to P' \to P \to P'' \to 0$ where $P'$, $P''$ are direct sums of shifts of $A$. \end{enumerate}


dga-lemma-resolve

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $P \to M$ of differential graded $A$-modules such that \begin{enumerate} \item $P \to M$ is a quasi-isomorphism, and \item $P$ has property (P). \end{enumerate}


dga-lemma-property-I-sequence

Let $(A, \text{d})$ be a differential graded algebra. Let $I$ be a differential graded $A$-module. If $F_\bullet$ is a filtration as in property (I), then we obtain an admissible short exact sequence $$ 0 \to I \to \prod\nolimits I/F_iI \to \prod\nolimits I/F_iI \to 0 $$ of differential graded $A$-modules.


dga-lemma-property-I-K-injective

Let $(A, \text{d})$ be a differential graded algebra. Let $I$ be a differential graded $A$-module with property (I). Then $$ \Hom_{K(\text{Mod}_{(A, \text{d})})}(N, I) = 0 $$ for all acyclic differential graded $A$-modules $N$.


dga-lemma-good-sub

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $M \to I$ of differential graded $A$-modules with the following properties \begin{enumerate} \item $M \to I$ is injective, \item $\Coker(\text{d}_M) \to \Coker(\text{d}_I)$ is injective, and \item $I$ sits in an admissible short exact sequence $0 \to I' \to I \to I'' \to 0$ where $I'$, $I''$ are products of shifts of $A^\vee$. \end{enumerate}


dga-lemma-right-resolution

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $M \to I$ of differential graded $A$-modules such that \begin{enumerate} \item $M \to I$ is a quasi-isomorphism, and \item $I$ has property (I). \end{enumerate}


dga-lemma-acyclic

Let $(A, \text{d})$ be a differential graded algebra. The full subcategory $\text{Ac}$ of $K(\text{Mod}_{(A, \text{d})})$ consisting of acyclic modules is a strictly full saturated triangulated subcategory of $K(\text{Mod}_{(A, \text{d})})$. The corresponding saturated multiplicative system (see Derived Categories, Lemma \ref{derived-lemma-operations}) of $K(\text{Mod}_{(A, \text{d})})$ is the class $\text{Qis}$ of quasi-isomorphisms. In particular, the kernel of the localization functor $$ Q : K(\text{Mod}_{(A, \text{d})}) \to \text{Qis}^{-1}K(\text{Mod}_{(A, \text{d})}) $$ is $\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$.


dga-lemma-hom-derived

Let $(A, \text{d})$ be a differential graded algebra. Let $M$ and $N$ be differential graded $A$-modules. \begin{enumerate} \item Let $P \to M$ be a P-resolution as in Lemma \ref{lemma-resolve}. Then $$ \Hom_{D(A, \text{d})}(M, N) = \Hom_{K(\text{Mod}_{(A, \text{d})})}(P, N) $$ \item Let $N \to I$ be an I-resolution as in Lemma \ref{lemma-right-resolution}. Then $$ \Hom_{D(A, \text{d})}(M, N) = \Hom_{K(\text{Mod}_{(A, \text{d})})}(M, I) $$ \end{enumerate}


dga-lemma-derived-products

Let $(A, \text{d})$ be a differential graded algebra. Then \begin{enumerate} \item $D(A, \text{d})$ has both direct sums and products, \item direct sums are obtained by taking direct sums of differential graded modules, \item products are obtained by taking products of differential graded modules. \end{enumerate}


dga-lemma-derived-canonical-delta-functor

Let $(A, \text{d})$ be a differential graded algebra. The functor $\text{Mod}_{(A, \text{d})} \to D(A, \text{d})$ defined has the natural structure of a $\delta$-functor, with $$ \delta_{K \to L \to M} = - p \circ q^{-1} $$ with $p$ and $q$ as explained above.


dga-lemma-homotopy-colimit

Let $(A, \text{d})$ be a differential graded algebra. Let $M_n$ be a system of differential graded modules. Then the derived colimit $\text{hocolim} M_n$ in $D(A, \text{d})$ is represented by the differential graded module $\colim M_n$.


dga-lemma-functorial

Let $R$ be a ring. A functor $F : \mathcal{A} \to \mathcal{B}$ of differential graded categories over $R$ induces functors $\text{Comp}(\mathcal{A}) \to \text{Comp}(\mathcal{B})$ and $K(\mathcal{A}) \to K(\mathcal{B})$.


dga-lemma-additive-functor-induces-dga-functor

Let $F : \mathcal{B} \to \mathcal{B}'$ be an additive functor between additive categories. Then $F$ induces a functor of differential graded categories $$ F : \text{Comp}^{dg}(\mathcal{B}) \to \text{Comp}^{dg}(\mathcal{B}') $$ of Example \ref{example-category-complexes} inducing the usual functors on the category of complexes and the homotopy categories.


dga-lemma-homomorphism-induces-dga-functor

Let $\varphi : (A, \text{d}) \to (E, \text{d})$ be a homomorphism of differential graded algebras. Then $\varphi$ induces a functor of differential graded categories $$ F : \text{Mod}^{dg}_{(E, \text{d})} \longrightarrow \text{Mod}^{dg}_{(A, \text{d})} $$ of Example \ref{example-dgm-dg-cat} inducing obvious restriction functors on the categories of differential graded modules and homotopy categories.


dga-lemma-construction

Let $R$ be a ring. Let $\mathcal{A}$ be a differential graded category over $R$. Let $x$ be an object of $\mathcal{A}$. Let $$ (E, \text{d}) = \Hom_\mathcal{A}(x, x) $$ be the differential graded $R$-algebra of endomorphisms of $x$. We obtain a functor $$ \mathcal{A} \longrightarrow \text{Mod}^{dg}_{(E, \text{d})},\quad y \longmapsto \Hom_\mathcal{A}(x, y) $$ of differential graded categories by letting $E$ act on $\Hom_\mathcal{A}(x, y)$ via composition in $\mathcal{A}$. This functor induces functors $$ \text{Comp}(\mathcal{A}) \to \text{Mod}_{(A, \text{d})} \quad\text{and}\quad K(\mathcal{A}) \to K(\text{Mod}_{(A, \text{d})}) $$ by an application of Lemma \ref{lemma-functorial}.


dga-lemma-get-triangle

Let $\mathcal{A}$ be a differential graded category satisfying axioms (A) and (B). Given an admissible short exact sequence $x \to y \to z$ we obtain (see proof) a triangle $$ x \to y \to z \to x[1] $$ in $\text{Comp}(\mathcal{A})$ with the property that any two compositions in $z[-1] \to x \to y \to z \to x[1]$ are zero in $K(\mathcal{A})$.


dga-lemma-cone

\begin{slogan} The homotopy category is a triangulated category. This lemma proves a part of the axioms of a triangulated category. \end{slogan} In Situation \ref{situation-ABC} suppose that $$ \xymatrix{ x_1 \ar[r]_{f_1} \ar[d]_a & y_1 \ar[d]^b \\ x_2 \ar[r]^{f_2} & y_2 } $$ is a diagram of $\text{Comp}(\mathcal{A})$ commutative up to homotopy. Then there exists a morphism $c : c(f_1) \to c(f_2)$ which gives rise to a morphism of triangles $$ (a, b, c) : (x_1, y_1, c(f_1)) \to (x_1, y_1, c(f_1)) $$ in $K(\mathcal{A})$.


dga-lemma-id-cone-null

In Situation \ref{situation-ABC} given any object $x$ of $\mathcal{A}$, and the cone $C(1_x)$ of the identity morphism $1_x : x \to x$, the identity morphism on $C(1_x)$ is homotopic to zero.


dga-lemma-homo-change

In Situation \ref{situation-ABC} given a diagram $$ \xymatrix{x\ar[r]^f\ar[d]_a & y\ar[d]^b\\ z\ar[r]^g & w} $$ in $\text{Comp}(\mathcal{A})$ commuting up to homotopy. Then \begin{enumerate} \item If $f$ is an admissible monomorphism, then $b$ is homotopic to a morphism $b'$ which makes the diagram commute. \item If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism $a'$ which makes the diagram commute. \end{enumerate}


dga-lemma-factor

In Situation \ref{situation-ABC} let $\alpha : x \to y$ be a morphism in $\text{Comp}(\mathcal{A})$. Then there exists a factorization in $\text{Comp}(\mathcal{A})$: $$ \xymatrix{ x \ar[r]^{\tilde{\alpha}}  & \tilde{y} \ar@<0.5ex>[r]^{\pi} & y\ar@<0.5ex>[l]^s } $$ such that \begin{enumerate} \item $\tilde{\alpha}$ is an admissible monomorphism, and $\pi\tilde{\alpha}=\alpha$. \item There exists a morphism $s:y\to\tilde{y}$ in $\text{Comp}(\mathcal{A})$ such that $\pi s=1_y$ and $s\pi$ is homotopic to $1_{\tilde{y}}$.  \end{enumerate}


dga-lemma-analogue-sequence-maps-split

In Situation \ref{situation-ABC} let $x_1 \to x_2 \to \ldots \to x_n$ be a sequence of composable morphisms in $\text{Comp}(\mathcal{A})$. Then there exists a commutative diagram in $\text{Comp}(\mathcal{A})$: $$ \xymatrix{x_1\ar[r] & x_2\ar[r] & \ldots\ar[r] & x_n\\ y_1\ar[r]\ar[u] & y_2\ar[r]\ar[u] & \ldots\ar[r] & y_n\ar[u]} $$ such that each $y_i\to y_{i+1}$ is an admissible monomorphism and each $y_i\to x_i$ is a homotopy equivalence.


