perfect-lemma-K-flat

Let $f : X \to Y$ be a morphism of schemes with $Y = \Spec(A)$ affine. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be a finite affine open covering such that all the finite intersections $U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ are affine. Let $\mathcal{F}^\bullet$ be a bounded complex of $f^{-1}\mathcal{O}_Y$-modules. Assume for all $n \in \mathbf{Z}$ the sheaf $\mathcal{F}^n$ is a flat $f^{-1}\mathcal{O}_Y$-module and $\mathcal{F}^n$ has the structure of a quasi-coherent $\mathcal{O}_X$-module compatible with the given $p^{-1}\mathcal{O}_Y$-module structure (but the differentials in the complex $\mathcal{F}^\bullet$ need not be $\mathcal{O}_X$-linear). Then the complex $\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))$ is K-flat as a complex of $A$-modules.


perfect-lemma-kunneth-single-sheaf

In the situation above the map (\ref{equation-kunneth-single-sheaves}) is an isomorphism if $S$ is affine, $\mathcal{F}$ and $\mathcal{G}$ are $S$-flat and quasi-coherent and $X$ and $Y$ are quasi-compact with affine diagonal.


perfect-lemma-kunneth-special

In the situation above the cup product (\ref{equation-de-rham-kunneth}) is an isomorphism in $D(A)$ if the following assumptions hold \begin{enumerate} \item $S = \Spec(A)$ is affine, \item $X$ and $Y$ are quasi-compact with affine diagonal, \item $\mathcal{F}^\bullet$ is bounded, \item $\mathcal{G}^\bullet$ is bounded below, \item $\mathcal{F}^n$ is $S$-flat, and \item $\mathcal{G}^m$ is $S$-flat. \end{enumerate}


perfect-lemma-single-complex-base-change-condition

Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. The following are equivalent \begin{enumerate} \item for any $x' \in X'$ and $i \in \mathbf{Z}$ the map (\ref{equation-bc}) is an isomorphism, \item for $U \subset X$, $V' \subset S'$ affine open both mapping into the affine open $V \subset S$ with $U' = V' \times_V U$ the composition $$ R\Gamma(U, K) \otimes_{\mathcal{O}_S(U)}^\mathbf{L} \mathcal{O}_{S'}(V') \to R\Gamma(U, K) \otimes_{\mathcal{O}_X(U)}^\mathbf{L} \mathcal{O}_{X'}(U') \to R\Gamma(U', K') $$ is an isomorphism in $D(\mathcal{O}_{S'}(V'))$, and \item there is a set $I$ of quadruples $U_i, V_i', V_i, U_i'$, $i \in I$ as in (2) with $X' = \bigcup U'_i$. \end{enumerate}


perfect-lemma-single-complex-base-change

Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. If \begin{enumerate} \item the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold, and \item $f$ is quasi-compact and quasi-separated, \end{enumerate} then the composition $Lg^*Rf_*K \to Rf'_*L(g')^*K \to Rf'_*K'$ is an isomorphism.


perfect-lemma-single-complex-base-change-condition-inherited

Let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ be a cartesian diagram of schemes. Let $K \in D_\QCoh(\mathcal{O}_X)$ and let $L(g')^*K \to K'$ be a map in $D_\QCoh(\mathcal{O}_{X'})$. If the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold, then \begin{enumerate} \item for $E \in D_\QCoh(\mathcal{O}_X)$ the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*(E \otimes^\mathbf{L} K) \to L(g')^*E \otimes^\mathbf{L} K'$, \item if $E$ in $D(\mathcal{O}_X)$ is perfect the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')$, and \item if $K$ is bounded below and $E$ in $D(\mathcal{O}_X)$ pseudo-coherent the equivalent conditions of Lemma \ref{lemma-single-complex-base-change-condition} hold for $L(g')^*R\SheafHom(E, K) \to R\SheafHom(L(g')^*E, K')$. \end{enumerate}


perfect-lemma-base-change-tensor

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a bounded above complex of quasi-coherent $\mathcal{O}_X$-modules flat over $S$. Then formation of $$ Rf_*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}^\bullet) $$ commutes with arbitrary base change (see proof for precise statement).


perfect-lemma-base-change-RHom

Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $E$ be an object of $D(\mathcal{O}_X)$. Let $\mathcal{G}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. If \begin{enumerate} \item $E$ is perfect, $\mathcal{G}^\bullet$ is a bounded above, and $\mathcal{G}^n$ is flat over $S$, or \item $E$ is pseudo-coherent, $\mathcal{G}^\bullet$ is bounded, and $\mathcal{G}^n$ is flat over $S$, \end{enumerate} then formation of $$ Rf_*R\SheafHom(E, \mathcal{G}^\bullet) $$ commutes with arbitrary base change (see proof for precise statement).


perfect-lemma-perfect-direct-image

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_X)$ such that \begin{enumerate} \item $E \in D^b_{\textit{Coh}}(\mathcal{O}_X)$, \item the support of $H^i(E)$ is proper over $S$ for all $i$, and \item $E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_S)$. \end{enumerate} Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$.


perfect-lemma-tensor-perfect

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet)$ is a perfect object of $D(\mathcal{O}_S)$.


perfect-lemma-ext-perfect

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_X)$ be perfect. Let $\mathcal{G}^\bullet$ be a bounded complex of coherent $\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet)$ is a perfect object of $D(\mathcal{O}_S)$.


perfect-lemma-flat-proper-perfect-direct-image

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a flat proper morphism of schemes. Let $E \in D(\mathcal{O}_X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$.


perfect-lemma-compute-tensor-perfect

Assumptions and notation as in Lemma \ref{lemma-tensor-perfect}. Then there are functorial isomorphisms $$ H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \longrightarrow H^i(X, E \otimes_{\mathcal{O}_X}^\mathbf{L} (\mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F})) $$ for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof).


perfect-lemma-compute-ext-perfect

Assumptions and notation as in Lemma \ref{lemma-ext-perfect}. Then there are functorial isomorphisms $$ H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof).


perfect-lemma-compute-ext

Let $f : X \to S$ be a morphism of schemes, $E \in D(\mathcal{O}_X)$ and $\mathcal{G}^\bullet$ a complex of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $S$ is Noetherian, \item $f$ is locally of finite type, \item $E \in D^-_{\textit{Coh}}(\mathcal{O}_X)$, \item $\mathcal{G}^\bullet$ is a bounded complex of coherent $\mathcal{O}_X$-modules flat over $S$ with support proper over $S$. \end{enumerate} Then the following two statements are true \begin{enumerate} \item[(A)] for every $m \in \mathbf{Z}$ there exists a perfect object $K$ of $D(\mathcal{O}_S)$ and functorial maps $$ \alpha^i_\mathcal{F} : \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) \longrightarrow H^i(S, K \otimes^\mathbf{L}_{\mathcal{O}_S} \mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps (see proof) such that $\alpha^i_\mathcal{F}$ is an isomorphism for $i \leq m$ \item[(B)] there exists a pseudo-coherent $L \in D(\mathcal{O}_S)$ and functorial isomorphisms $$ \Ext^i_{\mathcal{O}_S}(L, \mathcal{F}) \longrightarrow \Ext^i_{\mathcal{O}_X}(E, \mathcal{G}^\bullet \otimes_{\mathcal{O}_X} f^*\mathcal{F}) $$ for $\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps. \end{enumerate}


perfect-lemma-descend-homomorphisms

In Situation \ref{situation-descent}. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{S_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map $$ \colim_{i \geq 0} \Hom_{D(\mathcal{O}_{S_i})}(E_i, K_i) \longrightarrow \Hom_{D(\mathcal{O}_S)}(E, K) $$ is an isomorphism if either \begin{enumerate} \item $E_0$ is perfect and $K_0 \in D_\QCoh(\mathcal{O}_{S_0})$, or \item $E_0$ is pseudo-coherent and $K_0 \in D_\QCoh(\mathcal{O}_{S_0})$ has finite tor dimension. \end{enumerate}


perfect-lemma-descend-perfect

In Situation \ref{situation-descent} the category of perfect objects of $D(\mathcal{O}_S)$ is the colimit of the categories of perfect objects of $D(\mathcal{O}_{S_i})$.


perfect-lemma-base-change-tensor-perfect

Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet) $$ is a perfect object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change.


perfect-lemma-base-change-tensor-pseudo-coherent

Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_X)$ be a pseudo-coherent object. Let $\mathcal{G}^\bullet$ be a bounded above complex of finitely presented $\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}^\bullet) $$ is a pseudo-coherent object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change.


perfect-lemma-flat-proper-perfect-direct-image-general

Let $S$ be a scheme. Let $f : X \to S$ be a proper morphism of finite presentation. \begin{enumerate} \item Let $E \in D(\mathcal{O}_X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change. \item Let $\mathcal{G}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$. Then $Rf_*\mathcal{G}$ is a perfect object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change. \end{enumerate}


perfect-lemma-flat-proper-pseudo-coherent-direct-image-general

Let $S$ be a scheme. Let $f : X \to S$ be a flat proper morphism of finite presentation. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change.


perfect-lemma-base-change-RHom-perfect

Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_X)$ be a perfect object. Let $\mathcal{G}^\bullet$ be a bounded complex of finitely presented $\mathcal{O}_X$-modules, flat over $S$, with support proper over $S$. Then $$ K = Rf_*R\SheafHom(E, \mathcal{G}^\bullet) $$ is a perfect object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change.


perfect-lemma-jump-loci

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function $$ \beta_i : X \longrightarrow \{0, 1, 2, \ldots\},\quad x \longmapsto \dim_{\kappa(x)} H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x)) $$ Then we have \begin{enumerate} \item formation of $\beta_i$ commutes with arbitrary base change, \item the functions $\beta_i$ are upper semi-continuous, and \item the level sets of $\beta_i$ are locally constructible in $X$. \end{enumerate}


perfect-lemma-chi-locally-constant

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be perfect. The function $$ \chi_E : X \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^i \dim_{\kappa(x)} H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \kappa(x)) $$ is locally constant on $X$.


perfect-lemma-open-where-cohomology-in-degree-i-rank-r

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be perfect. Given $i, r \in \mathbf{Z}$, there exists an open subscheme $U \subset X$ characterized by the following \begin{enumerate} \item $E|_U \cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free $\mathcal{O}_U$-module of rank $r$, \item a morphism $f : Y \to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$. \end{enumerate}


perfect-lemma-locally-closed-where-H0-locally-free

Let $X$ be a scheme. Let $E \in D(\mathcal{O}_X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $r \geq 0$. Then there exists a locally closed subscheme $j : Z \to X$ characterized by the following \begin{enumerate} \item $H^a(Lj^*E)$ is a locally free $\mathcal{O}_Z$-module of rank $r$, and \item a morphism $f : Y \to X$ factors through $Z$ if and only if for all morphisms $g : Y' \to Y$ the $\mathcal{O}_{Y'}$-module $H^a(L(f \circ g)^*E)$ is locally free of rank $r$. \end{enumerate} Moreover, $j : Z \to X$ is of finite presentation and we have \begin{enumerate} \item[(3)] if $f : Y \to X$ factors as $Y \xrightarrow{g} Z \to X$, then $H^a(Lf^*E) = g^*H^a(Lj^*E)$, \item[(4)] if $\beta_a(x) \leq r$ for all $x \in X$, then $j$ is a closed immersion and given $f : Y \to X$ the following are equivalent \begin{enumerate} \item $f : Y \to X$ factors through $Z$, \item $H^0(Lf^*E)$ is a locally free $\mathcal{O}_Y$-module of rank $r$, \end{enumerate} and if $r = 1$ these are also equivalent to \begin{enumerate} \item[(c)] $\mathcal{O}_Y \to \SheafHom_{\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective. \end{enumerate} \end{enumerate}


perfect-lemma-jump-loci-geometric

Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$. For fixed $i \in \mathbf{Z}$ consider the function $$ \beta_i : S \to \{0, 1, 2, \ldots\},\quad s \longmapsto \dim_{\kappa(s)} H^i(X_s, \mathcal{F}_s) $$ Then we have \begin{enumerate} \item formation of $\beta_i$ commutes with arbitrary base change, \item the functions $\beta_i$ are upper semi-continuous, and \item the level sets of $\beta_i$ are locally constructible in $S$. \end{enumerate}


perfect-lemma-chi-locally-constant-geometric

Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$. The function $$ s \longmapsto \chi(X_s, \mathcal{F}_s) $$ is locally constant on $S$. Formation of this function commutes with base change.


