etale-cohomology-proposition-etale-morphisms

Facts on \'etale morphisms. \begin{enumerate} \item Let $k$ be a field. A morphism of schemes $U \to \Spec(k)$ is \'etale if and only if $U \cong \coprod_{i \in I} \Spec(k_i)$ such that for each $i \in I$ the ring $k_i$ is a field which is a finite separable extension of $k$. \item Let $\varphi : U \to S$ be a morphism of schemes. The following conditions are equivalent: \begin{enumerate} \item $\varphi$ is \'etale, \item $\varphi$ is locally finitely presented, flat, and all its fibres are \'etale, \item $\varphi$ is flat, unramified and locally of finite presentation. \end{enumerate} \item A ring map $A \to B$ is \'etale if and only if $B \cong A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$ such that $\Delta = \det \left( \frac{\partial f_i}{\partial x_j} \right)$ is invertible in $B$. \item The base change of an \'etale morphism is \'etale. \item Compositions of \'etale morphisms are \'etale. \item Fibre products and products of \'etale morphisms are \'etale. \item An \'etale morphism has relative dimension 0. \item Let $Y \to X$ be an \'etale morphism. If $X$ is reduced (respectively regular) then so is $Y$. \item \'Etale morphisms are open. \item If $X \to S$ and $Y \to S$ are \'etale, then any $S$-morphism $X \to Y$ is also \'etale. \end{enumerate}


etale-cohomology-proposition-topological-invariance

Let $X_0 \to X$ be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then \begin{enumerate} \item the \'etale sites $X_\etale$ and $(X_0)_\etale$ are isomorphic, \item the \'etale topoi $\Sh(X_\etale)$ and $\Sh((X_0)_\etale)$ are equivalent, and \item $H^q_\etale(X, \mathcal{F}) = H^q_\etale(X_0, \mathcal{F}|_{X_0})$ for all $q$ and for any abelian sheaf $\mathcal{F}$ on $X_\etale$. \end{enumerate}


etale-cohomology-proposition-closed-immersion-pushforward

Let $i : Z \to X$ be a closed immersion of schemes. \begin{enumerate} \item The functor $$ i_{small, *} : \Sh(Z_\etale) \longrightarrow \Sh(X_\etale) $$ is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_\etale$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and \item the functor $$ i_{small, *} : \textit{Ab}(Z_\etale) \longrightarrow \textit{Ab}(X_\etale) $$ is fully faithful and its essential image is those abelian sheaves on $X_\etale$ whose support is contained in $Z$. \end{enumerate} In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.


etale-cohomology-proposition-integral-universally-injective-pushforward

Let $f : X \to Y$ be a morphism of schemes which is integral and universally injective. \begin{enumerate} \item The functor $$ f_{small, *} : \Sh(X_\etale) \longrightarrow \Sh(Y_\etale) $$ is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $Y_\etale$ whose restriction to $Y \setminus f(X)$ is isomorphic to $*$, and \item the functor $$ f_{small, *} : \textit{Ab}(X_\etale) \longrightarrow \textit{Ab}(Y_\etale) $$ is fully faithful and its essential image is those abelian sheaves on $Y_\etale$ whose support is contained in $f(X)$. \end{enumerate} In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$.


etale-cohomology-proposition-leray

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ an \'etale sheaf on $X$. Then there is a spectral sequence $$ E_2^{p, q} = H_\etale^p(Y, R^qf_*\mathcal{F}) \Rightarrow H_\etale^{p+q}(X, \mathcal{F}). $$