dga-lemma-triseq

In Situation \ref{situation-ABC} let $x_i \to y_i \to z_i$ be morphisms in $\mathcal{A}$ ($i=1,2,3$) such that $x_2 \to y_2\to z_2$ is an admissible short exact sequence. Let $b : y_1 \to y_2$ and $b' : y_2\to y_3$ be morphisms in $\text{Comp}(\mathcal{A})$ such that $$ \vcenter{ \xymatrix{ x_1 \ar[d]_0 \ar[r] & y_1 \ar[r] \ar[d]_b & z_1 \ar[d]_0 \\ x_2 \ar[r] & y_2 \ar[r] & z_2 } } \quad\text{and}\quad \vcenter{ \xymatrix{ x_2 \ar[d]^0 \ar[r] & y_2 \ar[r] \ar[d]^{b'} & z_2 \ar[d]^0 \\ x_3 \ar[r] & y_3 \ar[r] & z_3 } } $$ commute up to homotopy. Then $b'\circ b$ is homotopic to $0$.


dga-lemma-restate-axiom-c

In Situation \ref{situation-ABC} let $f: x \to y$ be a morphism in $\text{Comp}(\mathcal{A})$. The triangle $(y, c(f), x[1], i, p, f[1])$ is the triangle associated to the admissible short exact sequence  $$ \xymatrix{y\ar[r] & c(f) \ar[r] & x[1]} $$ where the cone $c(f)$ is defined as in Lemma \ref{lemma-get-triangle}.


dga-lemma-cone-rotate-isom

In Situation \ref{situation-ABC} let $\alpha : x \to y$ and $\beta : y \to z$ define an admissible short exact sequence $$ \xymatrix{ x \ar[r] & y\ar[r] & z } $$ in $\text{Comp}(\mathcal{A})$. Let $(x, y, z, \alpha, \beta, \delta)$ be the associated triangle in $K(\mathcal{A})$. Then, the triangles $$ (z[-1], x, y, \delta[-1], \alpha, \beta) \quad\text{and}\quad (z[-1], x, c(\delta[-1]), \delta[-1], i, p) $$ are isomorphic.


dga-lemma-analogue-third-isomorphism

In Situation \ref{situation-ABC} let $f_1 : x_1 \to y_1$ and $f_2 : x_2 \to y_2$ be morphisms in $\text{Comp}(\mathcal{A})$. Let  $$ (a,b,c): (x_1,y_1,c(f_1), f_1, i_1, p_1) \to (x_2,y_2, c(f_2), f_2, i_1, p_1) $$ be any morphism of triangles in $K(\mathcal{A})$. If $a$ and $b$ are homotopy equivalences, then so is $c$.


dga-lemma-cone-homotopy

In Situation \ref{situation-ABC}. \begin{enumerate} \item Given an admissible short exact sequence $x\xrightarrow{\alpha} y\xrightarrow{\beta} z$. Then there exists a homotopy equivalence $e:C(\alpha)\to z$ such that the diagram \begin{equation} \label{equation-cone-isom-triangle} \vcenter{ \xymatrix{ x\ar[r]^{\alpha}\ar[d] & y\ar[r]^{b}\ar[d] & C(\alpha)\ar[r]^{-c}\ar@{.>}[d]^{e} & x[1]\ar[d] \\ x\ar[r]^{\alpha} & y\ar[r]^{\beta} & z\ar[r]^{\delta} & x[1] } } \end{equation} defines an isomorphism of triangles in $K(\mathcal{A})$. Here $y\xrightarrow{b}C(\alpha)\xrightarrow{c}x[1]$ is the admissible short exact sequence given as in axiom (C). \item Given a morphism $\alpha : x \to y$ in $\text{Comp}(\mathcal{A})$, let $x \xrightarrow{\tilde{\alpha}} \tilde{y} \to y$ be the factorization given as in Lemma \ref{lemma-factor}, where the admissible monomorphism $x \xrightarrow{\tilde{\alpha}} y$ extends to the admissible short exact sequence $$ \xymatrix{ x \ar[r]^{\tilde{\alpha}} & \tilde{y} \ar[r] & z } $$ Then there exists an isomorphism of triangles $$ \xymatrix{ x \ar[r]^{\tilde{\alpha}} \ar[d] & \tilde{y} \ar[r] \ar[d] & z \ar[r]^{\delta} \ar@{.>}[d]^{e} & x[1] \ar[d] \\ x \ar[r]^{\alpha} & y \ar[r] & C(\alpha) \ar[r]^{-c} & x[1] } $$ where the upper triangle is the triangle associated to the sequence $x \xrightarrow{\tilde{\alpha}} \tilde{y} \to z$. \end{enumerate}


dga-lemma-analogue-homotopy-category-pre-triangulated

In Situation \ref{situation-ABC} the homotopy category $K(\mathcal{A})$ with its natural translation functors and distinguished triangles is a pre-triangulated category.


dga-lemma-dgc-analogue-tr4

In Situation \ref{situation-ABC} given admissible monomorphisms $x \xrightarrow{\alpha} y$, $y \xrightarrow{\beta} z$ in $\mathcal{A}$, there exist distinguished triangles $(x,y,q_1,\alpha,p_1,\delta_1)$, $(x,z,q_2,\beta\alpha,p_2,\delta_2)$ and $(y,z,q_3,\beta,p_3,\delta_3)$ for which TR4 holds.


dga-lemma-functor-between-ABC

Let $R$ be a ring. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor between differential graded categories over $R$ satisfying axioms (A), (B), and (C) such that $F(x[1]) = F(x)[1]$. Then $F$ induces an exact functor $K(\mathcal{A}) \to K(\mathcal{B})$ of triangulated categories.


dga-lemma-bimodule-over-tensor

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. The construction above defines an equivalence of categories $$ \begin{matrix} \text{differential graded}\\ (A, B)\text{-bimodules} \end{matrix} \longleftrightarrow \begin{matrix} \text{right differential graded }\\ A^{opp} \otimes_R B\text{-modules} \end{matrix} $$


dga-lemma-bimodule-resolve

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $M$ be a differential graded $(A, B)$-bimodule. There exists a homomorphism $P \to M$ of differential graded $(A, B)$-bimodules which is a quasi-isomorphism such that $P$ has property (P) as defined above.


dga-lemma-bimodule-property-P-sequence

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $P$ be a differential graded $(A, B)$-bimodule having property (P) with corresponding filtration $F_\bullet$, then we obtain a short exact sequence $$ 0 \to \bigoplus\nolimits F_iP \to \bigoplus\nolimits F_iP \to P \to 0 $$ of differential graded $(A, B)$-bimodules which is split as a sequence of graded $(A, B)$-bimodules.


dga-lemma-tensor

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then $M \mapsto M \otimes_A N$ defines a functor $$ - \otimes_A N : \text{Mod}^{dg}_{(A, \text{d})} \longrightarrow \text{Mod}^{dg}_{(B, \text{d})} $$ of differential graded categories. This functor induces functors $$ \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(B, \text{d})} \quad\text{and}\quad K(\text{Mod}_{(A, \text{d})}) \to K(\text{Mod}_{(B, \text{d})}) $$ by an application of Lemma \ref{lemma-functorial}.


dga-lemma-hom

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. The construction above defines a functor $$ \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(N, -) : \text{Mod}^{dg}_{(B, \text{d})} \longrightarrow \text{Mod}^{dg}_{(A, \text{d})} $$ of differential graded categories. This functor induces functors $$ \text{Mod}_{(B, \text{d})} \to \text{Mod}_{(A, \text{d})} \quad\text{and}\quad K(\text{Mod}_{(B, \text{d})}) \to K(\text{Mod}_{(A, \text{d})}) $$ by an application of Lemma \ref{lemma-functorial}.


dga-lemma-tensor-hom-adjunction

Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. Let $M$ be a right $A$-module, $N$ an $(A, B)$-bimodule, and $N'$ a right $B$-module. Then we have a canonical isomorphism $$ \Hom_B(M \otimes_A N, N') = \Hom_A(M, \Hom_B(N, N')) $$ of $R$-modules. If $A$, $B$, $M$, $N$, $N'$ are compatibly graded, then we have a canonical isomorphism $$ \Hom_{\text{Mod}_B^{gr}}(M \otimes_A N, N') = \Hom_{\text{Mod}_A^{gr}}(M, \Hom_{\text{Mod}_B^{gr}}(N, N')) $$ of graded $R$-modules If $A$, $B$, $M$, $N$, $N'$ are compatibly differential graded, then we have a canonical isomorphism $$ \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(M \otimes_A N, N') = \Hom_{\text{Mod}^{dg}_{(A, \text{d})}}(M, \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(N, N')) $$ of complexes of $R$-modules.


dga-lemma-derived-restriction

In the situation above, the right derived functor of $F$ exists. We denote it $R\Hom(N, -) : D(B, \text{d}) \to D(A, \text{d})$.


dga-lemma-functoriality-derived-restriction

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $f : N \to N'$ be a homomorphism of differential graded $(A, B)$-bimodules. Then $f$ induces a morphism of functors $$ - \circ f : R\Hom(N', -) \longrightarrow R\Hom(N, -) $$ If $f$ is a quasi-isomorphism, then $f \circ -$ is an isomorphism of functors.


dga-lemma-derived-restriction-exts

Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over a ring $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then for every $n \in \mathbf{Z}$ there are isomorphisms $$ H^n(R\Hom(N, M)) = \Ext^n_{D(B, \text{d})}(N, M) $$ of $R$-modules functorial in $M$. It is also functorial in $N$ with respect to the operation described in Lemma \ref{lemma-functoriality-derived-restriction}.


dga-lemma-compute-derived-restriction

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. If $\Hom_{D(B, \text{d})}(N, N') = \Hom_{K(\text{Mod}_{(B, \text{d})})}(N, N')$ for all $N' \in K(B, \text{d})$, for example if $N$ has property (P) as a differential graded $B$-module, then $$ R\Hom(N, M) = \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(N, M) $$ functorially in $M$ in $D(B, \text{d})$.


dga-lemma-existence-of-derived

In the situation above. If the right derived functor $R\Hom(K^\bullet, -)$ of $\Hom(K^\bullet, -) : K(\mathcal{A}) \to D(\textit{Ab})$ is everywhere defined on $D(\mathcal{A})$, then we obtain a canonical exact functor $$ R\Hom(K^\bullet, -) : D(\mathcal{A}) \longrightarrow D(E, \text{d}) $$ of triangulated categories which reduces to the usual one  on taking associated complexes of abelian groups.