perfect-lemma-open-where-cohomology-in-degree-i-rank-r-geometric

Let $f : X \to S$ be a flat, proper morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$. Fix $i, r \in \mathbf{Z}$. Then there exists an open subscheme $U \subset S$ with the following property: A morphism $T \to S$ factors through $U$ if and only if $Rf_{T, *}\mathcal{F}_T$ is isomorphic to a finite locally free module of rank $r$ placed in degree $i$.


perfect-lemma-vanishing-implies-locally-free

Let $f : X \to S$ be a morphism of finite presentation. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $S$ with support proper over $S$. If $R^if_*\mathcal{F} = 0$ for $i > 0$, then $f_*\mathcal{F}$ is locally free and its formation commutes with arbitrary base change (see proof for explanation).


perfect-lemma-proper-flat-h0

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item for all $s \in S$ we have $\kappa(s) = H^0(X_s, \mathcal{O}_{X_s})$. \end{enumerate} Then we have \begin{enumerate} \item[(a)] $f_*\mathcal{O}_X = \mathcal{O}_S$ and this holds after any base change, \item[(b)] locally on $S$ we have $$ Rf_*\mathcal{O}_X = \mathcal{O}_S \oplus P $$ in $D(\mathcal{O}_S)$ where $P$ is perfect of tor amplitude in $[1, \infty)$. \end{enumerate}


perfect-lemma-proper-flat-geom-red-connected

Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item the geometric fibres of $f$ are reduced and connected. \end{enumerate} Then $f_*\mathcal{O}_X = \mathcal{O}_S$ and this holds after any base change.


perfect-lemma-proper-idempotent-on-fibre

Let $f : X \to S$ be a proper morphism of schemes. Let $s \in S$ and let $e \in H^0(X_s, \mathcal{O}_{X_s})$ be an idempotent. Then $e$ is in the image of the map $(f_*\mathcal{O}_X)_s \to H^0(X_s, \mathcal{O}_{X_s})$.


perfect-lemma-proper-flat-geom-red

Let $f : X \to S$ be a morphism of schemes. Let $s \in S$. Assume \begin{enumerate} \item $f$ is proper, flat, and of finite presentation, and \item the fibre $X_s$ is geometrically reduced. \end{enumerate} Then, after replacing $S$ by an open neighbourhood of $s$, there exists a direct sum decomposition $Rf_*\mathcal{O}_X = f_*\mathcal{O}_X \oplus P$ in $D(\mathcal{O}_S)$ where $f_*\mathcal{O}_X$ is a finite \'etale $\mathcal{O}_S$-algebra and $P$ is a perfect of tor amplitude in $[1, \infty)$.


perfect-lemma-cohomology-over-coherent-ring

Let $R$ be a coherent ring. Let $X$ be a scheme of finite presentation over $R$. Let $\mathcal{G}$ be an $\mathcal{O}_X$-module of finite presentation, flat over $R$, with support proper over $R$. Then $H^i(X, \mathcal{G})$ is a coherent $R$-module.


perfect-lemma-countable-cohomology

Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be an object of $D_\QCoh(\mathcal{O}_X)$ such that the cohomology sheaves $H^i(K)$ have countable sets of sections over affine opens. Then for any quasi-compact open $U \subset X$ and any perfect object $E$ in $D(\mathcal{O}_X)$ the sets $$ H^i(U, K \otimes^\mathbf{L} E),\quad \Ext^i(E|_U, K|_U) $$ are countable.


perfect-lemma-countable

Let $X$ be a quasi-compact and quasi-separated scheme such that the sets of sections of $\mathcal{O}_X$ over affine opens are countable. Let $K$ be an object of $D_\QCoh(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $K = \text{hocolim} E_n$ with $E_n$ a perfect object of $D(\mathcal{O}_X)$, and \item the cohomology sheaves $H^i(K)$ have countable sets of sections over affine opens. \end{enumerate}


perfect-lemma-computing-sections-as-colim

Let $A$ be a ring. Let $X$ be a scheme of finite presentation over $A$. Let $f : U \to X$ be a flat morphism of finite presentation. Then \begin{enumerate} \item there exists an inverse system of perfect objects $L_n$ of $D(\mathcal{O}_X)$ such that $$ R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Hom_X(L_n, K) $$ in $D(A)$ functorially in $K$ in $D_\QCoh(\mathcal{O}_X)$, and \item there exists a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$ such that $$ R\Gamma(U, Lf^*K) = \text{hocolim}\ R\Gamma(X, E_n \otimes^\mathbf{L} K) $$ in $D(A)$ functorially in $K$ in $D_\QCoh(\mathcal{O}_X)$. \end{enumerate}


perfect-lemma-pseudo-coherent-over-algebra

Let $A$ be a ring. Let $R$ be a (possibly noncommutative) $A$-algebra which is finite free as an $A$-module. Then any object $M$ of $D(R)$ which is pseudo-coherent in $D(A)$ can be represented by a bounded above complex of finite free (right) $R$-modules.


perfect-lemma-pseudo-coherent-on-projective-space

Let $A$ be a ring. Let $n \geq 0$. Let $K \in D_\QCoh(\mathcal{O}_{\mathbf{P}^n_A})$. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, \item $R\Gamma(\mathbf{P}^n_A, E \otimes^\mathbf{L} K)$ is a pseudo-coherent object of $D(A)$ for each pseudo-coherent object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_A})$, \item $R\Gamma(\mathbf{P}^n_A, E \otimes^\mathbf{L} K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_A})$, \item $R\Hom_{\mathbf{P}^n_A}(E, K)$ is a pseudo-coherent object of $D(A)$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_A})$, \item $R\Gamma(\mathbf{P}^n_A, K \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_A}(d))$ is pseudo-coherent object of $D(A)$ for $d = 0, 1, \ldots, n$. \end{enumerate}


perfect-lemma-perfect-enough

Let $A$ be a ring. Let $X$ be a scheme over $A$ which is quasi-compact and quasi-separated. Let $K \in D^-_\QCoh(\mathcal{O}_X)$. If $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\mathcal{O}_X)$, then $R\Gamma(X, E \otimes^\mathbf{L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_X)$.


perfect-lemma-affine-locally-rel-perfect

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E$ is $S$-perfect, \item for any affine open $U \subset X$ mapping into an affine open $V \subset S$ the complex $R\Gamma(U, E)$ is $\mathcal{O}_S(V)$-perfect. \item there exists an affine open covering $S = \bigcup V_i$ and for each $i$ an affine open covering $f^{-1}(V_i) = \bigcup U_{ij}$ such that the complex $R\Gamma(U_{ij}, E)$ is $\mathcal{O}_S(V_i)$-perfect. \end{enumerate}


perfect-lemma-perfect-relatively-perfect

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. A perfect object of $D(\mathcal{O}_X)$ is $S$-perfect. If $K, M \in D(\mathcal{O}_X)$, then $K \otimes_{\mathcal{O}_X}^\mathbf{L} M$ is $S$-perfect if $K$ is perfect and $M$ is $S$-perfect.


perfect-lemma-base-change-relatively-perfect

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $g : S' \to S$ be a morphism of schemes. Set $X' = S' \times_S X$ and denote $g' : X' \to X$ the projection. If $K \in D(\mathcal{O}_X)$ is $S$-perfect, then $L(g')^*K$ is $S'$-perfect.


perfect-lemma-relative-descend-homomorphisms

In Situation \ref{situation-relative-descent}. Let $K_0$ and $L_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \geq 0$ and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map $$ \colim_{i \geq 0} \Hom_{D(\mathcal{O}_{X_i})}(K_i, L_i) \longrightarrow \Hom_{D(\mathcal{O}_X)}(K, L) $$ is an isomorphism if $K_0$ is pseudo-coherent and $L_0 \in D_\QCoh(\mathcal{O}_{X_0})$ has (locally) finite tor dimension as an object of $D((X_0 \to S_0)^{-1}\mathcal{O}_{S_0})$


perfect-lemma-descend-relatively-perfect

In Situation \ref{situation-relative-descent} the category of $S$-perfect objects of $D(\mathcal{O}_X)$ is the colimit of the categories of $S_i$-perfect objects of $D(\mathcal{O}_{X_i})$.


perfect-lemma-derived-pushforward-rel-perfect

Let $f : X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. Let $E \in D(\mathcal{O}_X)$ be $S$-perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$ and its formation commutes with arbitrary base change.


perfect-lemma-compute-ext-rel-perfect

Let $f : X \to S$ be a morphism of schemes. Let $E, K \in D(\mathcal{O}_X)$. Assume \begin{enumerate} \item $S$ is quasi-compact and quasi-separated, \item $f$ is proper, flat, and of finite presentation, \item $E$ is $S$-perfect, \item $K$ is pseudo-coherent. \end{enumerate} Then there exists a pseudo-coherent $L \in D(\mathcal{O}_S)$ such that $$ Rf_*R\SheafHom(K, E) = R\SheafHom(L, \mathcal{O}_S) $$ and the same is true after arbitrary base change: given $$ \vcenter{ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } } \quad\quad \begin{matrix} \text{cartesian, then we have } \\ Rf'_*R\SheafHom(L(g')^*K, L(g')^*E) \\ = R\SheafHom(Lg^*L, \mathcal{O}_{S'}) \end{matrix} $$


perfect-lemma-bounded-on-fibres

Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Let $E$ be a pseudo-coherent object of $D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E$ is $S$-perfect, and \item $E$ is locally bounded below and for every point $s \in S$ the object $L(X_s \to X)^*E$ of $D(\mathcal{O}_{X_s})$ is locally bounded below. \end{enumerate}


perfect-lemma-resolution-property-ample

Let $X$ be a scheme. If $X$ has an ample invertible $\mathcal{O}_X$-module, then $X$ has the resolution property.


perfect-lemma-resolution-property-ample-relative

Let $f : X \to Y$ be a morphism of schemes. Assume \begin{enumerate} \item $Y$ is quasi-compact and quasi-separated and has the resolution property, \item there exists an $f$-ample invertible module on $X$. \end{enumerate} Then $X$ has the resolution property.


perfect-lemma-resolution-property-goes-up-affine

Let $f : X \to Y$ be an affine morphism of schemes with $Y$ quasi-compact and quasi-separated. If $Y$ has the resolution property, so does $X$.


perfect-lemma-resolution-property-finite-number

Let $X$ be a scheme. Suppose given \begin{enumerate} \item a finite affine open covering $X = U_1 \cup \ldots \cup U_m$ \item finite type quasi-coherent ideals $\mathcal{I}_j$ with $V(\mathcal{I}_j) = X \setminus U_j$ \end{enumerate} Then $X$ has the resolution property if and only if $\mathcal{I}_j$ is the quotient of a finite locally free $\mathcal{O}_X$-module for $j = 1, \ldots, m$.


perfect-lemma-regular-resolution-property

Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has the resolution property.


perfect-lemma-resolution-property-descends

Let $X = \lim X_i$ be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $X$ has the resolution property if and only if $X_i$ has the resolution properties for some $i$.


perfect-lemma-resolution-property-affine-diagonal

\begin{reference} Special case of \cite[Proposition 1.3]{totaro_resolution}. \end{reference} Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then $X$ has affine diagonal.


perfect-lemma-construct-strictly-perfect

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{F}^\bullet$ be a bounded below complex of quasi-coherent $\mathcal{O}_X$-modules representing a perfect object of $D(\mathcal{O}_X)$. Then there exists a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules and a quasi-isomorphism $\mathcal{E}^\bullet \to \mathcal{F}^\bullet$.


perfect-lemma-resolution-property-perfect-complex

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of $D(\mathcal{O}_X)$ can be represented by a bounded complex of finite locally free $\mathcal{O}_X$-modules.


perfect-lemma-resolution-property-map-perfect-complex

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ be finite complexes of finite locally free $\mathcal{O}_X$-modules. Then any $\alpha \in \Hom_{D(\mathcal{O}_X)}(\mathcal{E}^\bullet, \mathcal{F}^\bullet)$ can be represented by a diagram $$ \mathcal{E}^\bullet \leftarrow \mathcal{G}^\bullet \to \mathcal{F}^\bullet $$ where $\mathcal{G}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_X$-modules and where $\mathcal{G}^\bullet \to \mathcal{E}^\bullet$ is a quasi-isomorphism.


perfect-lemma-resolution-property-homotopy-map-perfect-complex

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Let $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$ be finite complexes of finite locally free $\mathcal{O}_X$-modules. Let $\alpha^\bullet, \beta^\bullet :\mathcal{E}^\bullet \to \mathcal{F}^\bullet$ be two maps of complexes defining the same map in $D(\mathcal{O}_X)$. Then there exists a quasi-isomorphism $\gamma^\bullet : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ where $\mathcal{G}^\bullet$ is a bounded complex of finite locally free $\mathcal{O}_X$-modules such that $\alpha^\bullet \circ \gamma^\bullet$ and $\beta^\bullet \circ \gamma^\bullet$ are homotopic maps of complexes.


perfect-lemma-Noetherian-Kprime

Let $X$ be a Noetherian scheme. Then $$ K_0(\textit{Coh}(\mathcal{O}_X)) = K_0(D^b(\textit{Coh}(\mathcal{O}_X)) = K_0(D^b_{\textit{Coh}}(\mathcal{O}_X)) $$


perfect-lemma-Kprime-K

Let $X$ be a Noetherian regular scheme of finite dimension. Then the map $K_0(X) \to K'_0(X)$ is an isomorphism.


perfect-lemma-K-is-old-K

Let $X$ be a quasi-compact and quasi-separated scheme with the resolution property. Then the map $K_0(\textit{Vect}(X)) \to K_0(X)$ is an isomorphism.


perfect-lemma-projection-formula

Let $f : X \to Y$ be a proper morphism of locally Noetherian schemes. Then we have $f_*(\alpha \cdot f^*\beta) = f_*\alpha \cdot \beta$ for $\alpha \in K'_0(X)$ and $\beta \in K_0(Y)$.


perfect-lemma-determinant-two-term-complexes

Let $X$ be a scheme. There is a functor $$ \det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{with tor amplitude in }[-1, 0] \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} $$ In addition, given a rank $0$ perfect object $L$ of $D(\mathcal{O}_X)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\delta(L) \in \Gamma(X, \det(L))$ such that for any isomorphism $a : L \to K$ in $D(\mathcal{O}_X)$ we have $\det(a)(\delta(L)) = \delta(K)$. Moreover, the construction is affine locally given by the construction of More on Algebra, Section \ref{more-algebra-section-determinants-complexes}.


perfect-lemma-orthogonal-koszul-first-variant

In Situation \ref{situation-complex} denote $j : U \to X$ the open immersion and let $K$ be the perfect object of $D(\mathcal{O}_X)$ corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$. Let $E \in D_\QCoh(\mathcal{O}_X)$ and $a \in \mathbf{Z}$. Consider the following conditions \begin{enumerate} \item The canonical map $\tau_{\geq a}E \to \tau_{\geq a} Rj_*(E|_U)$ is an isomorphism. \item We have $\Hom_{D(\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \geq a$. \end{enumerate} Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a + 1$.


perfect-lemma-orthogonal-koszul-second-variant

In Situation \ref{situation-complex} denote $j : U \to X$ the open immersion and let $K$ be the perfect object of $D(\mathcal{O}_X)$ corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$. Let $E \in D_\QCoh(\mathcal{O}_X)$ and $a \in \mathbf{Z}$. Consider the following conditions \begin{enumerate} \item The canonical map $\tau_{\leq a}E \to \tau_{\leq a} Rj_*(E|_U)$ is an isomorphism, and \item $\Hom_{D(\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \leq a$. \end{enumerate} Then (2) implies (1) and (1) implies (2) with $a$ replaced by $a - 1$.


perfect-lemma-bounded-truncation

Let $X$ be a quasi-compact and quasi-separated scheme. Let $P \in D_{perf}(\mathcal{O}_X)$ and $E \in D_{\QCoh}(\mathcal{O}_X)$.  Let $a \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0$ for $i \gg 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\geq a} E) = 0$ for $i \gg 0$. \end{enumerate}


perfect-lemma-bounded-below-truncation

Let $X$ be a quasi-compact and quasi-separated scheme. Let $P \in D_{perf}(\mathcal{O}_X)$ and $E \in D_{\QCoh}(\mathcal{O}_X)$. Let $a \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $\Hom_{D(\mathcal{O}_X)}(P[-i], E) = 0$ for $i \ll 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[-i], \tau_{\leq a} E) = 0$ for $i \ll 0$. \end{enumerate}


perfect-proposition-quasi-compact-affine-diagonal

Let $X$ be a quasi-compact scheme with affine diagonal. Then the functor (\ref{equation-compare}) $$ D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$.


perfect-proposition-Noetherian

Let $X$ be a Noetherian scheme. Then the functor (\ref{equation-compare}) $$ D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$.


perfect-proposition-represent-cohomology-class-on-open

In Situation \ref{situation-complex}. For every object $E$ of $D_\QCoh(\mathcal{O}_X)$ the map (\ref{equation-comparison}) is an isomorphism.


perfect-proposition-DCoh

Let $X$ be a Noetherian scheme. Then the functors $$ D^-(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^-_{\textit{Coh}}(\mathcal{O}_X) \quad\text{and}\quad D^b(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_X) $$ are equivalences.


perfect-proposition-compact-is-perfect

Let $X$ be a quasi-compact and quasi-separated scheme. An object of $D_\QCoh(\mathcal{O}_X)$ is compact if and only if it is perfect.


perfect-proposition-detecting-bounded-above

Let $X$ be a quasi-compact and quasi-separated scheme. Let $G \in D_{perf}(\mathcal{O}_X)$ be a perfect complex which generates  $D_\QCoh (\mathcal{O}_X)$. Let $E \in D_\QCoh (\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E \in D^-_\QCoh (\mathcal{O}_X)$, \item $\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0$ for $i \gg 0$, \item $\Ext^i_X(G, E) = 0$ for $i \gg 0$, \item $R\Hom_X(G, E)$ is in $D^-(\mathbf{Z})$, \item $H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \gg 0$, \item $R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^-(\mathbf{Z})$, \item for every perfect object $P$ of $D(\mathcal{O}_X)$ \begin{enumerate} \item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \item $H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \gg 0$, \item $R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^-(\mathbf{Z})$. \end{enumerate} \end{enumerate}


perfect-proposition-detecting-bounded-below

Let $X$ be a quasi-compact and quasi-separated scheme. Let $G \in D_{perf}(\mathcal{O}_X)$ be a perfect complex which generates $D_\QCoh (\mathcal{O}_X)$. Let $E \in D_\QCoh (\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $E \in D^+_\QCoh (\mathcal{O}_X)$, \item $\Hom_{D(\mathcal{O}_X)}(G[-i], E) = 0$ for $i \ll 0$, \item $\Ext^i_X(G, E) = 0$ for $i \ll 0$, \item $R\Hom_X(G, E)$ is in $D^+(\mathbf{Z})$, \item $H^i(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \ll 0$, \item $R\Gamma(X, G^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^+(\mathbf{Z})$, \item for every perfect object $P$ of $D(\mathcal{O}_X)$ \begin{enumerate} \item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and \item $H^i(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E) = 0$ for $i \ll 0$, \item $R\Gamma(X, P \otimes_{\mathcal{O}_X}^\mathbf{L} E)$ is in $D^+(\mathbf{Z})$. \end{enumerate} \end{enumerate}


spaces-flat-theorem-finite-type-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $X \to Y$ is locally of finite presentation, \item $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type, and \item the set of weakly associated points of $Y$ is locally finite in $Y$. \end{enumerate} Then $U = \{x \in |X| : \mathcal{F}\text{ flat at }x\text{ over }Y\}$ is open in $X$ and $\mathcal{F}|_U$ is an $\mathcal{O}_U$-module of finite presentation and flat over $Y$.


spaces-flat-theorem-check-flatness-at-associated-points

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 $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $x \in |X|$ with image $y \in |Y|$. Set $F = f^{-1}(\{y\}) \subset |X|$. Consider the conditions \begin{enumerate} \item $\mathcal{F}$ is flat at $x$ over $Y$, and \item for every $x' \in F \cap \text{Ass}_{X/Y}(\mathcal{F})$ which specializes to $x$ we have that $\mathcal{F}$ is flat at $x'$ over $Y$. \end{enumerate} Then we always have (2) $\Rightarrow$ (1). If $X$ and $Y$ are decent, then (1) $\Rightarrow$ (2).


spaces-flat-theorem-flat-dimension-n-representable

In Situation \ref{situation-flat-dimension-n}. Assume moreover that $f$ is of finite presentation, that $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and that $\mathcal{F}$ is pure relative to $Y$. Then $F_n$ is an algebraic space and $F_n \to Y$ is a monomorphism of finite presentation.


spaces-flat-theorem-existence

In Situation \ref{situation-existence} there exists a finitely presented $\mathcal{O}_X$-module $\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\varphi_n$.


spaces-flat-theorem-existence-derived

In Situation \ref{situation-existence-derived} there exists a pseudo-coherent $K$ in $D(\mathcal{O}_X)$ such that $K_n = K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_{X_n}$ for all $n$ compatibly with the maps $\varphi_n$.


spaces-flat-lemma-impure-limit

In Situation \ref{situation-pre-pure}. Let $(g : T \to S, t' \leadsto t, \xi)$ be an impurity of $\mathcal{F}$ above $y$. Assume $T = \lim_{i \in I} T_i$ is a directed limit of affine schemes over $Y$. Then for some $i$ the triple $(T_i \to Y, t'_i \leadsto t_i, \xi_i)$ is an impurity of $\mathcal{F}$ above $y$.


spaces-flat-lemma-flat-ascent-impurity

In Situation \ref{situation-pre-pure}. Let $(Y_1, y_1) \to (Y, y)$ be a morphism of pointed algebraic spaces over $S$. Assume $Y_1 \to Y$ is flat at $y_1$. If $(T \to Y, t' \leadsto t, \xi)$ is an impurity of $\mathcal{F}$ above $y$, then there exists an impurity $(T_1 \to Y_1, t_1' \leadsto t_1, \xi_1)$ of the pullback $\mathcal{F}_1$ of $\mathcal{F}$ to $X_1 = Y_1 \times_Y X$ over $y_1$ such that $T_1$ is \'etale over $Y_1 \times_Y T$.