dga-lemma-derived-bc

In the situation above, the left derived functor of $F$ exists. We denote it $- \otimes_A^\mathbf{L} N : D(A, \text{d}) \to D(B, \text{d})$.


dga-lemma-functoriality-bc

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $f : N \to N'$ be a homomorphism of differential graded $(A, B)$-bimodules. Then $f$ induces a morphism of functors $$ 1\otimes f : - \otimes_A^\mathbf{L} N \longrightarrow - \otimes_A^\mathbf{L} N' $$ If $f$ is a quasi-isomorphism, then $1 \otimes f$ is an isomorphism of functors.


dga-lemma-tensor-hom-adjoint

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. Then the functor $$ - \otimes_A^\mathbf{L} N : D(A, \text{d}) \longrightarrow D(B, \text{d}) $$ of Lemma \ref{lemma-derived-bc} is a left adjoint to the functor $$ R\Hom(N, -) : D(B, \text{d}) \longrightarrow D(A, \text{d}) $$ of Lemma \ref{lemma-derived-restriction}.


dga-lemma-tensor-with-compact-fully-faithful

With notation and assumptions as in Lemma \ref{lemma-tensor-hom-adjoint}. Assume \begin{enumerate} \item $N$ defines a compact object of $D(B, \text{d})$, and \item the map $H^k(A) \to \Hom_{D(B, \text{d})}(N, N[k])$ is an isomorphism for all $k \in \mathbf{Z}$. \end{enumerate} Then the functor $-\otimes_A^\mathbf{L} N$ is fully faithful.


dga-lemma-RHom-is-tensor

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $T$ be a differential graded $(A, B)$-bimodule. Assume \begin{enumerate} \item $T$ defines a compact object of $D(B, \text{d})$, and \item $S = \Hom_{\text{Mod}^{dg}_{(B, \text{d})}}(T, B)$ represents $R\Hom(T, B)$ in $D(A, \text{d})$. \end{enumerate} Then $S$ has a structure of a differential graded $(B, A)$-bimodule and there is an isomorphism $$ N \otimes_B^\mathbf{L} S \longrightarrow R\Hom(T, N) $$ functorial in $N$ in $D(B, \text{d})$.


dga-lemma-compose-tensor-functors-general

Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-module. Assume (\ref{equation-plain-versus-derived}) is an isomorphism. Then the composition $$ \xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes_A^\mathbf{L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes_B^\mathbf{L} N'} & & D(C, \text{d}) } $$ is isomorphic to $- \otimes_A^\mathbf{L} N''$ with $N'' = N \otimes_B N'$ viewed as $(A, C)$-bimodule.


dga-lemma-compose-tensor-functors-general-algebra

Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Assume that (\ref{equation-plain-versus-derived-algebras}) is an isomorphism. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-bimodule. Then the composition $$ \xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes_A^\mathbf{L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes_B^\mathbf{L} N'} & & D(C, \text{d}) } $$ is isomorphic to $- \otimes_A^\mathbf{L} N''$ for a differential graded $(A, C)$-bimodule $N''$ described in the proof.


dga-lemma-compose-tensor-functors

Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. If $C$ is K-flat as a complex of $R$-modules, then (\ref{equation-plain-versus-derived-algebras}) is an isomorphism and the conclusion of Lemma \ref{lemma-compose-tensor-functors-general-algebra} is valid.


dga-lemma-tensor-with-complex

In the situation above there is a functor $$ - \otimes_E K^\bullet : \text{Mod}^{dg}_{(E, \text{d})} \longrightarrow \text{Comp}^{dg}(\mathcal{O}) $$ of differential graded categories. This functor sends $E$ to $K^\bullet$ and commutes with direct sums.


dga-lemma-tensor-with-complex-homotopy

The functor of Lemma \ref{lemma-tensor-with-complex} defines an exact functor of triangulated categories $K(\text{Mod}_{(E \text{d})}) \to K(\mathcal{O})$.


dga-lemma-tensor-with-complex-derived

The functor $K(\text{Mod}_{(E, \text{d})}) \to K(\mathcal{O})$ of Lemma \ref{lemma-tensor-with-complex-homotopy} has a left derived version defined on all of $D(E, \text{d})$. We denote it $- \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \to D(\mathcal{O})$.


dga-lemma-upgrade-tensor-with-complex-derived

Let $R$ be a ring. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of commutative $R$-algebras. Let $K^\bullet$ be a complex of $\mathcal{O}$-modules. The functor of Lemma \ref{lemma-tensor-with-complex-derived} has the following property: For every $M$, $N$ in $D(E, \text{d})$ there is a canonical map $$ R\Hom(M, N) \longrightarrow R\Hom_\mathcal{O}(M \otimes_E^\mathbf{L} K^\bullet, N \otimes_E^\mathbf{L} K^\bullet) $$ in $D(R)$ which on cohomology modules gives the maps $$ \Ext^n_{D(E, \text{d})}(M, N) \to \Ext^n_{D(\mathcal{O})} (M \otimes_E^\mathbf{L} K^\bullet, N \otimes_E^\mathbf{L} K^\bullet) $$ induced by the functor $- \otimes_E^\mathbf{L} K^\bullet$.


dga-lemma-tensor-with-complex-hom-adjoint

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K^\bullet$ be a complex of $\mathcal{O}$-modules. Then the functor $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}) $$ of Lemma \ref{lemma-tensor-with-complex-derived} is a left adjoint of the functor $$ R\Hom(K^\bullet, -) : D(\mathcal{O}) \longrightarrow D(E, \text{d}) $$ of Lemma \ref{lemma-existence-of-derived}.


dga-lemma-fully-faithful-in-compact-case

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K^\bullet$ be a complex of $\mathcal{O}$-modules. Assume \begin{enumerate} \item $K^\bullet$ represents a compact object of $D(\mathcal{O})$, and \item $E = \Hom_{\text{Comp}^{dg}(\mathcal{O})}(K^\bullet, K^\bullet)$ computes the ext groups of $K^\bullet$ in $D(\mathcal{O})$. \end{enumerate} Then the functor $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}) $$ of Lemma \ref{lemma-tensor-with-complex-derived} is fully faithful.


dga-lemma-factor-through-nicer

Let $(A, \text{d})$ be a differential graded algebra. Let $E$ be a compact object of $D(A, \text{d})$. Let $P$ be a differential graded $A$-module which has a finite filtration $$ 0 = F_{-1}P \subset F_0P \subset F_1P \subset \ldots \subset F_nP = P $$ by differential graded submodules such that $$ F_{i + 1}P/F_iP \cong \bigoplus\nolimits_{j \in J_i} A[k_{i, j}] $$ as differential graded $A$-modules for some sets $J_i$ and integers $k_{i, j}$. Let $E \to P$ be a morphism of $D(A, \text{d})$. Then there exists a differential graded submodule $P' \subset P$ such that $F_{i + 1}P \cap P'/(F_iP \cap P')$ is equal to $\bigoplus_{j \in J'_i} A[k_{i, j}]$ for some finite subsets $J'_i \subset J_i$ and such that $E \to P$ factors through $P'$.


dga-lemma-compact-implies-bounded

Let $(A, \text{d})$ be a differential graded algebra. For every compact object $E$ of $D(A, \text{d})$ there exist integers $a \leq b$ such that $\Hom_{D(A, \text{d})}(E, M) = 0$ if $H^i(M) = 0$ for $i \in [a, b]$.


dga-lemma-compact

Let $(A, \text{d})$ be a differential graded algebra. Assume that $A^n = 0$ for $|n| \gg 0$. Let $E$ be an object of $D(A, \text{d})$. The following are equivalent \begin{enumerate} \item $E$ is a compact object, and \item $E$ can be represented by a differential graded $A$-module $P$ which is finite projective as a graded $A$-module and satisfies $\Hom_{K(A, \text{d})}(P, M) = \Hom_{D(A, \text{d})}(P, M)$ for every differential graded $A$-module $M$. \end{enumerate}


dga-lemma-qis-equivalence

Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded algebras over $R$, which induces an isomorphism on cohomology algebras. Then $$ - \otimes_A^\mathbf{L} B : D(A, \text{d}) \to D(B, \text{d}) $$ gives an $R$-linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \mapsto N_A$.


dga-lemma-tilting-equivalence

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be  differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Assume that \begin{enumerate} \item $N$ defines a compact object of $D(B, \text{d})$, \item if $N' \in D(B, \text{d})$ and $\Hom_{D(B, \text{d})}(N, N'[n]) = 0$ for $n \in \mathbf{Z}$, then $N' = 0$, and \item the map $H^k(A) \to \Hom_{D(B, \text{d})}(N, N[k])$ is an isomorphism for all $k \in \mathbf{Z}$. \end{enumerate} Then $$ - \otimes_A^\mathbf{L} N : D(A, \text{d}) \to D(B, \text{d}) $$ gives an $R$-linear equivalence of triangulated categories.


dga-lemma-rickard

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Assume that $A = H^0(A)$. The following are equivalent \begin{enumerate} \item $D(A, \text{d})$ and $D(B, \text{d})$ are equivalent as $R$-linear triangulated categories, and \item there exists an object $P$ of $D(B, \text{d})$ such that \begin{enumerate} \item $P$ is a compact object of $D(B, \text{d})$, \item if $N \in D(B, \text{d})$ with $\Hom_{D(B, \text{d})}(P, N[i]) = 0$ for $i \in \mathbf{Z}$, then $N = 0$, \item $\Hom_{D(B, \text{d})}(P, P[i]) = 0$ for $i \not = 0$ and equal to $A$ for $i = 0$. \end{enumerate} \end{enumerate} The equivalence $D(A, \text{d}) \to D(B, \text{d})$ constructed in (2) sends $A$ to $P$.