spaces-flat-lemma-pure-along-X-y

In Situation \ref{situation-pre-pure}. Let $\overline{y}$ be a geometric point lying over $y$. Let $\mathcal{O} = \mathcal{O}_{Y, \overline{y}}$ be the \'etale local ring of $Y$ at $\overline{y}$. Denote $Y^{sh} = \Spec(\mathcal{O})$, $X^{sh} = X \times_Y Y^{sh}$, and $\mathcal{F}^{sh}$ the pullback of $\mathcal{F}$ to $X^{sh}$. The following are equivalent \begin{enumerate} \item there exists an impurity $(Y^{sh} \to Y, y' \leadsto \overline{y}, \xi)$ of $\mathcal{F}$ above $y$, \item every point of $\text{Ass}_{X^{sh}/Y^{sh}}(\mathcal{F}^{sh})$ specializes to a point of the closed fibre $X_{\overline{y}}$, \item there exists an impurity $(T \to Y, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $y$ such that $(T, t) \to (Y, y)$ is an \'etale neighbourhood, and \item there exists an impurity $(T \to Y, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $y$ such that $T \to Y$ is quasi-finite at $t$. \end{enumerate}


spaces-flat-lemma-base-change-universally

In Situation \ref{situation-pre-pure}. \begin{enumerate} \item $\mathcal{F}$ is universally pure above $y$, and \item for every morphism $(Y', y') \to (Y, y)$ of pointed algebraic spaces the pullback $\mathcal{F}_{Y'}$ is pure above $y'$. \end{enumerate} In particular, $\mathcal{F}$ is universally pure relative to $Y$ if and only if every base change $\mathcal{F}_{Y'}$ of $\mathcal{F}$ is pure relative to $Y'$.


spaces-flat-lemma-quasi-finite-base-change

In Situation \ref{situation-pre-pure}. Let $(Y', y') \to (Y, y)$ be a morphism of pointed algebraic spaces. If $Y' \to Y$ is quasi-finite at $y'$ and $\mathcal{F}$ is pure above $y$, then $\mathcal{F}_{Y'}$ is pure above $y'$.


spaces-flat-lemma-flat-descend-pure

In Situation \ref{situation-pre-pure}. Let $(Y_1, y_1) \to (Y, y)$ be a morphism of pointed algebraic spaces. Assume $Y_1 \to Y$ is flat at $y_1$. \begin{enumerate} \item If $\mathcal{F}_{Y_1}$ is pure above $y_1$, then $\mathcal{F}$ is pure above $y$. \item If $\mathcal{F}_{Y_1}$ is universally pure above $y_1$, then $\mathcal{F}$ is universally pure above $y$. \end{enumerate}


spaces-flat-lemma-supported-on-closed

In Situation \ref{situation-pre-pure}. Let $i : Z \to X$ be a closed immersion and assume that $\mathcal{F} = i_*\mathcal{G}$ for some finite type, quasi-coherent sheaf $\mathcal{G}$ on $Z$. Then $\mathcal{G}$ is (universally) pure above $y$ if and only if $\mathcal{F}$ is (universally) pure above $y$.


spaces-flat-lemma-proper-pure

In Situation \ref{situation-pre-pure}. \begin{enumerate} \item If the support of $\mathcal{F}$ is proper over $Y$, then $\mathcal{F}$ is universally pure relative to $Y$. \item If $f$ is proper, then $\mathcal{F}$ is universally pure relative to $Y$. \item If $f$ is proper, then $X$ is universally pure relative to $Y$. \end{enumerate}


spaces-flat-lemma-existence-complete

Let $S$ be a scheme. Let $X \to Y$ be a finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $y \in |Y|$ be a point. There exists an \'etale morphism $(Y', y') \to (Y, y)$ with $Y'$ an affine scheme and \'etale morphisms $h_i : W_i \to X_{Y'}$, $i = 1, \ldots, n$ such that for each $i$ there exists a complete d\'evissage of $\mathcal{F}_i/W_i/Y'$ over $y'$, where $\mathcal{F}_i$ is the pullback of $\mathcal{F}$ to $W_i$ and such that $|(X_{Y'})_{y'}| \subset \bigcup h_i(W_i)$.


spaces-flat-lemma-bourbaki-finite-type-general-base-at-point

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 $x \in |X|$ with image $y \in |Y|$. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. If $\mathcal{F}$ is flat at $x$ over $Y$, then $$ x \in \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}_Y(\mathcal{G}) \text{ and } x \in \text{Ass}_{X/Y}(\mathcal{F}). $$


spaces-flat-lemma-bourbaki-finite-type-general-base

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 $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$ which is flat over $Y$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then we have $$ \text{WeakAss}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \text{Ass}_{X/Y}(\mathcal{F}) \cap |f|^{-1}(\text{WeakAss}_Y(\mathcal{G})) $$


spaces-flat-lemma-finite-type-flat-along-fibre-free-variant

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $y \in |Y|$. Set $F = f^{-1}(\{y\}) \subset |X|$. Assume that \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat over $Y$ at all $x \in F$. \end{enumerate} Then there exists an \'etale morphism $(Y', y') \to (Y, y)$ where $Y'$ is a scheme and a commutative diagram of algebraic spaces $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ Y & \Spec(\mathcal{O}_{Y', y'}) \ar[l] } $$ such that $X' \to X \times_Y \Spec(\mathcal{O}_{Y', y'})$ is \'etale, $|X'_{y'}| \to F$ is surjective, $X'$ is affine, and $\Gamma(X', g^*\mathcal{F})$ is a free $\mathcal{O}_{Y', y'}$-module.


spaces-flat-lemma-finite-presentation-flat-along-fibre

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $y \in |Y|$. Set $F = f^{-1}(\{y\}) \subset |X|$. Assume that \begin{enumerate} \item $f$ is of finite presentation, \item $\mathcal{F}$ is of finite presentation, and \item $\mathcal{F}$ is flat over $Y$ at all $x \in F$. \end{enumerate} Then there exists a commutative diagram of algebraic spaces $$ \xymatrix{ X \ar[d] & X' \ar[l]^g \ar[d] \\ Y & Y' \ar[l]_h } $$ such that $h$ and $g$ are \'etale, there is a point $y' \in |Y'|$ mapping to $y$, we have $F \subset g(|X'|)$, the algebraic spaces $X'$, $Y'$ are affine, and $\Gamma(X', g^*\mathcal{F})$ is a projective $\Gamma(Y', \mathcal{O}_{Y'})$-module.


spaces-flat-lemma-associated-point-specializes

Let $S$ be a scheme. Let $X$ be a decent algebraic space locally of finite type over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$ such that $\mathcal{F}$ is flat over $S$ at all points of $X_s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$. If the closure of $\{x'\}$ in $|X|$ meets $|X_s|$, then the closure meets $\text{Ass}_{X/S}(\mathcal{F}) \cap |X_s|$.


spaces-flat-lemma-contains-relative-ass-after-base-change

Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be a morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$, then for any morphism $Z \to Y$ we have $\text{Ass}_{X_Z/Z}(\mathcal{F}_Z) \subset g_Z(|X'_Z|)$.


spaces-flat-lemma-pure-on-top

Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be an \'etale morphism of algebraic spaces over $Y$. Assume the structure morphisms $X' \to Y$ and $X \to Y$ are decent and of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $y \in |Y|$. Set $F = f^{-1}(\{y\}) \subset |X|$. \begin{enumerate} \item If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$ and $g^*\mathcal{F}$ is (universally) pure above $y$, then $\mathcal{F}$ is (universally) pure above $y$. \item If $\mathcal{F}$ is pure above $y$, $g(|X'|)$ contains $F$, and $Y$ is affine local with closed point $y$, then $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$. \item If $\mathcal{F}$ is pure above $y$, $\mathcal{F}$ is flat at all points of $F$, $g(|X'|)$ contains $\text{Ass}_{X/Y}(\mathcal{F}) \cap F$, and $Y$ is affine local with closed point $y$, then $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$. \item Add more here. \end{enumerate}


spaces-flat-lemma-finite-type-flat-pure-along-fibre-is-universal

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $y \in |Y|$. Assume \begin{enumerate} \item $f$ is decent and of finite type, \item $\mathcal{F}$ is of finite type, \item $\mathcal{F}$ is flat over $Y$ at all points lying over $y$, and \item $\mathcal{F}$ is pure above $y$. \end{enumerate} Then $\mathcal{F}$ is universally pure above $y$.


spaces-flat-lemma-finite-type-flat-pure-is-universal

Let $S$ be a scheme. Let $f : X \to Y$ be a decent, finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Assume $\mathcal{F}$ is flat over $Y$. In this case $\mathcal{F}$ is pure relative to $Y$ if and only if $\mathcal{F}$ is universally pure relative to $Y$.


spaces-flat-lemma-universally-separating

Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \to X$ be a flat morphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module which is flat over $Y$. If $\text{Ass}_{X/Y}(\mathcal{F}) \subset g(|X'|)$ then the canonical map $$ \mathcal{F} \longrightarrow g_*g^*\mathcal{F} $$ is injective, and remains injective after any base change.


spaces-flat-lemma-iso-sheaf

In Situation \ref{situation-iso}. Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology.


spaces-flat-lemma-iso-go-up

In Situation \ref{situation-iso} let $X' \to X$ be a flat morphism of algebraic spaces. Denote $u' : \mathcal{F}' \to \mathcal{G}'$ the pullback of $u$ to $X'$. Denote $F'_{iso}$, $F'_{inj}$, $F'_{surj}$, $F'_{zero}$ the functors on $\Sch/B$ associated to $u'$. \begin{enumerate} \item If $\mathcal{G}$ is of finite type and the image of $|X'| \to |X|$ contains the support of $\mathcal{G}$, then $F_{surj} = F'_{surj}$ and $F_{zero} = F'_{zero}$. \item If $\mathcal{F}$ is of finite type and the image of $|X'| \to |X|$ contains the support of $\mathcal{F}$, then $F_{inj} = F'_{inj}$ and $F_{zero} = F'_{zero}$. \item If $\mathcal{F}$ and $\mathcal{G}$ are of finite type and the image of $|X'| \to |X|$ contains the supports of $\mathcal{F}$ and $\mathcal{G}$, then $F_{iso} = F'_{iso}$. \end{enumerate}


spaces-flat-lemma-iso-limits

In Situation \ref{situation-iso}. \begin{enumerate} \item If $\mathcal{G}$ is of finite type and the scheme theoretic support of $\mathcal{G}$ is quasi-compact over $B$, then $F_{surj}$ is limit preserving. \item If $\mathcal{F}$ of finite type and the scheme theoretic support of $\mathcal{F}$ is quasi-compact over $B$, then $F_{zero}$ is limit preserving. \item If $\mathcal{F}$ is of finite type, $\mathcal{G}$ is of finite presentation, and the scheme theoretic supports of $\mathcal{F}$ and $\mathcal{G}$ are quasi-compact over $B$, then $F_{iso}$ is limit preserving. \end{enumerate}


spaces-flat-lemma-relate-zero-iso

In Situation \ref{situation-iso} suppose given an exact sequence $$ \mathcal{F} \xrightarrow{u} \mathcal{G} \xrightarrow{v} \mathcal{H} \to 0 $$ Then we have $F_{v, iso} = F_{u, zero}$ with obvious notation.