dga-lemma-K-flat-resolution

Let $R$ be a ring. Let $(B, \text{d})$ be a differential graded $R$-algebra. There exists a quasi-isomorphism $(A, \text{d}) \to (B, \text{d})$ of differential graded $R$-algebras with the following properties \begin{enumerate} \item $A$ is K-flat as a complex of $R$-modules, \item $A$ is a free graded $R$-algebra. \end{enumerate}


dga-lemma-countable

Let $(A, \text{d})$ be a differential graded algebra with $H^i(A)$ countable for each $i$. Let $M$ be an object of $D(A, \text{d})$. Then the following are equivalent \begin{enumerate} \item $M = \text{hocolim} E_n$ with $E_n$ compact in $D(A, \text{d})$, and \item $H^i(M)$ is countable for each $i$. \end{enumerate}


dga-proposition-homotopy-category-triangulated

Let $(A, \text{d})$ be a differential graded algebra. The homotopy category $K(\text{Mod}_{(A, \text{d})})$ of differential graded $A$-modules with its natural translation functors and distinguished triangles is a triangulated category.


dga-proposition-ABC-homotopy-category-triangulated

In Situation \ref{situation-ABC} the homotopy category $K(\mathcal{A})$ with its natural translation functors and distinguished triangles is a triangulated category.


dga-proposition-compact

Let $(A, \text{d})$ be a differential graded algebra. Let $E$ be an object of $D(A, \text{d})$. Then the following are equivalent \begin{enumerate} \item $E$ is a compact object, \item $E$ is a direct summand of an object of $D(A, \text{d})$ which is represented by a differential graded module $P$ which has a finite filtration $F_\bullet$ by differential graded submodules such that $F_iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. \end{enumerate}


dga-proposition-rickard

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $F : D(A, \text{d}) \to D(B, \text{d})$ be an $R$-linear equivalence of triangulated categories. Assume that \begin{enumerate} \item $A = H^0(A)$, and \item $B$ is K-flat as a complex of $R$-modules. \end{enumerate} Then there exists an $(A, B)$-bimodule $N$ as in Lemma \ref{lemma-tilting-equivalence}.


spaces-more-groupoids-lemma-first-order-structure-c

The map $I/I^2 \to J/J^2$ induced by $c$ is the composition $$ I/I^2 \xrightarrow{(1, 1)} I/I^2 \oplus I/I^2 \to J/J^2 $$ where the second arrow comes from the equality $J = (I \otimes B + B \otimes I)C$. The map $i : B \to B$ induces the map $-1 : I/I^2 \to I/I^2$.


spaces-more-groupoids-lemma-idenity-on-conormal

In the situation discussed in this section, let $\delta \in \Gamma_0$ and $f = t \circ \delta : U \to U$. If $s, t$ are flat, then the canonical map $\mathcal{C}_{U_0/U} \to \mathcal{C}_{U_0/U}$ induced by $f$ (More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-conormal-functorial}) is the identity map.


spaces-more-groupoids-lemma-composition-is-addition

The bijection (\ref{equation-isomorphism}) is an isomorphism of groups.


spaces-more-groupoids-lemma-property-G-invariant

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be its stabilizer group algebraic space. Let $\tau \in \{fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\tau$-local on the target. Assume $\{s : R \to U\}$ and $\{t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subspace such that $G_W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids in Spaces, Definition \ref{spaces-groupoids-definition-invariant-open}).


spaces-more-groupoids-lemma-two-fibres

Let $B \to S$ be as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $K$ be a field and let $r, r' : \Spec(K) \to R$ be morphisms such that $t \circ r = t \circ r' : \Spec(K) \to U$. Set $u = s \circ r$, $u' = s \circ r'$ and denote $F_u = \Spec(K) \times_{u, U, s} R$ and $F_{u'} = \Spec(K) \times_{u', U, s} R$ the fibre products. Then $F_u \cong F_{u'}$ as algebraic spaces over $K$.


spaces-more-groupoids-lemma-restrict-preserves-type

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $g : U' \to U$ be a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. \begin{enumerate} \item If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type. \item If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation. \item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat. \item Add more here. \end{enumerate}


spaces-more-groupoids-lemma-groupoid-on-field-open-multiplication

In Situation \ref{situation-groupoid-on-field} the composition morphism $c : R \times_{s, U, t} R \to R$ is flat and universally open. In Situation \ref{situation-group-over-field} the group law $m : G \times_k G \to G$ is flat and universally open.


spaces-more-groupoids-lemma-group-scheme-over-field-separated

In Situation \ref{situation-groupoid-on-field} assume $R$ is a decent space. Then $R$ is a separated algebraic space. In Situation \ref{situation-group-over-field} assume that $G$ is a decent algebraic space. Then $G$ is separated algebraic space.


spaces-more-groupoids-lemma-restrict-groupoid-on-field

In Situation \ref{situation-groupoid-on-field}. Let $k \subset k'$ be a field extension, $U' = \Spec(k')$ and let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $U' \to U$. In the defining diagram $$ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'} \ar@{..>}[rd] & R \times_{s, U} U' \ar[r] \ar[d] & U' \ar[d] \\ U' \times_{U, t} R \ar[d] \ar[r] & R \ar[r]^s \ar[d]_t & U \\ U' \ar[r] & U } $$ all the morphisms are surjective, flat, and universally open. The dotted arrow $R' \to R$ is in addition affine.


spaces-more-groupoids-lemma-groupoid-on-field-explain-points

In Situation \ref{situation-groupoid-on-field}. For any point $r \in |R|$ there exist \begin{enumerate} \item a field extension $k \subset k'$ with $k'$ algebraically closed, \item a point $r' : \Spec(k') \to R'$ where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $\Spec(k') \to \Spec(k)$ \end{enumerate} such that \begin{enumerate} \item the point $r'$ maps to $r$ under the morphism $R' \to R$, and \item the maps $s' \circ r', t' \circ r' : \Spec(k') \to \Spec(k')$ are automorphisms. \end{enumerate}


spaces-more-groupoids-lemma-groupoid-on-field-move-point

In Situation \ref{situation-groupoid-on-field}. If $r : \Spec(k) \to R$ is a morphism such that $s \circ r, t \circ r$ are automorphisms of $\Spec(k)$, then the map $$ R \longrightarrow R, \quad x \longmapsto c(r, x) $$ is an automorphism $R \to R$ which maps $e$ to $r$.


spaces-more-groupoids-lemma-groupoid-on-field-locally-finite-type-dimension

In Situation \ref{situation-groupoid-on-field} assume $s, t$ are locally of finite type. For all $r \in |R|$ \begin{enumerate} \item $\dim(R) = \dim_r(R)$, \item the transcendence degree of $r$ over $\Spec(k)$ via $s$ equals the transcendence degree of $r$ over $\Spec(k)$ via $t$, and \item if the transcendence degree mentioned in (2) is $0$, then $\dim(R) = \dim(\mathcal{O}_{R, \overline{r}})$. \end{enumerate}


spaces-more-groupoids-lemma-group-over-field-locally-finite-type-dimension

In Situation \ref{situation-group-over-field} assume $G$ locally of finite type. For all $g \in |G|$ \begin{enumerate} \item $\dim(G) = \dim_g(G)$, \item if the transcendence degree of $g$ over $k$ is $0$, then $\dim(G) = \dim(\mathcal{O}_{G, \overline{g}})$. \end{enumerate}


spaces-more-groupoids-lemma-groupoid-on-field-dimension-equal-stabilizer

In Situation \ref{situation-groupoid-on-field} assume $s, t$ are locally of finite type. Let $G = \Spec(k) \times_{\Delta, \Spec(k) \times_B \Spec(k), t \times s} R$ be the stabilizer group algebraic space. Then we have $\dim(R) = \dim(G)$.


spaces-more-groupoids-lemma-group-space-scheme-over-kbar

Let $k$ be a field with algebraic closure $\overline{k}$. Let $G$ be a group algebraic space over $k$ which is separated\footnote{It is enough to assume $G$ is decent, e.g., locally separated or quasi-separated by Lemma \ref{lemma-group-scheme-over-field-separated}.}. Then $G_{\overline{k}}$ is a scheme.


spaces-more-groupoids-lemma-group-space-scheme-locally-finite-type-over-k

Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated and locally of finite type over $k$, then $G$ is a scheme.


spaces-more-groupoids-lemma-factor-through-over-open

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $f : X \to Y$ and $g : X \to Z$ be morphisms of algebraic spaces over $B$. Assume \begin{enumerate} \item $Y \to B$ is separated, \item $g$ is surjective, flat, and locally of finite presentation, \item there is a scheme theoretically dense open $V \subset Z$ such that $f|_{g^{-1}(V)} : g^{-1}(V) \to Y$ factors through $V$. \end{enumerate} Then $f$ factors through $g$.


spaces-more-groupoids-lemma-quotient-power-P1

\begin{slogan} A morphism from a nonempty product of projective lines over a field to a separated finite type algebraic space over a field factors as a finite morphism after a projection to a product of projective lines. \end{slogan} Let $k$ be a field. Let $n \geq 1$ and let $(\mathbf{P}^1_k)^n$ be the $n$-fold self product over $\Spec(k)$. Let $f : (\mathbf{P}^1_k)^n \to Z$ be a morphism of algebraic spaces over $k$. If $Z$ is separated of finite type over $k$, then $f$ factors as $$ (\mathbf{P}^1_k)^n \xrightarrow{projection} (\mathbf{P}^1_k)^m \xrightarrow{finite} Z. $$


spaces-more-groupoids-lemma-no-nonconstant-morphism-from-P1-to-group

\begin{slogan} No complete rational curves on groups. \end{slogan} Let $k$ be a field. Let $G$ be a separated group algebraic space locally of finite type over $k$. There does not exist a nonconstant morphism $f : \mathbf{P}^1_k \to G$ over $\Spec(k)$.