spaces-flat-lemma-relate-zero-affine

In Situation \ref{situation-iso} suppose given an affine morphism $i : Z \to X$ and a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{H}$ such that $\mathcal{G} = i_*\mathcal{H}$. Let $v : i^*\mathcal{F} \to \mathcal{H}$ be the map adjoint to $u$. Then \begin{enumerate} \item $F_{v, zero} = F_{u, zero}$, and \item if $i$ is a closed immersion, then $F_{v, surj} = F_{u, surj}$. \end{enumerate}


spaces-flat-lemma-F-zero-somewhat-closed

In Situation \ref{situation-somewhat-closed}. Let $T \to S$ be a quasi-compact morphism of schemes such that the base change $u_T$ is zero. Then exists a closed subscheme $Z \subset S$ such that (a) $T \to S$ factors through $Z$ and (b) the base change $u_Z$ is zero. If $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module and the scheme theoretic support of $\mathcal{F}$ is quasi-compact, then we can take $Z \to S$ of finite presentation.


spaces-flat-lemma-F-zero-module-map

Let $A$ be a ring. Let $u : M \to N$ be a map of $A$-modules. If $N$ is projective as an $A$-module, then there exists an ideal $I \subset A$ such that for any ring map $\varphi : A \to B$ the following are equivalent \begin{enumerate} \item $u \otimes 1 : M \otimes_A B \to N \otimes_A B$ is zero, and \item $\varphi(I) = 0$. \end{enumerate}


spaces-flat-lemma-F-zero-closed-pure

In Situation \ref{situation-iso}. Assume \begin{enumerate} \item $f$ is of finite presentation, and \item $\mathcal{G}$ is of finite presentation, flat over $B$, and pure relative to $B$. \end{enumerate} Then $F_{zero}$ is an algebraic space and $F_{zero} \to B$ is a closed immersion. If $\mathcal{F}$ is of finite type, then $F_{zero} \to B$ is of finite presentation.


spaces-flat-lemma-F-zero-closed-proper

In Situation \ref{situation-iso}. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{G}$ is an $\mathcal{O}_X$-module of finite presentation flat over $B$, \item the support of $\mathcal{G}$ is proper over $B$. \end{enumerate} Then the functor $F_{zero}$ is an algebraic space and $F_{zero} \to B$ is a closed immersion. If $\mathcal{F}$ is of finite type, then $F_{zero} \to B$ is of finite presentation.


spaces-flat-lemma-F-surj-open

In Situation \ref{situation-iso}. Assume \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{G}$ is of finite type, \item the support of $\mathcal{G}$ is proper over $B$. \end{enumerate} Then $F_{surj}$ is an algebraic space and $F_{surj} \to B$ is an open immersion.


spaces-flat-lemma-pre-flat-dimension-n

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Let $n \geq 0$. The following are equivalent \begin{enumerate} \item for some commutative diagram $$ \xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with surjective, \'etale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$ (More on Flatness, Definition \ref{flat-definition-flat-dimension-n}), \item for every commutative diagram $$ \xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y } $$ with \'etale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$, and \item for $x \in |X|$ such that $\mathcal{F}$ is not flat at $x$ over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}). \end{enumerate} If this is true, then it remains true after any base change $Y' \to Y$.


spaces-flat-lemma-flat-dimension-n

In Situation \ref{situation-flat-dimension-n}. \begin{enumerate} \item The functor $F_n$ satisfies the sheaf property for the fpqc topology. \item If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_n$ is limit preserving. \end{enumerate}


spaces-flat-lemma-localize-flat-dimension-n

In Situation \ref{situation-flat-dimension-n}. Let $h : X' \to X$ be an \'etale morphism. Set $\mathcal{F}' = h^*\mathcal{F}$ and $f' = f \circ h$. Let $F_n'$ be (\ref{equation-flat-dimension-n}) associated to $(f' : X' \to Y, \mathcal{F}')$. Then $F_n$ is a subfunctor of $F_n'$ and if $h(X') \supset \text{Ass}_{X/Y}(\mathcal{F})$, then $F_n = F'_n$.


spaces-flat-lemma-when-universal-flattening

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $f$ is of finite presentation, $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, and $\mathcal{F}$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation. \item If $f$ is of finite presentation and $X$ is pure relative to $Y$, then there exists a universal flattening $Y' \to Y$ of $X$. Moreover $Y' \to Y$ is a monomorphism of finite presentation. \item If $f$ is proper and of finite presentation and $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite presentation, then there exists a universal flattening $Y' \to Y$ of $\mathcal{F}$. Moreover $Y' \to Y$ is a monomorphism of finite presentation. \item If $f$ is proper and of finite presentation then there exists a universal flattening $Y' \to Y$ of $X$. \end{enumerate}


spaces-flat-lemma-compute-what-it-should-be

In Situation \ref{situation-existence} consider $$ K = R\lim_{D_\QCoh(\mathcal{O}_X)}(\mathcal{F}_n) = DQ_X(R\lim_{D(\mathcal{O}_X)}\mathcal{F}_n) $$ Then $K$ is in $D^b_{\QCoh}(\mathcal{O}_X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\geq 0$.


spaces-flat-lemma-compute-against-perfect

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. For any perfect object $E$ of $D(\mathcal{O}_X)$ we have \begin{enumerate} \item $M = R\Gamma(X, K \otimes^\mathbf{L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Gamma(X_n, \mathcal{F}_n \otimes^\mathbf{L} E|_{X_n}) = M \otimes_A^\mathbf{L} A_n$ in $D(A_n)$, \item $N = R\Hom_X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Hom_{X_n}(E|_{X_n}, \mathcal{F}_n) = N \otimes_A^\mathbf{L} A_n$ in $D(A_n)$. \end{enumerate} In both statements $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.


spaces-flat-lemma-relative-pseudo-coherence

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Then $K$ is pseudo-coherent relative to $A$.


spaces-flat-lemma-compute-over-affine

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. For any \'etale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have $$ R\Gamma(U, K) \otimes_A^\mathbf{L} A_n = R\Gamma(U_n, \mathcal{F}_n) $$ in $D(A_n)$ where $U_n = U \times_X X_n$.


spaces-flat-lemma-finitely-presented

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Denote $X_0 \subset |X|$ the closed subset consisting of points lying over the closed subset $\Spec(A_1) = \Spec(A_2) = \ldots$ of $\Spec(A)$. There exists an open subspace $W \subset X$ containing $X_0$ such that \begin{enumerate} \item $H^i(K)|_W$ is zero unless $i = 0$, \item $\mathcal{F} = H^0(K)|_W$ is of finite presentation, and \item $\mathcal{F}_n = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_{X_n}$. \end{enumerate}


spaces-flat-lemma-proper-support

In Situation \ref{situation-existence} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be}. Let $W \subset X$ be as in Lemma \ref{lemma-finitely-presented}. Set $\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$, the support of $\mathcal{F}$ is proper over $A$.


spaces-flat-lemma-compute-what-it-should-be-derived

In Situation \ref{situation-existence-derived} consider $$ K = R\lim_{D_\QCoh(\mathcal{O}_X)}(K_n) = DQ_X(R\lim_{D(\mathcal{O}_X)} K_n) $$ Then $K$ is in $D^-_{\QCoh}(\mathcal{O}_X)$.


spaces-flat-lemma-compute-against-perfect-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. For any perfect object $E$ of $D(\mathcal{O}_X)$ the cohomology $$ M = R\Gamma(X, K \otimes^\mathbf{L} E) $$ is a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism $$ R\Gamma(X_n, K_n \otimes^\mathbf{L} E|_{X_n}) = M \otimes_A^\mathbf{L} A_n $$ in $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.


spaces-flat-lemma-relative-pseudo-coherence-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. Then $K$ is pseudo-coherent on $X$.


spaces-flat-lemma-compute-over-affine-derived

In Situation \ref{situation-existence-derived} let $K$ be as in Lemma \ref{lemma-compute-what-it-should-be-derived}. For any \'etale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have $$ R\Gamma(U, K) \otimes_A^\mathbf{L} A_n = R\Gamma(U_n, K_n) $$ in $D(A_n)$ where $U_n = U \times_X X_n$.


spaces-flat-proposition-finite-presentation-flat-at-point

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$ with image $y \in |Y|$. Assume that \begin{enumerate} \item $f$ is locally of finite presentation, \item $\mathcal{F}$ is of finite presentation, and \item $\mathcal{F}$ is flat at $x$ over $Y$. \end{enumerate} Then there exists a commutative diagram of pointed schemes $$ \xymatrix{ (X, x) \ar[d] & (X', x') \ar[l]^g \ar[d] \\ (Y, y) & (Y', y') \ar[l] } $$ whose horizontal arrows are \'etale such that $X'$, $Y'$ are affine and such that $\Gamma(X', g^*\mathcal{F})$ is a projective $\Gamma(Y', \mathcal{O}_{Y'})$-module.


spaces-chow-lemma-length

Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. Let $d \in \{0, 1, 2, \ldots, \infty\}$. The following are equivalent \begin{enumerate} \item $\text{length}_{\mathcal{O}_{X, \overline{x}}} \mathcal{F}_{\overline{x}} = d$ \item for some \'etale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_U)_u = d$ \item for any \'etale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_U)_u = d$ \end{enumerate}


spaces-chow-lemma-length-closed-immersion

Let $S$ be a scheme. Let $i : Y \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $y \in |Y|$ with image $x \in |X|$. Let $d \in \{0, 1, 2, \ldots, \infty\}$. The following are equivalent \begin{enumerate} \item $\mathcal{G}$ has length $d$ at $y$, and \item $i_*\mathcal{G}$ has length $d$ at $x$. \end{enumerate}


spaces-chow-lemma-length-finite

Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item for some \'etale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $u$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F}|_U)$, \item for any \'etale morphism $U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$ we have $u$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F}|_U)$, \item the length of $\mathcal{F}$ at $x$ is finite and nonzero. \end{enumerate} If $X$ is decent (equivalently quasi-separated) then these are also equivalent to \begin{enumerate} \item[(4)] $x$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F})$. \end{enumerate}


spaces-chow-lemma-point-of-max-dimension

In Situation \ref{situation-setup} let $X/B$ be good. Let $T \subset |X|$ be a closed subset and $t \in T$. If $\dim_\delta(T) \leq k$ and $\delta(t) = k$, then $t$ is a generic point of an irreducible component of $T$.


spaces-chow-lemma-reformulate-coeff-coherent

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module with $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Let $Z$ be an integral closed subspace of $X$ with $\dim_\delta(Z) = k$. Let $\xi \in |Z|$ be the generic point. Then the coefficient of $Z$ in $[\mathcal{F}]_k$ is the length of $\mathcal{F}$ at $\xi$.


spaces-chow-lemma-cycle-closed-coherent

In Situation \ref{situation-setup} let $X/B$ be good. Let $Y \subset X$ be a closed subspace. If $\dim_\delta(Y) \leq k$, then $[Y]_k = [i_*\mathcal{O}_Y]_k$ where $i : Y \to X$ is the inclusion morphism.


spaces-chow-lemma-proper-image

In Situation \ref{situation-setup} let $X,Y/B$ be good and let $f : X \to Y$ be a morphism over $B$. If $Z \subset X$ is an integral closed subspace, then there exists a unique integral closed subspace $Z' \subset Y$ such that there is a commutative diagram $$ \xymatrix{ Z \ar[r] \ar[d] & X \ar[d]^f \\ Z' \ar[r] & Y } $$ with $Z \to Z'$ dominant. If $f$ is proper, then $Z \to Z'$ is proper and surjective.


spaces-chow-lemma-equal-dimension

In Situation \ref{situation-setup} let $X, Y/B$ be good and let $f : X \to Y$ be a morphism over $B$. Assume $X$, $Y$ integral and $\dim_\delta(X) = \dim_\delta(Y)$. Then either $f$ factors through a proper closed subspace of $Y$, or $f$ is dominant and the extension of function fields $R(X) / R(Y)$ is finite.