spaces-more-groupoids-lemma-finite-sheaf

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then we have \begin{enumerate} \item The presheaf $(X/Y)_{fin}$ satisfies the sheaf condition for the fppf topology. \item If $T$ is an algebraic space over $S$, then there is a canonical bijection $$ \Mor_{\Sh((\Sch/S)_{fppf})}(T, (X/Y)_{fin}) = \{(a, Z)\text{ satisfying \ref{equation-finite-conditions}}\} $$ \end{enumerate}


spaces-more-groupoids-lemma-finite-open

Let $S$ be a scheme. Consider a commutative diagram $$ \xymatrix{ X' \ar[rr]_j \ar[rd] & & X \ar[ld] \\ & Y } $$ of algebraic spaces over $S$. If $j$ is an open immersion, then there is a canonical injective map of sheaves $j : (X'/Y)_{fin} \to (X/Y)_{fin}$.


spaces-more-groupoids-lemma-finite-lives-on-locally-quasi-finite-part

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $X' \subset X$ be the maximal open subspace over which $f$ is locally quasi-finite, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}. Then $(X/Y)_{fin} = (X'/Y)_{fin}$.


spaces-more-groupoids-lemma-finite-separated

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $T$ be an algebraic space over $S$, and let $(a, Z)$ be a pair as in \ref{equation-finite-conditions}. If $f$ is separated, then $Z$ is closed in $T \times_Y X$.


spaces-more-groupoids-lemma-finite-diagonal

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The diagonal of $(X/Y)_{fin} \to Y$ $$ (X/Y)_{fin} \longrightarrow (X/Y)_{fin} \times_Y (X/Y)_{fin} $$ is representable (by schemes) and an open immersion and the ``absolute'' diagonal $$ (X/Y)_{fin} \longrightarrow (X/Y)_{fin} \times (X/Y)_{fin} $$ is representable (by schemes).


spaces-more-groupoids-lemma-finite-criterion-etale

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Suppose that $U$ is a scheme, $U \to Y$ is an \'etale morphism and $Z \subset U \times_Y X$ is an open subspace finite over $U$. Then the induced morphism $U \to (X/Y)_{fin}$ is \'etale.


spaces-more-groupoids-lemma-finite-pullback

Let $S$ be a scheme. Let $$ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } $$ be a fibre product square of algebraic spaces over $S$. Then $$ \xymatrix{ (X'/Y')_{fin} \ar[d] \ar[r] & (X/Y)_{fin} \ar[d] \\ Y' \ar[r] & Y } $$ is a fibre product square of sheaves on $(\Sch/S)_{fppf}$.


spaces-more-groupoids-lemma-finite-surjective-etale-cover

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated and locally quasi-finite, then there exists a scheme $U$ \'etale over $Y$ and a surjective \'etale morphism $U \to (X/Y)_{fin}$ over $Y$.


spaces-more-groupoids-lemma-finite-separated-flat-locally-finite-presentation

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is separated, flat, and locally of finite presentation. In this case \begin{enumerate} \item $(X/Y)_{fin} \to Y$ is separated, representable, and \'etale, and \item if $Y$ is a scheme, then $(X/Y)_{fin}$ is (representable by) a scheme. \end{enumerate}


spaces-more-groupoids-lemma-finite-plus-section

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\sigma : Y \to X$ be a section of $f$. Consider the transformation of functors $$ t : (X/Y, \sigma)_{fin} \longrightarrow (X/Y)_{fin}. $$ defined above. Then \begin{enumerate} \item $t$ is representable by open immersions, \item if $f$ is separated, then $t$ is representable by open and closed immersions, \item if $(X/Y)_{fin}$ is an algebraic space, then $(X/Y, \sigma)_{fin}$ is an algebraic space and an open subspace of $(X/Y)_{fin}$, and \item if $(X/Y)_{fin}$ is a scheme, then $(X/Y, \sigma)_{fin}$ is an open subscheme of it. \end{enumerate}


spaces-more-groupoids-lemma-finite-part-groupoid

Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$. Assume the morphisms $s, t$ are separated and locally of finite type. There exists a canonical morphism $$ (U', Z_{univ}, s', t', c', e', i') \longrightarrow (U, R, s, t, c, e, i) $$ of groupoids in algebraic spaces over $B$ where \begin{enumerate} \item $g : U' \to U$ is identified with $(R_s/U, e)_{fin} \to U$, and \item $Z_{univ} \subset R \times_{s, U, g} U'$ is the universal open (and closed) subspace finite over $U'$ which contains the base change of the unit $e$. \end{enumerate}


spaces-more-groupoids-lemma-strong-splitting

In Situation \ref{situation-strong-splitting} there exists an algebraic space $U'$, an \'etale morphism $U' \to U$, and a point $u' : \Spec(\kappa(u)) \to U'$ lying over $u : \Spec(\kappa(u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.


spaces-more-groupoids-lemma-splitting

In Situation \ref{situation-splitting} there exists an algebraic space $U'$, an \'etale morphism $U' \to U$, and a point $u' : \Spec(\kappa(u)) \to U'$ lying over $u : \Spec(\kappa(u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.


spaces-more-groupoids-lemma-quasi-splitting

In Situation \ref{situation-quasi-splitting} there exists an algebraic space $U'$, an \'etale morphism $U' \to U$, and a point $u' : \Spec(\kappa(u)) \to U'$ lying over $u : \Spec(\kappa(u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.


spaces-more-groupoids-lemma-strong-splitting-scheme

In Situation \ref{situation-strong-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.


spaces-more-groupoids-lemma-splitting-scheme

In Situation \ref{situation-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.


spaces-more-groupoids-lemma-quasi-splitting-scheme

In Situation \ref{situation-quasi-splitting} assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.


spaces-more-groupoids-lemma-strong-splitting-affine-scheme

In Situation \ref{situation-strong-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.


spaces-more-groupoids-lemma-splitting-affine-scheme

In Situation \ref{situation-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.


spaces-more-groupoids-lemma-quasi-splitting-affine-scheme

In Situation \ref{situation-quasi-splitting} assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an \'etale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa(u) = \kappa(u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.


spaces-more-groupoids-proposition-group-space-scheme-over-field

Let $k$ be a field. Let $G$ be a group algebraic space over $k$. If $G$ is separated, then $G$ is a scheme.


spaces-more-groupoids-proposition-finite-algebraic-space

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is separated and locally of finite type. Then $(X/Y)_{fin}$ is an algebraic space. Moreover, the morphism $(X/Y)_{fin} \to Y$ is \'etale.


modules-lemma-abelian

Let $(X, \mathcal{O}_X)$ be a ringed space. The category $\textit{Mod}(\mathcal{O}_X)$ is an abelian category. Moreover a complex $$ \mathcal{F} \to \mathcal{G} \to \mathcal{H} $$ is exact at $\mathcal{G}$ if and only if for all $x \in X$ the complex $$ \mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x $$ is exact at $\mathcal{G}_x$.


modules-lemma-limits-colimits

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item All limits exist in $\textit{Mod}(\mathcal{O}_X)$. Limits are the same as the corresponding limits of presheaves of $\mathcal{O}_X$-modules (i.e., commute with taking sections over opens). \item All colimits exist in $\textit{Mod}(\mathcal{O}_X)$. Colimits are the sheafification of the corresponding colimit in the category of presheaves. Taking colimits commutes with taking stalks. \item Filtered colimits are exact. \item Finite direct sums are the same as the corresponding finite direct sums of presheaves of $\mathcal{O}_X$-modules. \end{enumerate}


modules-lemma-exactness-pushforward-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. \begin{enumerate} \item The functor $f_* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$ is left exact. In fact it commutes with all limits. \item The functor $f^* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ is right exact. In fact it commutes with all colimits. \item Pullback $f^{-1} : \textit{Ab}(Y) \to \textit{Ab}(X)$ on abelian sheaves is exact. \end{enumerate}


modules-lemma-j-shriek-exact

Let $j : U \to X$ be an open immersion of topological spaces. The functor $j_! : \textit{Ab}(U) \to \textit{Ab}(X)$ is exact.


modules-lemma-globally-generated

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $I$ be a set. Let $s_i \in \Gamma(X, \mathcal{F})$, $i \in I$ be global sections. The sections $s_i$ generate $\mathcal{F}$ if and only if for all $x\in X$ the elements $s_{i, x} \in \mathcal{F}_x$ generate the $\mathcal{O}_{X, x}$-module $\mathcal{F}_x$.


modules-lemma-tensor-product-globally-generated

\begin{slogan} The tensor product of globally generated sheaves of modules is globally generated. \end{slogan} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_X$-modules. If $\mathcal{F}$ and $\mathcal{G}$ are generated by global sections then so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$.


modules-lemma-generated-by-local-sections

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $I$ be a set. Let $s_i$, $i \in I$ be a collection of local sections of $\mathcal{F}$, i.e., $s_i \in \mathcal{F}(U_i)$ for some opens $U_i \subset X$. There exists a unique smallest subsheaf of $\mathcal{O}_X$-modules $\mathcal{G}$ such that each $s_i$ corresponds to a local section of $\mathcal{G}$.


modules-lemma-generated-by-local-sections-stalk

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Given a set $I$, and local sections $s_i$, $i \in I$ of $\mathcal{F}$. Let $\mathcal{G}$ be the subsheaf generated by the $s_i$ and let $x\in X$. Then $\mathcal{G}_x$ is the $\mathcal{O}_{X, x}$-submodule of $\mathcal{F}_x$ generated by the elements $s_{i, x}$ for those $i$ such that $s_i$ is defined at $x$.


modules-lemma-support-section-closed

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset X$ open. \begin{enumerate} \item The support of $s \in \mathcal{F}(U)$ is closed in $U$. \item The support of $fs$ is contained in the intersections of the supports of $f \in \mathcal{O}_X(U)$ and $s \in \mathcal{F}(U)$. \item The support of $s + s'$ is contained in the union of the supports of $s, s' \in \mathcal{F}(U)$. \item The support of $\mathcal{F}$ is the union of the supports of all local sections of $\mathcal{F}$. \item If $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of $\mathcal{O}_X$-modules, then the support of $\varphi(s)$ is contained in the support of $s \in \mathcal{F}(U)$. \end{enumerate}


modules-lemma-support-sheaf-rings-closed

Let $X$ be a topological space. The support of a sheaf of rings is closed.