spaces-chow-lemma-quasi-compact-locally-finite

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is quasi-compact, and $\{T_i\}_{i \in I}$ is a locally finite collection of closed subsets of $|X|$. Then $\{\overline{|f|(T_i)}\}_{i \in I}$ is a locally finite collection of closed subsets of $|Y|$.


spaces-chow-lemma-compose-pushforward

In Situation \ref{situation-setup} let $X, Y, Z/B$ be good. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms over $B$. Then $g_* \circ f_* = (g \circ f)_*$ as maps $Z_k(X) \to Z_k(Z)$.


spaces-chow-lemma-cycle-push-sheaf

In Situation \ref{situation-setup} let $f : X \to Y$ be a proper morphism of good algebraic spaces over $B$. \begin{enumerate} \item Let $Z \subset X$ be a closed subspace with $\dim_\delta(Z) \leq k$. Then $$ f_*[Z]_k = [f_*{\mathcal O}_Z]_k. $$ \item Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then $$ f_*[\mathcal{F}]_k = [f_*{\mathcal F}]_k. $$ \end{enumerate} Note that the statement makes sense since $f_*\mathcal{F}$ and $f_*\mathcal{O}_Z$ are coherent $\mathcal{O}_Y$-modules by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent}.


spaces-chow-lemma-flat-inverse-image-dimension

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $f$ is flat of relative dimension $r$. For any closed subset $T \subset |Y|$ we have $$ \dim_\delta(|f|^{-1}(T)) = \dim_\delta(T) + r. $$ provided $|f|^{-1}(T)$ is nonempty. If $Z \subset Y$ is an integral closed subscheme and $Z' \subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\dim_\delta(Z') = \dim_\delta(Z) + r$.


spaces-chow-lemma-inverse-image-locally-finite

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a morphism over $B$. Assume $\{T_i\}_{i \in I}$ is a locally finite collection of closed subsets of $|Y|$. Then $\{|f|^{-1}(T_i)\}_{i \in I}$ is a locally finite collection of closed subsets of $X$.


spaces-chow-lemma-exact-sequence-open

In Situation \ref{situation-setup} let $X/B$ be good. Let $U \subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \setminus |U|$ and denote $i : Y \to X$ the inclusion morphism. For every $k \in \mathbf{Z}$ the sequence $$ \xymatrix{ Z_k(Y) \ar[r]^{i_*} & Z_k(X) \ar[r]^{j^*} & Z_k(U) \ar[r] & 0 } $$ is an exact complex of abelian groups.


spaces-chow-lemma-etale-pullback

In Situation \ref{situation-setup} let $f : X \to Y$ be an \'etale morphism of good algebraic spaces over $B$. If $Z \subset Y$ is an integral closed subspace, then $f^*[Z] = \sum [Z']$ where the sum is over the irreducible components (Remark \ref{remark-irreducible-component}) of $f^{-1}(Z)$.


spaces-chow-lemma-compose-flat-pullback

In Situation \ref{situation-setup} let $X, Y, Z/B$ be good. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of relative dimensions $r$ and $s$ over $B$. Then $g \circ f$ is flat of relative dimension $r + s$ and $$ f^* \circ g^* = (g \circ f)^* $$ as maps $Z_k(Z) \to Z_{k + r + s}(X)$.


spaces-chow-lemma-pullback-coherent

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. \begin{enumerate} \item Let $Z \subset Y$ be a closed subspace with $\dim_\delta(Z) \leq k$. Then we have $\dim_\delta(f^{-1}(Z)) \leq k + r$ and $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$. \item Let $\mathcal{F}$ be a coherent sheaf on $Y$ with $\dim_\delta(\text{Supp}(\mathcal{F})) \leq k$. Then we have $\dim_\delta(\text{Supp}(f^*\mathcal{F})) \leq k + r$ and $$ f^*[{\mathcal F}]_k = [f^*{\mathcal F}]_{k+r} $$ in $Z_{k + r}(X)$. \end{enumerate}


spaces-chow-lemma-flat-pullback-proper-pushforward

In Situation \ref{situation-setup} let $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ be a fibre product diagram of good algebraic spaces over $B$. Assume $f : X \to Y$ proper and $g : Y' \to Y$ flat of relative dimension $r$. Then also $f'$ is proper and $g'$ is flat of relative dimension $r$. For any $k$-cycle $\alpha$ on $X$ we have $$ g^*f_*\alpha = f'_*(g')^*\alpha $$ in $Z_{k + r}(Y')$.


spaces-chow-lemma-divisor-delta-dimension

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ is integral. \begin{enumerate} \item If $Z \subset X$ is an integral closed subspace, then the following are equivalent: \begin{enumerate} \item $Z$ is a prime divisor, \item $|Z|$ has codimension $1$ in $|X|$, and \item $\dim_\delta(Z) = \dim_\delta(X) - 1$. \end{enumerate} \item If $Z$ is an irreducible component of an effective Cartier divisor on $X$, then $\dim_\delta(Z) = \dim_\delta(X) - 1$. \end{enumerate}


spaces-chow-lemma-etale-pullback-principal-divisor

In Situation \ref{situation-setup} let $f : X \to Y$ be an \'etale morphism of good algebraic spaces over $B$. Assume $Y$ is integral. Let $g \in R(Y)^*$. As cycles on $X$ we have $$ f^*(\text{div}_Y(g)) = \sum\nolimits_{X'} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'}) $$ where the sum is over the irreducible components of $X$ (Remark \ref{remark-irreducible-component}).


spaces-chow-lemma-proper-pushforward-alteration

In Situation \ref{situation-setup} let $X, Y/B$ be good. Assume $X$, $Y$ are integral and $n = \dim_\delta(X) = \dim_\delta(Y)$. Let $p : X \to Y$ be a dominant proper morphism. Let $f \in R(X)^*$. Set $$ g = \text{Nm}_{R(X)/R(Y)}(f). $$ Then we have $p_*\text{div}(f) = \text{div}(g)$.


spaces-chow-lemma-restrict-to-open

In Situation \ref{situation-setup} let $X/B$ be good. Let $U \subset X$ be an open subspace. Let $Y$ be the reduced closed subspace of $X$ with $|Y| = |X| \setminus |U|$ and denote $i : Y \to X$ the inclusion morphism. Let $k \in \mathbf{Z}$. Suppose $\alpha, \beta \in Z_k(X)$. If $\alpha|_U \sim_{rat} \beta|_U$ then there exist a cycle $\gamma \in Z_k(Y)$ such that $$ \alpha \sim_{rat} \beta + i_*\gamma. $$ In other words, the sequence $$ \xymatrix{ \CH_k(Y) \ar[r]^{i_*} & \CH_k(X) \ar[r]^{j^*} & \CH_k(U) \ar[r] & 0 } $$ is an exact complex of abelian groups.


spaces-chow-lemma-prepare-flat-pullback-rational-equivalence

In Situation \ref{situation-setup} let $X, Y/B$ be good. Assume $Y$ integral with $\dim_\delta(Y) = k$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Then for $g \in R(Y)^*$ we have $$ f^*\text{div}_Y(g) = \sum m_{X', X} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'}) $$ as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible components $X'$ of $X$ and $m_{X', X}$ is the multiplicity of $X'$ in $X$.


spaces-chow-lemma-flat-pullback-rational-equivalence

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha \sim_{rat} \beta$ be rationally equivalent $k$-cycles on $Y$. Then $f^*\alpha \sim_{rat} f^*\beta$ as $(k + r)$-cycles on $X$.


spaces-chow-lemma-proper-pushforward-rational-equivalence

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $p : X \to Y$ be a proper morphism. Suppose $\alpha, \beta \in Z_k(X)$ are rationally equivalent. Then $p_*\alpha$ is rationally equivalent to $p_*\beta$.


spaces-chow-lemma-compute-c1

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ is integral and $n = \dim_\delta(X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$ be a nonzero global section. Then $$ \text{div}_\mathcal{L}(s) = [Z(s)]_{n - 1} $$ in $Z_{n - 1}(X)$ and $$ c_1(\mathcal{L}) \cap [X] = [Z(s)]_{n - 1} $$ in $\CH_{n - 1}(X)$.


spaces-chow-lemma-Gm-torsor

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The morphism $$ q : T = \underline{\Spec}\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\right) \longrightarrow X $$ has the following properties: \begin{enumerate} \item $q$ is surjective, smooth, affine, of relative dimension $1$, \item there is an isomorphism $\alpha : q^*\mathcal{L} \cong \mathcal{O}_T$, \item formation of $(q : T \to X, \alpha)$ commutes with base change, \item $q^* : Z_k(X) \to Z_{k + 1}(T)$ is injective, \item if $Z \subset X$ is an integral closed subspace, then $q^{-1}(Z) \subset T$ is an integral closed subspace, \item if $Z \subset X$ is a closed subspace of $X$ of $\delta$-dimension $\leq k$, then $q^{-1}(Z)$ is a closed subspace of $T$ of $\delta$-dimension $\leq k + 1$ and $q^*[Z]_k = [q^{-1}(Z)]_{k + 1}$, \item if $\xi' \in |T|$ is the generic point of the fibre of $|T| \to |X|$ over $\xi$, then the ring map $\mathcal{O}_{X, \xi}^h \to \mathcal{O}_{T, \xi'}^h$ is flat, we have $\mathfrak m_{\xi'}^h = \mathfrak m_\xi^h \mathcal{O}_{T, \xi'}^h$, and the residue field extension is purely transcendental of transcendence degree $1$, and \item add more here as needed. \end{enumerate}


spaces-chow-lemma-c1-cap-additive

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$, $\mathcal{N}$ be an invertible sheaves on $X$. Then $$ c_1(\mathcal{L}) \cap \alpha  + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}) \cap \alpha $$ in $\CH_k(X)$ for every $\alpha \in Z_{k - 1}(X)$. Moreover, $c_1(\mathcal{O}_X) \cap \alpha = 0$ for all $\alpha$.


spaces-chow-lemma-prepare-geometric-cap

In Situation \ref{situation-setup} let $Y/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module. Let $s \in \Gamma(Y, \mathcal{L})$ be a regular section and assume $\dim_\delta(Y) \leq k + 1$. Write $[Y]_{k + 1} = \sum n_i[Y_i]$ where $Y_i \subset Y$ are the irreducible components of $Y$ of $\delta$-dimension $k + 1$. Set $s_i = s|_{Y_i} \in \Gamma(Y_i, \mathcal{L}|_{Y_i})$. Then \begin{equation} \label{equation-equal-as-cycles} [Z(s)]_k =  \sum n_i[Z(s_i)]_k \end{equation} as $k$-cycles on $Y$.


spaces-chow-lemma-geometric-cap

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y \subset X$ be a closed subscheme with $\dim_\delta(Y) \leq k + 1$ and let $s \in \Gamma(Y, \mathcal{L}|_Y)$ be a regular section. Then $$ c_1(\mathcal{L}) \cap [Y]_{k + 1} = [Z(s)]_k $$ in $\CH_k(X)$.


spaces-chow-lemma-prepare-flat-pullback-cap-c1

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \dim_\delta(Y)$. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Then we have $$ f^*\text{div}_\mathcal{L}(s) = \sum n_i\text{div}_{f^*\mathcal{L}|_{X_i}}(s_i) $$ in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_i \subset X$ of $\delta$-dimension $n + r$, the section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and $n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$.