modules-lemma-i-star-exact

Let $X$ be a topological space. Let $Z \subset X$ be a closed subset. Denote $i : Z \to X$ the inclusion map. The functor $$ i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X) $$ is exact, fully faithful, with essential image exactly those abelian sheaves whose support is contained in $Z$. The functor $i^{-1}$ is a left inverse to $i_*$.


modules-lemma-i-star-right-adjoint

Let $i : Z \to X$ be the inclusion of a closed subset into the topological space $X$. The functor $\textit{Ab}(X) \to \textit{Ab}(Z)$, $\mathcal{F} \mapsto \mathcal{H}_Z(\mathcal{F})$ of Remark \ref{remark-sections-support-in-closed} is a right adjoint to $i_* : \textit{Ab}(Z) \to \textit{Ab}(Z)$. In particular $i_*$ commutes with arbitrary colimits.


modules-lemma-canonical-exact-sequence

Let $X$ be a topological space. Let $U \subset X$ be an open subset with complement $Z \subset X$. Denote $j : U \to X$ the open immersion and $i : Z \to X$ the closed immersion. For any sheaf of abelian groups $\mathcal{F}$ on $X$ the adjunction mappings $j_{!}j^*\mathcal{F} \to \mathcal{F}$ and $\mathcal{F} \to i_*i^*\mathcal{F}$ give a short exact sequence $$ 0 \to j_{!}j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \to 0 $$ of sheaves of abelian groups. For any morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of abelian sheaves on $X$ we obtain a morphism of short exact sequences $$ \xymatrix{ 0 \ar[r] & j_{!}j^*\mathcal{F} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & i_*i^*\mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & j_{!}j^*\mathcal{G} \ar[r] & \mathcal{G} \ar[r] & i_*i^*\mathcal{G} \ar[r] & 0 } $$


modules-lemma-pullback-locally-generated

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ is locally generated by sections if $\mathcal{G}$ is locally generated by sections.


modules-lemma-pullback-finite-type

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a finite type $\mathcal{O}_Y$-module is a finite type $\mathcal{O}_X$-module.


modules-lemma-extension-finite-type

Let $X$ be a ringed space. The image of a morphism of $\mathcal{O}_X$-modules of finite type is of finite type. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of $\mathcal{O}_X$-modules. If $\mathcal{F}_1$ and $\mathcal{F}_3$ are of finite type, so is $\mathcal{F}_2$.


modules-lemma-finite-type-surjective-on-stalk

Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_X$-modules. Let $x \in X$. Assume $\mathcal{F}$ of finite type and the map on stalks $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ surjective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi|_U$ is surjective.


modules-lemma-finite-type-stalk-zero

Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $x \in X$. Assume $\mathcal{F}$ of finite type and $\mathcal{F}_x = 0$. Then there exists an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_U$ is zero.


modules-lemma-support-finite-type-closed

\begin{slogan} Over any ringed space, sheaves of modules of finite type have closed support. \end{slogan} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. If $\mathcal{F}$ is of finite type then support of $\mathcal{F}$ is closed.


modules-lemma-finite-type-quasi-compact-colimit

Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_i, f_{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_X$-modules (see Categories, Section \ref{categories-section-posets-limits}). Let $\mathcal{F} = \colim \mathcal{F}_i$ be the colimit. Assume (a) $I$ is directed, (b) $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, and (c) $X$ is quasi-compact. Then there exists an $i$ such that $\mathcal{F}_i \to \mathcal{F}$ is surjective. If the transition maps $f_{ii'}$ are injective then we conclude that $\mathcal{F} = \mathcal{F}_i$ for some $i \in I$.


modules-lemma-set-isomorphism-classes-finite-type-modules

Let $X$ be a ringed space. There exists a set of $\mathcal{O}_X$-modules $\{\mathcal{F}_i\}_{i \in I}$ of finite type such that each finite type $\mathcal{O}_X$-module on $X$ is isomorphic to exactly one of the $\mathcal{F}_i$.


modules-lemma-direct-sum-quasi-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. The direct sum of two quasi-coherent $\mathcal{O}_X$-modules is a quasi-coherent $\mathcal{O}_X$-module.


modules-lemma-pullback-quasi-coherent

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a quasi-coherent $\mathcal{O}_Y$-module is quasi-coherent.


modules-lemma-construct-quasi-coherent-sheaves

Let $(X, \mathcal{O}_X)$ be ringed space. Let $\alpha : R \to \Gamma(X, \mathcal{O}_X)$ be a ring homomorphism from a ring $R$ into the ring of global sections on $X$. Let $M$ be an $R$-module. The following three constructions give canonically isomorphic sheaves of $\mathcal{O}_X$-modules: \begin{enumerate} \item Let $\pi : (X, \mathcal{O}_X) \longrightarrow (\{*\}, R)$ be the morphism of ringed spaces with $\pi : X \to \{*\}$ the unique map and with $\pi$-map $\pi^\sharp$ the given map $\alpha : R \to \Gamma(X, \mathcal{O}_X)$. Set $\mathcal{F}_1 = \pi^*M$. \item Choose a presentation $\bigoplus_{j \in J} R \to \bigoplus_{i \in I} R \to M \to 0$. Set $$ \mathcal{F}_2 = \Coker\left( \bigoplus\nolimits_{j \in J} \mathcal{O}_X \to \bigoplus\nolimits_{i \in I} \mathcal{O}_X \right). $$ Here the map on the component $\mathcal{O}_X$ corresponding to $j \in J$ given by the section $\sum_i \alpha(r_{ij})$ where the $r_{ij}$ are the matrix coefficients of the map in the presentation of $M$. \item Set $\mathcal{F}_3$ equal to the sheaf associated to the presheaf $U \mapsto \mathcal{O}_X(U) \otimes_R M$, where the map $R \to \mathcal{O}_X(U)$ is the composition of $\alpha$ and the restriction map $\mathcal{O}_X(X) \to \mathcal{O}_X(U)$. \end{enumerate} This construction has the following properties: \begin{enumerate} \item The resulting sheaf of $\mathcal{O}_X$-modules $\mathcal{F}_M = \mathcal{F}_1 = \mathcal{F}_2 = \mathcal{F}_3$ is quasi-coherent. \item The construction gives a functor from the category of $R$-modules to the category of quasi-coherent sheaves on $X$ which commutes with arbitrary colimits. \item For any $x \in X$ we have $\mathcal{F}_{M, x} = \mathcal{O}_{X, x} \otimes_R M$ functorial in $M$. \item Given any $\mathcal{O}_X$-module $\mathcal{G}$ we have $$ \Mor_{\mathcal{O}_X}(\mathcal{F}_M, \mathcal{G}) = \Hom_R(M, \Gamma(X, \mathcal{G})) $$ where the $R$-module structure on $\Gamma(X, \mathcal{G})$ comes from the $\Gamma(X, \mathcal{O}_X)$-module structure via $\alpha$. \end{enumerate}


modules-lemma-restrict-quasi-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Set $R = \Gamma(X, \mathcal{O}_X)$. Let $M$ be an $R$-module. Let $\mathcal{F}_M$ be the quasi-coherent sheaf of $\mathcal{O}_X$-modules associated to $M$. If $g : (Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)$ is a morphism of ringed spaces, then $g^*\mathcal{F}_M$ is the sheaf associated to the $\Gamma(Y, \mathcal{O}_Y)$-module $\Gamma(Y, \mathcal{O}_Y) \otimes_R M$.


modules-lemma-quasi-coherent-module

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $x \in X$ be a point. Assume that $x$ has a fundamental system of quasi-compact neighbourhoods. Consider any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$. Then there exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|_U$ is isomorphic to the sheaf of modules $\mathcal{F}_M$ on $(U, \mathcal{O}_U)$ associated to some $\Gamma(U, \mathcal{O}_U)$-module $M$.


modules-lemma-finite-presentation-quasi-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Any $\mathcal{O}_X$-module of finite presentation is quasi-coherent.


modules-lemma-kernel-surjection-finite-free-onto-finite-presentation

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module of finite presentation. \begin{enumerate} \item If $\psi : \mathcal{O}_X^{\oplus r} \to \mathcal{F}$ is a surjection, then $\Ker(\psi)$ is of finite type. \item If $\theta : \mathcal{G} \to \mathcal{F}$ is surjective with $\mathcal{G}$ of finite type, then $\Ker(\theta)$ is of finite type. \end{enumerate}


modules-lemma-pullback-finite-presentation

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. The pullback $f^*\mathcal{G}$ of a module of finite presentation is of finite presentation.


modules-lemma-finite-presentation-quasi-compact-colimit

Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_i, \varphi_{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_X$-modules (see Categories, Section \ref{categories-section-posets-limits}). Assume \begin{enumerate} \item $I$ is directed, \item $\mathcal{G}$ is an $\mathcal{O}_X$-module of finite presentation, and \item $X$ has a cofinal system of open coverings $\mathcal{U} : X = \bigcup_{j\in J} U_j$ with $J$ finite and $U_j \cap U_{j'}$ quasi-compact for all $j, j' \in J$. \end{enumerate} Then we have $$ \colim_i \Hom_X(\mathcal{G}, \mathcal{F}_i) = \Hom_X(\mathcal{G}, \colim_i \mathcal{F}_i). $$


modules-lemma-finite-presentation-stalk-free

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Let $x \in X$ such that $\mathcal{F}_x \cong \mathcal{O}_{X, x}^{\oplus r}$. Then there exists an open neighbourhood $U$ of $x$ such that $\mathcal{F}|_U \cong \mathcal{O}_U^{\oplus r}$.


modules-lemma-coherent-finite-presentation

Let $(X, \mathcal{O}_X)$ be a ringed space. Any coherent $\mathcal{O}_X$-module is of finite presentation and hence quasi-coherent.


modules-lemma-coherent-abelian

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item Any finite type subsheaf of a coherent sheaf is coherent. \item Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism from a finite type sheaf $\mathcal{F}$ to a coherent sheaf $\mathcal{G}$. Then $\Ker(\varphi)$ is finite type. \item Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of coherent $\mathcal{O}_X$-modules. Then $\Ker(\varphi)$ and $\Coker(\varphi)$ are coherent. \item Given a short exact sequence of $\mathcal{O}_X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ if two out of three are coherent so is the third. \item The category $\textit{Coh}(\mathcal{O}_X)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_X)$. In particular, the category of coherent modules is abelian and the inclusion functor $\textit{Coh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ is exact. \end{enumerate}


modules-lemma-coherent-structure-sheaf

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume $\mathcal{O}_X$ is a coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is coherent if and only if it is of finite presentation.


modules-lemma-finite-type-to-coherent-injective-on-stalk

Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_X$-modules. Let $x \in X$. Assume $\mathcal{G}$ of finite type, $\mathcal{F}$ coherent and the map on stalks $\varphi_x : \mathcal{G}_x \to \mathcal{F}_x$ injective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi|_U$ is injective.


modules-lemma-i-star-reflects-finite-type

Let $i : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)$ be a morphism of ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and that $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. Let $\mathcal{F}$ be an $\mathcal{O}_Z$-module. Then $i_*\mathcal{F}$ is of finite type if and only if $\mathcal{F}$ is of finite type.


modules-lemma-i-star-equivalence

Let $i : (Z, \mathcal{O}_Z) \to (X, \mathcal{O}_X)$ be a morphism of ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. Denote $\mathcal{I} \subset \mathcal{O}_X$ the kernel of $i^\sharp$. The functor $$ i_* : \textit{Mod}(\mathcal{O}_Z) \longrightarrow \textit{Mod}(\mathcal{O}_X) $$ is exact, fully faithful, with essential image those $\mathcal{O}_X$-modules $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$.


modules-lemma-adjoint-section-with-support

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $\mathcal{H}_Z : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X|_Z)$ of Remark \ref{remark-sections-support-in-closed-modules} is right adjoint to $i_* : \textit{Mod}(\mathcal{O}_X|_Z) \to \textit{Mod}(\mathcal{O}_X)$.


modules-lemma-locally-free-quasi-coherent

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. If $\mathcal{F}$ is locally free then it is quasi-coherent.


modules-lemma-pullback-locally-free

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. If $\mathcal{G}$ is a locally free $\mathcal{O}_Y$-module, then $f^*\mathcal{G}$ is a locally free $\mathcal{O}_X$-module.


modules-lemma-rank

Let $(X, \mathcal{O}_X)$ be a ringed space. Suppose that the support of $\mathcal{O}_X$ is $X$, i.e., all stalk of $\mathcal{O}_X$ are nonzero rings. Let $\mathcal{F}$ be a locally free sheaf of $\mathcal{O}_X$-modules. There exists a locally constant function $$ \text{rank}_\mathcal{F} : X \longrightarrow \{0, 1, 2, \ldots\}\cup\{\infty\} $$ such that for any point $x \in X$ the cardinality of any set $I$ such that $\mathcal{F}$ is isomorphic to $\bigoplus_{i\in I} \mathcal{O}_X$ in a neighbourhood of $x$ is $\text{rank}_\mathcal{F}(x)$.


modules-lemma-map-finite-locally-free

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $r \geq 0$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of finite locally free $\mathcal{O}_X$-modules of rank $r$. Then $\varphi$ is an isomorphism if and only if $\varphi$ is surjective.


modules-lemma-direct-summand-of-locally-free-is-locally-free

Let $(X, \mathcal{O}_X)$ be a ringed space. If all stalks $\mathcal{O}_{X, x}$ are local rings, then any direct summand of a finite locally free $\mathcal{O}_X$-module is finite locally free.


modules-lemma-stalk-tensor-product

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. Let $x \in X$. There is a canonical isomorphism of $\mathcal{O}_{X, x}$-modules $$ (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})_x = \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{G}_x $$ functorial in $\mathcal{F}$ and $\mathcal{G}$.


modules-lemma-tensor-product-sheafification

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}'$, $\mathcal{G}'$ be presheaves of $\mathcal{O}_X$-modules with sheafifications $\mathcal{F}$, $\mathcal{G}$. Then $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} = (\mathcal{F}' \otimes_{p, \mathcal{O}_X} \mathcal{G}')^\#$.


modules-lemma-tensor-product-exact

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{G}$ be an $\mathcal{O}_X$-module. If $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is an exact sequence of $\mathcal{O}_X$-modules then the induced sequence $$ \mathcal{F}_1 \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{F}_2 \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{F}_3 \otimes_{\mathcal{O}_X} \mathcal{G} \to 0 $$ is exact.


modules-lemma-tensor-product-pullback

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_Y$-modules. Then $f^*(\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{G}) = f^*\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}$ functorially in $\mathcal{F}$, $\mathcal{G}$.


modules-lemma-tensor-product-permanence

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}$, $\mathcal{G}$ are locally generated by sections, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are of finite type, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are quasi-coherent, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are of finite presentation, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \item If $\mathcal{F}$ is of finite presentation and $\mathcal{G}$ is coherent, then $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ is coherent. \item If $\mathcal{F}$, $\mathcal{G}$ are coherent, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \item If $\mathcal{F}$, $\mathcal{G}$ are locally free, so is $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$. \end{enumerate}


modules-lemma-tensor-commute-colimits

Let $(X, \mathcal{O}_X)$ be a ringed space. For any $\mathcal{O}_X$-module $\mathcal{F}$ the functor $$ \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_X) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes_\mathcal{O} \mathcal{G} $$ commutes with arbitrary colimits.


modules-lemma-flat-stalks-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. An $\mathcal{O}_X$-module $\mathcal{F}$ is flat if and only if the stalk $\mathcal{F}_x$ is a flat $\mathcal{O}_{X, x}$-module for all $x \in X$.


modules-lemma-colimits-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. A filtered colimit of flat $\mathcal{O}_X$-modules is flat. A direct sum of flat $\mathcal{O}_X$-modules is flat.


modules-lemma-j-shriek-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $U \subset X$ be open. The sheaf $j_{U!}\mathcal{O}_U$ is a flat sheaf of $\mathcal{O}_X$-modules.


modules-lemma-module-quotient-flat

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item Any sheaf of $\mathcal{O}_X$-modules is a quotient of a direct sum $\bigoplus j_{U_i!}\mathcal{O}_{U_i}$. \item Any $\mathcal{O}_X$-module is a quotient of a flat $\mathcal{O}_X$-module. \end{enumerate}


modules-lemma-flat-tor-zero

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $$ 0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0 $$ be a short exact sequence of $\mathcal{O}_X$-modules. Assume $\mathcal{F}$ is flat. Then for any $\mathcal{O}_X$-module $\mathcal{G}$ the sequence $$ 0 \to \mathcal{F}'' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F}' \otimes_\mathcal{O} \mathcal{G} \to \mathcal{F} \otimes_\mathcal{O} \mathcal{G} \to 0 $$ is exact.


modules-lemma-flat-ses

\begin{slogan} Kernels of epimorphisms and extensions of flat sheaves of modules over a ringed space are again flat. \end{slogan} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $$ 0 \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to 0 $$ be a short exact sequence of $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}_2$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_1$. \item If $\mathcal{F}_1$ and $\mathcal{F}_0$ are flat so is $\mathcal{F}_2$. \end{enumerate}


modules-lemma-flat-eq

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a flat $\mathcal{O}_X$-module. Let $U \subset X$ be open and let $$ \mathcal{O}_U \xrightarrow{(f_1, \ldots, f_n)} \mathcal{O}_U^{\oplus n} \xrightarrow{(s_1, \ldots, s_n)} \mathcal{F}|_U $$ be a complex of $\mathcal{O}_U$-modules. For every $x \in U$ there exists an open neighbourhood $V \subset U$ of $x$ and a factorization $$ \mathcal{O}_V^{\oplus n} \xrightarrow{A} \mathcal{O}_V^{\oplus m} \xrightarrow{(t_1, \ldots, t_m)} \mathcal{F}|_V $$ of $(s_1, \ldots, s_n)|_V$ such that $A \circ (f_1, \ldots, f_n)|_V = 0$.


modules-lemma-left-dual-module

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $\mathcal{G}, \eta, \epsilon$ be a left dual of $\mathcal{F}$ in the monoidal category of $\mathcal{O}_X$-modules, see Categories, Definition \ref{categories-definition-dual}. Then \begin{enumerate} \item $\mathcal{F}$ is locally a direct summand of a finite free $\mathcal{O}_X$-module, \item the map $e : \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{O}_X) \to \mathcal{G}$ sending a local section $\lambda$ to $(\lambda \otimes 1)(\eta)$ is an isomorphism, \item we have $\epsilon(f, g) = e^{-1}(g)(f)$ for local sections $f$ and $g$ of $\mathcal{F}$ and $\mathcal{G}$. \end{enumerate}


modules-lemma-flat-locally-finite-presentation

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a flat $\mathcal{O}_X$-module of finite presentation. Then $\mathcal{F}$ is locally a direct summand of a finite free $\mathcal{O}_X$-module.


modules-lemma-surjection

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be a sheaf of sets on $X$. There exists a set $I$ and for each $i \in I$ an element $U_i \in \mathcal{B}$ and a finite set $S_i$ such that there exists a surjection $\coprod_{i \in I} j_{U_i!}\underline{S_i} \to \mathcal{F}$.


modules-lemma-filtered-colimit-constructibles

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$ and assume that each $U \in \mathcal{B}$ is quasi-compact. Then every sheaf of sets on $X$ is a filtered colimit of sheaves of the form \begin{equation} \label{equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod\nolimits_{b = 1, \ldots, m} j_{V_b!}\underline{S_b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod\nolimits_{a = 1, \ldots, n} j_{U_a!}\underline{S_a} } \right) \end{equation} with $U_a$ and $V_b$ in $\mathcal{B}$ and $S_a$ and $S_b$ finite sets.