spaces-chow-lemma-flat-pullback-cap-c1

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $\alpha$ be a $k$-cycle on $Y$. Then $$ f^*(c_1(\mathcal{L}) \cap \alpha) = c_1(f^*\mathcal{L}) \cap f^*\alpha $$ in $\CH_{k + r - 1}(X)$.


spaces-chow-lemma-equal-c1-as-cycles

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a proper morphism. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Assume $X$, $Y$ integral, $f$ dominant, and $\dim_\delta(X) = \dim_\delta(Y)$. Let $s$ be a nonzero meromorphic section $s$ of $\mathcal{L}$ on $Y$. Then $$ f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) = [R(X) : R(Y)]\text{div}_\mathcal{L}(s). $$ as cycles on $Y$. In particular $$ f_*(c_1(f^*\mathcal{L}) \cap [X]) = c_1(\mathcal{L}) \cap f_*[Y]. $$


spaces-chow-lemma-pushforward-cap-c1

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $p : X \to Y$ be a proper morphism. Let $\alpha \in Z_{k + 1}(X)$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ p_*(c_1(p^*\mathcal{L}) \cap \alpha) = c_1(\mathcal{L}) \cap p_*\alpha $$ in $\CH_k(Y)$.


spaces-chow-lemma-key-formula

In the situation above the cycle $$ \sum (Z_i \to X)_*\left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \right) $$ is equal to the cycle $$ \sum (Z_i \to X)_*\text{div}(\partial_{B_i}(f_i, g_i)) $$


spaces-chow-lemma-commutativity-on-integral

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ integral and $\dim_\delta(X) = n$. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}$ and a nonzero meromorphic section $t$ of $\mathcal{N}$. Set $\alpha = \text{div}_\mathcal{L}(s)$ and $\beta = \text{div}_\mathcal{N}(t)$. Then $$ c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L}) \cap \beta $$ in $\CH_{n - 2}(X)$.


spaces-chow-lemma-factors

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be invertible on $X$. The operation $\alpha \mapsto c_1(\mathcal{L}) \cap \alpha$ factors through rational equivalence to give an operation $$ c_1(\mathcal{L}) \cap - : \CH_{k + 1}(X) \to \CH_k(X) $$


spaces-chow-lemma-cap-commutative

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$, $\mathcal{N}$ be invertible on $X$. For any $\alpha \in \CH_{k + 2}(X)$ we have $$ c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha $$ as elements of $\CH_k(X)$.


spaces-chow-lemma-support-cap-effective-Cartier

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Let $\alpha$ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\alpha = c_1(\mathcal{L}) \cap \alpha$ in $\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \cdot \alpha = c_1(\mathcal{O}_X(D)) \cap \alpha$.


spaces-chow-lemma-closed-in-X-gysin

In Situation \ref{situation-setup}. Let $f : X' \to X$ be a proper morphism of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Form the diagram $$ \xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X } $$ as in Remark \ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\alpha'$ on $X'$ we have $i^*f_*\alpha' = g_*(i')^*\alpha'$ in $\CH_k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).


spaces-chow-lemma-gysin-flat-pullback

In Situation \ref{situation-setup}. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Form the diagram $$ \xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X } $$ as in Remark \ref{remark-pullback-pairs}. For any $(k + 1)$-cycle $\alpha$ on $X$ we have $(i')^*f^*\alpha = g^*i^*\alpha'$ in $\CH_{k + r}(D)$ (this makes sense as $f^*$ is defined on the level of cycles).


spaces-chow-lemma-easy-gysin

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. Let $Z \subset X$ be a closed subscheme such that $\dim_\delta(Z) \leq k + 1$ and such that $D \cap Z$ is an effective Cartier divisor on $Z$. Then $i^*([Z]_{k + 1}) = [D \cap Z]_k$.


spaces-chow-lemma-gysin-factors-general

In Situation \ref{situation-setup} let $X/B$ be good. Assume $X$ integral and $n = \dim_\delta(X)$. Let $i : D \to X$ be an effective Cartier divisor. Let $\mathcal{N}$ be an invertible $\mathcal{O}_X$-module and let $t$ be a nonzero meromorphic section of $\mathcal{N}$. Then $i^*\text{div}_\mathcal{N}(t) = c_1(\mathcal{N}) \cap [D]_{n - 1}$ in $\CH_{n - 2}(D)$.


spaces-chow-lemma-gysin-factors

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition \ref{definition-gysin-homomorphism}. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \CH_{k + 1}(X) \to \CH_k(D)$.


spaces-chow-lemma-gysin-commutes-cap-c1

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition \ref{definition-gysin-homomorphism}. Let $\mathcal{N}$ be an invertible $\mathcal{O}_X$-module. Then $i^*(c_1(\mathcal{N}) \cap \alpha) = c_1(i^*\mathcal{N}) \cap i^*\alpha$ in $\CH_{k - 2}(D)$ for all $\alpha \in \CH_k(Z)$.


spaces-chow-lemma-gysin-commutes-gysin

In Situation \ref{situation-setup} let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ and $(\mathcal{L}', s', i' : D' \to X)$ be two triples as in Definition \ref{definition-gysin-homomorphism}. Then the diagram $$ \xymatrix{ \CH_k(X) \ar[r]_{i^*} \ar[d]_{(i')^*} & \CH_{k - 1}(D) \ar[d] \\ \CH_{k - 1}(D') \ar[r] & \CH_{k - 2}(D \cap D') } $$ commutes where each of the maps is a gysin map.


spaces-chow-lemma-relative-effective-cartier

In Situation \ref{situation-setup}. Let $X, Y/B$ be good. Let $p : X \to Y$ be a flat morphism of relative dimension $r$. Let $i : D \to X$ be a relative effective Cartier divisor (Divisors on Spaces, Definition \ref{spaces-divisors-definition-relative-effective-Cartier-divisor}). Let $\mathcal{L} = \mathcal{O}_X(D)$. For any $\alpha \in \CH_{k + 1}(Y)$ we have $$ i^*p^*\alpha = (p|_D)^*\alpha $$ in $\CH_{k + r}(D)$ and $$ c_1(\mathcal{L}) \cap p^*\alpha = i_* ((p|_D)^*\alpha) $$ in $\CH_{k + r}(X)$.


spaces-chow-lemma-pullback-affine-fibres-surjective

In Situation \ref{situation-setup} let $X, Y/B$ be good. Let $f : X \to Y$ be a quasi-compact flat morphism over $B$ of relative dimension $r$. Assume that for every $y \in Y$ we have $X_y \cong \mathbf{A}^r_{\kappa(y)}$. Then $f^* : \CH_k(Y) \to \CH_{k + r}(X)$ is surjective for all $k \in \mathbf{Z}$.


spaces-chow-lemma-linebundle

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $$ p : L = \underline{\Spec}(\text{Sym}^*(\mathcal{L})) \longrightarrow X $$ be the associated vector bundle over $X$. Then $p^* : \CH_k(X) \to \CH_{k + 1}(L)$ is an isomorphism for all $k$.


spaces-chow-lemma-cap-c1-bivariant

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then the rule that to $f : X' \to X$ assigns $c_1(f^*\mathcal{L}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')$ is a bivariant class of degree $1$.


spaces-chow-lemma-flat-pullback-bivariant

In Situation \ref{situation-setup} let $f : X \to Y$ be a morphism of good algebraic spaces over $B$ which is flat of relative dimension $r$. Then the rule that to $Y' \to Y$ assigns $(f')^* : \CH_k(Y') \to \CH_{k + r}(X')$ where $X' = X \times_Y Y'$ is a bivariant class of degree $-r$.


spaces-chow-lemma-push-proper-bivariant

In Situation \ref{situation-setup} let $f : X \to Y$ and $g : Y \to Z$ be morphisms of good algebraic spaces over $B$. Let $c \in A^p(X \to Z)$ and assume $f$ is proper. Then the rule that to $X' \to X$ assigns $\alpha \longmapsto f_*(c \cap \alpha)$ is a bivariant class of degree $p$.


spaces-chow-lemma-c1-center

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $c_1(\mathcal{L}) \in A^1(X)$ commutes with every element $c \in A^p(X)$.


spaces-chow-lemma-bivariant-zero

In Situation \ref{situation-setup} let $X/B$ be good. Let $c \in A^p(X)$. Then $c$ is zero if and only if $c \cap [Y] = 0$ in $\CH_*(Y)$ for every integral algebraic space $Y$ locally of finite type over $X$.


spaces-chow-lemma-cap-projective-bundle

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. For any $\alpha \in \CH_k(X)$ the element $$ \pi_*\left( c_1(\mathcal{O}_P(1))^s \cap \pi^*\alpha \right) \in \CH_{k + r - 1 - s}(X) $$ is $0$ if $s < r - 1$ and is equal to $\alpha$ when $s = r - 1$.


spaces-chow-lemma-chow-ring-projective-bundle

Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ of rank $r$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. The map $$ \bigoplus\nolimits_{i = 0}^{r - 1} \CH_{k + i}(X) \longrightarrow \CH_{k + r - 1}(P), $$ $$ (\alpha_0, \ldots, \alpha_{r-1}) \longmapsto \pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1} $$ is an isomorphism.


spaces-chow-lemma-segre-classes

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ be the projective space bundle associated to $\mathcal{E}$. For every morphism $X' \to X$ of good algebraic spaces over $B$ there are unique maps $$ c_i(\mathcal{E}) \cap - : \CH_k(X') \longrightarrow \CH_{k - i}(X'),\quad i = 0, \ldots, r $$ such that for $\alpha \in \CH_k(X')$ we have $c_0(\mathcal{E}) \cap \alpha = \alpha$ and $$ \sum\nolimits_{i = 0, \ldots, r} (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*\left(c_{r - i}(\mathcal{E}) \cap \alpha\right) = 0 $$ where $\pi' : P' \to X'$ is the base change of $\pi$. Moreover, these maps define a bivariant class $c_i(\mathcal{E})$ of degree $i$ on $X$.


spaces-chow-lemma-chern-classes-E-tensor-L

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then we have \begin{equation} \label{equation-twist} c_i({\mathcal E} \otimes {\mathcal L}) = \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c_{i - j}({\mathcal E}) c_1({\mathcal L})^j \end{equation} in $A^*(X)$.


spaces-chow-lemma-get-rid-of-trivial-subbundle

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{F} \to 0 $$ Then we have $$ c_r(\mathcal{E}) = 0, \quad c_j(\mathcal{E}) = c_j(\mathcal{F}), \quad j = 0, \ldots, r - 1 $$ in $A^*(X)$.


spaces-chow-lemma-additivity-invertible-subsheaf

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$, $\mathcal{F}$ be finite locally free sheaves on $X$ of ranks $r$, $r - 1$ which fit into a short exact sequence $$ 0 \to \mathcal{L} \to \mathcal{E} \to \mathcal{F} \to 0 $$ where $\mathcal{L}$ is an invertible sheaf. Then $$ c(\mathcal{E}) = c(\mathcal{L}) c(\mathcal{F}) $$ in $A^*(X)$.


spaces-chow-lemma-chern-filter-by-linebundles

In Situation \ref{situation-setup} let $X/B$ be good. Let ${\mathcal L}_i$, $i = 1, \ldots, r$ be invertible $\mathcal{O}_X$-modules. Let $\mathcal{E}$ be a locally free rank $\mathcal{O}_X$-module endowed with a filtration $$ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E} $$ such that $\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i$. Set $c_1({\mathcal L}_i) = x_i$. Then $$ c(\mathcal{E}) = \prod\nolimits_{i = 1}^r (1 + x_i) $$ in $A^*(X)$.