modules-lemma-constructible-comes-from-finite

Let $X$ be a spectral topological space. Let $\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\mathcal{F}$ be a sheaf of sets as in Equation (\ref{equation-towards-constructible-sets}). Then there exists a continuous spectral map $f : X \to Y$ to a finite sober topological space $Y$ and a sheaf of sets $\mathcal{G}$ on $Y$ with finite stalks such that $f^{-1}\mathcal{G} \cong \mathcal{F}$.


modules-lemma-constructible-in-constant

Let $X$ be a spectral topological space. Let $\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\mathcal{F}$ be a sheaf of sets as in Equation (\ref{equation-towards-constructible-sets}). Then there exist finitely many constructible closed subsets $Z_1, \ldots, Z_n \subset X$ and finite sets $S_i$ such that $\mathcal{F}$ is isomorphic to a subsheaf of $\prod (Z_i \to X)_*\underline{S_i}$.


modules-lemma-pullback-flat

Let $f : X \to Y$ be a flat morphism of ringed spaces. Then the pullback functor $f^* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ is exact.


modules-lemma-local-tensor-algebra

In the situation described above. The sheaf $\wedge^n\mathcal{F}$ is the sheafification of the presheaf $$ U \longmapsto \wedge^n_{\mathcal{O}_X(U)}(\mathcal{F}(U)). $$ See Algebra, Section \ref{algebra-section-tensor-algebra}. Similarly, the sheaf $\text{Sym}^n\mathcal{F}$ is the sheafification of the presheaf $$ U \longmapsto \text{Sym}^n_{\mathcal{O}_X(U)}(\mathcal{F}(U)). $$


modules-lemma-pullback-tensor-algebra

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_Y$-modules. Then $f^*\text{T}(\mathcal{F}) = \text{T}(f^*\mathcal{F})$, and similarly for the exterior and symmetric algebras associated to $\mathcal{F}$.


modules-lemma-presentation-sym-exterior

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ be an exact sequence of sheaves of $\mathcal{O}_X$-modules. For each $n \geq 1$ there is an exact sequence $$ \mathcal{F}_2 \otimes_{\mathcal{O}_X} \text{Sym}^{n - 1}(\mathcal{F}_1) \to \text{Sym}^n(\mathcal{F}_1) \to \text{Sym}^n(\mathcal{F}) \to 0 $$ and similarly an exact sequence $$ \mathcal{F}_2 \otimes_{\mathcal{O}_X} \wedge^{n - 1}(\mathcal{F}_1) \to \wedge^n(\mathcal{F}_1) \to \wedge^n(\mathcal{F}) \to 0 $$


modules-lemma-tensor-algebra-permanence

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}$ is locally generated by sections, then so is each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$. \item If $\mathcal{F}$ is of finite type, then so is each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$. \item If $\mathcal{F}$ is of finite presentation, then so is each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$. \item If $\mathcal{F}$ is coherent, then for $n > 0$ each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$ is coherent. \item If $\mathcal{F}$ is quasi-coherent, then so is each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$. \item If $\mathcal{F}$ is locally free, then so is each $\text{T}^n(\mathcal{F})$, $\wedge^n(\mathcal{F})$, and $\text{Sym}^n(\mathcal{F})$. \end{enumerate}


modules-lemma-internal-hom

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be $\mathcal{O}_X$-modules. There is a canonical isomorphism $$ \SheafHom_{\mathcal{O}_X} (\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}, \mathcal{H}) \longrightarrow \SheafHom_{\mathcal{O}_X} (\mathcal{F}, \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H})) $$ which is functorial in all three entries (sheaf Hom in all three spots). In particular, to give a morphism $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{H}$ is the same as giving a morphism $\mathcal{F} \to \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H})$.


modules-lemma-internal-hom-exact

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ is an exact sequence of $\mathcal{O}_X$-modules, then $$ 0 \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}_1, \mathcal{G}) \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}_2, \mathcal{G}) $$ is exact. \item If $0 \to \mathcal{G} \to \mathcal{G}_1 \to \mathcal{G}_2$ is an exact sequence of $\mathcal{O}_X$-modules, then $$ 0 \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}_1) \to \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}_2) $$ is exact. \end{enumerate}


modules-lemma-stalk-internal-hom

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. If $\mathcal{F}$ is finitely presented then the canonical map $$ \SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x \to \Hom_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x) $$ is an isomorphism.


modules-lemma-pullback-internal-hom

Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_Y$-modules. If $\mathcal{F}$ is finitely presented and $f$ is flat, then the canonical map $$ f^*\SheafHom_{\mathcal{O}_Y}(\mathcal{F}, \mathcal{G}) \longrightarrow \SheafHom_{\mathcal{O}_X}(f^*\mathcal{F}, f^*\mathcal{G}) $$ is an isomorphism.


modules-lemma-internal-hom-locally-kernel-direct-sum

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_X$-modules. If $\mathcal{F}$ is finitely presented then the sheaf $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is locally a kernel of a map between finite direct sums of copies of $\mathcal{G}$. In particular, if $\mathcal{G}$ is coherent then $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent too.


modules-lemma-adjoint-tensor-restrict

Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Then we have $$ \Hom_{\mathcal{O}_1}(\mathcal{F}_{\mathcal{O}_1}, \mathcal{G}) = \Hom_{\mathcal{O}_2}(\mathcal{F}, \SheafHom_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{G})) $$ bifunctorially in $\mathcal{F} \in \textit{Mod}(\mathcal{O}_2)$ and $\mathcal{G} \in \textit{Mod}(\mathcal{O}_1)$.


modules-lemma-invertible

Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{L}$ be an $\mathcal{O}_X$-module. Equivalent are \begin{enumerate} \item $\mathcal{L}$ is invertible, and \item there exists an $\mathcal{O}_X$-module $\mathcal{N}$ such that $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N} \cong \mathcal{O}_X$. \end{enumerate} In this case $\mathcal{L}$ is locally a direct summand of a finite free $\mathcal{O}_X$-module and the module $\mathcal{N}$ in (2) is isomorphic to $\SheafHom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X)$.


modules-lemma-invertible-is-locally-free-rank-1

Let $(X, \mathcal{O}_X)$ be a ringed space. Any locally free $\mathcal{O}_X$-module of rank $1$ is invertible. If all stalks $\mathcal{O}_{X, x}$ are local rings, then the converse holds as well (but in general this is not the case).


modules-lemma-constructions-invertible

Let $(X, \mathcal{O}_X)$ be a ringed space. \begin{enumerate} \item If $\mathcal{L}$, $\mathcal{N}$ are invertible $\mathcal{O}_X$-modules, then so is $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$. \item If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then so is $\SheafHom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X)$ and the evaluation map $\mathcal{L} \otimes_{\mathcal{O}_X} \SheafHom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X) \to \mathcal{O}_X$ is an isomorphism. \end{enumerate}


modules-lemma-pic-set

Let $(X, \mathcal{O}_X)$ be a ringed space. There exists a set of invertible modules $\{\mathcal{L}_i\}_{i \in I}$ such that each invertible module on $X$ is isomorphic to exactly one of the $\mathcal{L}_i$.


modules-lemma-s-open

\begin{slogan} A (local) trivialisation of a linebundle is the same as a (local) nonvanishing section. \end{slogan} Let $X$ be a ringed space. Assume that each stalk $\mathcal{O}_{X, x}$ is a local ring with maximal ideal $\mathfrak m_x$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. For any section $s \in \Gamma(X, \mathcal{L})$ the set $$ X_s = \{x \in X \mid \text{image }s \not\in \mathfrak m_x\mathcal{L}_x\} $$ is open in $X$. The map $s : \mathcal{O}_{X_s} \to \mathcal{L}|_{X_s}$ is an isomorphism, and there exists a section $s'$ of $\mathcal{L}^{\otimes -1}$ over $X_s$ such that $s' (s|_{X_s}) = 1$.


modules-lemma-det-ses

Let $X$ be a ringed space. Let $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ be a short exact sequence of finite locally free $\mathcal{O}_X$-modules, Then there is a canonical isomorphism $$ \det(\mathcal{E}') \otimes_{\mathcal{O}_X}\det(\mathcal{E}'') \longrightarrow \det(\mathcal{E}) $$ of $\mathcal{O}_X$-modules.


modules-lemma-simple-invert

Let $X$ be a topological space and let $\mathcal{O}_X$ be a presheaf of rings. Let $\mathcal{S} \subset \mathcal{O}_X$ be a pre-sheaf of sets contained in $\mathcal{O}_X$. Suppose that for every open $U \subset X$ the set $\mathcal{S}(U) \subset \mathcal{O}_X(U)$ is a multiplicative subset. \begin{enumerate} \item There is a map of presheaves of rings $\mathcal{O}_X \to \mathcal{S}^{-1}\mathcal{O}_X$ such that every local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{O}_X$. \item For any homomorphism of presheaves of rings $\mathcal{O}_X \to \mathcal{A}$ such that each local section of $\mathcal{S}$ maps to an invertible section of $\mathcal{A}$ there exists a unique factorization $\mathcal{S}^{-1}\mathcal{O}_X \to \mathcal{A}$. \item For any $x \in X$ we have $$ (\mathcal{S}^{-1}\mathcal{O}_X)_x = \mathcal{S}_x^{-1} \mathcal{O}_{X, x}. $$ \item The sheafification $(\mathcal{S}^{-1}\mathcal{O}_X)^\#$ is a sheaf of rings with a map of sheaves of rings $(\mathcal{O}_X)^\# \to (\mathcal{S}^{-1}\mathcal{O}_X)^\#$ which is universal for maps of $(\mathcal{O}_X)^\#$ into sheaves of rings such that each local section of $\mathcal{S}$ maps to an invertible section. \item For any $x \in X$ we have $$ (\mathcal{S}^{-1}\mathcal{O}_X)^\#_x = \mathcal{S}_x^{-1} \mathcal{O}_{X, x}. $$ \end{enumerate}