spaces-chow-lemma-splitting-principle

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}_i$ be a finite collection of locally free $\mathcal{O}_X$-modules of rank $r_i$. There exists a projective flat morphism $\pi : P \to X$ of relative dimension $d$ such that \begin{enumerate} \item for any morphism $f : Y \to X$ of good algebraic spaces over $B$ the map $\pi_Y^* : \CH_*(Y) \to \CH_{* + d}(Y \times_X P)$ is injective, and \item each $\pi^*\mathcal{E}_i$ has a filtration whose successive quotients $\mathcal{L}_{i, 1}, \ldots, \mathcal{L}_{i, r_i}$ are invertible ${\mathcal O}_P$-modules. \end{enumerate}


spaces-chow-lemma-chern-classes-dual

In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module with dual $\mathcal{E}^\vee$. Then $$ c_i(\mathcal{E}^\vee) = (-1)^i c_i(\mathcal{E}) $$ in $A^i(X)$.


sdga-theorem-qis-into-dg-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. For every differential graded $\mathcal{A}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{M} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Moreover, the construction is functorial in $\mathcal{M}$.


sdga-lemma-gm-abelian

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a graded $\mathcal{O}$-algebra. The category $\text{Mod}_\mathcal{A}$ is an abelian category with the following properties \begin{enumerate} \item $\text{Mod}_\mathcal{A}$ has arbitrary direct sums, \item $\text{Mod}_\mathcal{A}$ has arbitrary colimits, \item filtered colimit in $\text{Mod}_\mathcal{A}$ are exact, \item $\text{Mod}_\mathcal{A}$ has arbitrary products, \item $\text{Mod}_\mathcal{A}$ has arbitrary limits. \end{enumerate} The functor $$ \text{Mod}_\mathcal{A} \longrightarrow \textit{Mod}(\mathcal{O}),\quad \mathcal{M} \longmapsto \mathcal{M}^n $$ sending a graded $\mathcal{A}$-module to its $n$th term commutes with all limits and colimits.


sdga-lemma-tensor-hom-adjunction-gr

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right graded $\mathcal{A}$-module. Let $\mathcal{N}$ be a graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right graded $\mathcal{B}$-module. With conventions as above we have $$ \Hom_{\text{Mod}_\mathcal{B}^{gr}}( \mathcal{M} \otimes_\mathcal{A} \mathcal{N}, \mathcal{L}) = \Hom_{\text{Mod}_\mathcal{A}^{gr}}( \mathcal{M}, \SheafHom_\mathcal{B}^{gr}(\mathcal{N}, \mathcal{L})) $$ and $$ \SheafHom_\mathcal{B}^{gr}( \mathcal{M} \otimes_\mathcal{A} \mathcal{N}, \mathcal{L}) = \SheafHom_\mathcal{A}^{gr}( \mathcal{M}, \SheafHom_\mathcal{B}^{gr}(\mathcal{N}, \mathcal{L})) $$ functorially in $\mathcal{M}$, $\mathcal{N}$, $\mathcal{L}$.


sdga-lemma-adjunction-push-pull-gr

In the situation above we have $$ \Hom_{\text{Mod}_\mathcal{B}^{gr}}( \mathcal{N}, f_*\mathcal{M}) = \Hom_{\text{Mod}_\mathcal{A}^{gr}}( f^*\mathcal{N}, \mathcal{M}) $$


sdga-lemma-extension-by-zero-graded

In the situation above we have $$ \Hom_{\text{Mod}_\mathcal{A}^{gr}}( j_!\mathcal{M}, \mathcal{N}) = \Hom_{\text{Mod}_{\mathcal{A}_U}^{gr}}( \mathcal{M}, j^*\mathcal{N}) $$


sdga-lemma-tensor-with-extension-by-zero

In the situation above, let $\mathcal{M}$ be a right graded $\mathcal{A}_U$-module and let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. Then $$ j_!\mathcal{M} \otimes_\mathcal{A} \mathcal{N} = j_!(\mathcal{M} \otimes_{\mathcal{A}_U} \mathcal{N}|_U) $$ as graded $\mathcal{O}$-modules functorially in $\mathcal{M}$ and $\mathcal{N}$.


sdga-lemma-gm-grothendieck-abelian

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a graded $\mathcal{O}$-algebra. The category $\text{Mod}_\mathcal{A}$ is a Grothendieck abelian category.


sdga-lemma-dgm-abelian

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. The category $\text{Mod}_{(\mathcal{A}, \text{d})}$ is an abelian category with the following properties \begin{enumerate} \item $\text{Mod}_{(\mathcal{A}, \text{d})}$ has arbitrary direct sums, \item $\text{Mod}_{(\mathcal{A}, \text{d})}$ has arbitrary colimits, \item filtered colimit in $\text{Mod}_{(\mathcal{A}, \text{d})}$ are exact, \item $\text{Mod}_{(\mathcal{A}, \text{d})}$ has arbitrary products, \item $\text{Mod}_{(\mathcal{A}, \text{d})}$ has arbitrary limits. \end{enumerate} The forgetful functor $$ \text{Mod}_{(\mathcal{A}, \text{d})} \longrightarrow \text{Mod}_\mathcal{A} $$ sending a differential graded $\mathcal{A}$-module to its underlying graded module commutes with all limits and colimits.


sdga-lemma-what-makes-a-bimodule-dg

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a right differential graded $\mathcal{B}$-module. There is a $1$-to-$1$ correspondence between $(\mathcal{A}, \mathcal{B})$-bimodule structures on $\mathcal{N}$ compatible with the given differential graded $\mathcal{B}$-module structure and homomorphisms $$ \mathcal{A} \longrightarrow \SheafHom^{dg}_\mathcal{B}(\mathcal{N}, \mathcal{N}) $$ of differential graded $\mathcal{O}$-algebras.


sdga-lemma-tensor-hom-adjunction-dg

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right differential graded $\mathcal{A}$-module. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right differential graded $\mathcal{B}$-module. With conventions as above we have $$ \Hom_{\text{Mod}_{(\mathcal{B}, \text{d})}^{dg}}( \mathcal{M} \otimes_\mathcal{A} \mathcal{N}, \mathcal{L}) = \Hom_{\text{Mod}_{(\mathcal{A}, \text{d})}^{dg}}( \mathcal{M}, \SheafHom_\mathcal{B}^{dg}(\mathcal{N}, \mathcal{L})) $$ and $$ \SheafHom_\mathcal{B}^{dg}( \mathcal{M} \otimes_\mathcal{A} \mathcal{N}, \mathcal{L}) = \SheafHom_\mathcal{A}^{dg}( \mathcal{M}, \SheafHom_\mathcal{B}^{dg}(\mathcal{N}, \mathcal{L})) $$ functorially in $\mathcal{M}$, $\mathcal{N}$, $\mathcal{L}$.


sdga-lemma-adjunction-push-pull-dg

In the situation above we have $$ \Hom_{\text{Mod}_{(\mathcal{B}, \text{d})}^{dg}}( \mathcal{N}, f_*\mathcal{M}) = \Hom_{\text{Mod}_{(\mathcal{A}, \text{d})}^{dg}}( f^*\mathcal{N}, \mathcal{M}) $$


sdga-lemma-tensor-with-extension-by-zero-dg

In the situation above, let $\mathcal{M}$ be a right differential graded $\mathcal{A}_U$-module and let $\mathcal{N}$ be a left differential graded $\mathcal{A}$-module. Then $$ j_!\mathcal{M} \otimes_\mathcal{A} \mathcal{N} = j_!(\mathcal{M} \otimes_{\mathcal{A}_U} \mathcal{N}|_U) $$ as complexes of $\mathcal{O}$-modules functorially in $\mathcal{M}$ and $\mathcal{N}$.


sdga-lemma-homotopy-direct-sums

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The homotopy category $K(\text{Mod}_{(\mathcal{A}, \text{d})})$ has direct sums and products.


sdga-lemma-axioms-AB

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The differential graded category $\text{Mod}^{dg}_{(\mathcal{A}, \text{d})}$ satisfies axioms (A) and (B) of Differential Graded Algebra, Section \ref{dga-section-review}.


sdga-lemma-axiom-C

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The differential graded category $\text{Mod}^{dg}_{(\mathcal{A}, \text{d})}$ satisfies axiom (C) formulated in Differential Graded Algebra, Situation \ref{dga-situation-ABC}.


sdga-lemma-supply-good

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $U \in \Ob(\mathcal{C})$. Then $j_!\mathcal{A}_U$ is a good differential graded $\mathcal{A}$-module.


sdga-lemma-good-admissible-ses

et $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $0 \to \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to 0$ be an admissible short exact sequence of differential graded $\mathcal{A}$-modules. If two-out-of-three of these modules are good, so is the third.


sdga-lemma-good-direct-sum

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. An arbitrary direct sum of good differential graded $\mathcal{A}$-modules is good. A filtered colimit of good differential graded $\mathcal{A}$-modules is good.


sdga-lemma-free-graded-module-good

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a differential graded $\mathcal{A}$-algebra. Let $\mathcal{S}$ be a sheaf of graded sets on $\mathcal{C}$. Then the free graded module $\mathcal{A}[\mathcal{S}]$ on $\mathcal{S}$ endowed with differential as in Remark \ref{remark-sheaf-graded-sets} is a good differential graded $\mathcal{A}$-module.


sdga-lemma-resolve

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. There exists a homomorphism $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules with the following properties \begin{enumerate} \item $\mathcal{P} \to \mathcal{M}$ is a quasi-isomorphism, and \item $\mathcal{P}$ is good. \end{enumerate}


sdga-lemma-acyclic-good

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{P}$ be a good acyclic right differential graded $\mathcal{A}$-module. \begin{enumerate} \item for any differential graded left $\mathcal{A}$-module $\mathcal{N}$ the tensor product $\mathcal{P} \otimes_\mathcal{A} \mathcal{N}$ is acyclic, \item for any morphism $(f, f^\sharp) : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})$ of ringed topoi and any differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$ and any map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras the pullback $f^*\mathcal{P}$ is acyclic and good. \end{enumerate}


sdga-lemma-dg-hull

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. The forgetful functor $F : \text{Mod}_{(\mathcal{A}, \text{d})} \to \text{Mod}_\mathcal{A}$ has a left adjoint $G : \text{Mod}_\mathcal{A} \to \text{Mod}_{(\mathcal{A}, \text{d})}$.


sdga-lemma-dg-hull-acyclic

The functors $F, G$ of Lemma \ref{lemma-dg-hull} have the following properties. Given a graded $\mathcal{A}$-module $\mathcal{N}$ we have \begin{enumerate} \item the counit $\mathcal{N} \to F(G(\mathcal{N}))$ is injective, \item the map $\overline{\text{d}} : \mathcal{N} \to \Coker(\mathcal{N} \to F(G(\mathcal{N})))[1]$ is an isomorphism, and \item $G(\mathcal{N})$ is an acyclic differential graded $\mathcal{A}$-module. \end{enumerate}


sdga-lemma-characterize-injectives

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{N}_t \to \mathcal{N}'_t$ of graded $\mathcal{A}$-modules such that an object $\mathcal{I}$ of $\text{Mod}_\mathcal{A}$ is injective if and only if for every solid diagram $$ \xymatrix{ \mathcal{N}_t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{N}'_t \ar@{..>}[ru] } $$ a dotted arrow exists in $\text{Mod}_\mathcal{A}$ making the diagram commute.


sdga-lemma-product-graded-injective

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $T$ be a set and for each $t \in T$ let $\mathcal{I}_t$ be a graded injective diffential graded $\mathcal{A}$-module. Then $\prod \mathcal{I}_t$ is a graded injective differential graded $\mathcal{A}$-module.