etale-cohomology-proposition-finite-higher-direct-image-zero

Let $f : X \to Y$ be a finite morphism of schemes. \begin{enumerate} \item For any geometric point $\overline{y} : \Spec(k) \to Y$ we have $$ (f_*\mathcal{F})_{\overline{y}} = \prod\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. $$ for $\mathcal{F}$ in $\Sh(X_\etale)$ and $$ (f_*\mathcal{F})_{\overline{y}} = \bigoplus\nolimits_{\overline{x} : \Spec(k) \to X,\ f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}}. $$ for $\mathcal{F}$ in $\textit{Ab}(X_\etale)$. \item For any $q \geq 1$ we have $R^q f_*\mathcal{F} = 0$. \end{enumerate}


etale-cohomology-proposition-serre-galois

\begin{reference} \cite[Chapter II, Section 3, Proposition 5]{SerreGaloisCohomology} \end{reference} Let $K$ be a field with separable algebraic closure $K^{sep}$. Assume that for any finite extension $K'$ of $K$ we have $\text{Br}(K') = 0$. Then \begin{enumerate} \item $H^q(\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \geq 1$, and \item $H^q(\text{Gal}(K^{sep}/K), M) = 0$ for any torsion $\text{Gal}(K^{sep}/K)$-module $M$ and any $q \geq 2$, \end{enumerate}


etale-cohomology-proposition-describe-jshriek

Let $j : U \to X$ be an \'etale morphism of schemes. Let $\mathcal{F}$ in $\textit{Ab}(U_\etale)$. If $\overline{x} : \Spec(k) \to X$ is a geometric point of $X$, then  $$ (j_!\mathcal{F})_{\overline{x}} = \bigoplus\nolimits_{\overline{u} : \Spec(k) \to U,\ j(\overline{u}) = \overline{x}} \mathcal{F}_{\bar{u}}. $$ In particular, $j_!$ is an exact functor.


etale-cohomology-proposition-constructible-over-noetherian

Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring. \begin{enumerate} \item Any sub or quotient sheaf of a constructible sheaf of sets is constructible. \item The category of constructible abelian sheaves on $X_\etale$ is a (strong) Serre subcategory of $\textit{Ab}(X_\etale)$. In particular, every sub and quotient sheaf of a constructible abelian sheaf on $X_\etale$ is constructible. \item The category of constructible sheaves of $\Lambda$-modules on $X_\etale$ is a (strong) Serre subcategory of $\textit{Mod}(X_\etale, \Lambda)$. In particular, every submodule and quotient module of a constructible sheaf of $\Lambda$-modules on $X_\etale$ is constructible. \end{enumerate}


etale-cohomology-proposition-cd-affine

Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$. Then we have $\text{cd}(X) \leq \dim(X) + \text{cd}(K)$.


equiv-theorem-fully-faithful

\begin{reference} \cite[Theorem 2.2]{Orlov-K3}; this is shown in \cite{Noah} without the assumption that $X$ be projective \end{reference} Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$ with $X$ projective over $k$. Any $k$-linear fully faithful exact  functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is a Fourier-Mukai functor for some kernel in $D_{perf}(\mathcal{O}_{X \times Y})$.


equiv-lemma-Serre-functor-exists

Let $k$ be a field. Let $\mathcal{T}$ be a $k$-linear triangulated category such that $\dim_k \Hom_\mathcal{T}(X, Y) < \infty$ for all $X, Y \in \Ob(\mathcal{T})$. The following are equivalent \begin{enumerate} \item there exists a $k$-linear equivalence $S : \mathcal{T} \to \mathcal{T}$ and $k$-linear isomorphisms $c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee$ functorial in $X, Y \in \Ob(\mathcal{T})$, \item for every $X \in \Ob(\mathcal{T})$ the functor $Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee$ is representable and the functor $Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee$ is corepresentable. \end{enumerate}


equiv-lemma-perfect-for-R

With $k$, $n$, and $R$ as above, for an object $K$ of $D(R)$ the following are equivalent \begin{enumerate} \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(K) < \infty$, and \item $K$ is a compact object. \end{enumerate}


equiv-lemma-coherent-on-projective-space

Let $k$ be a field. Let $n \geq 0$. Let $K \in D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})$. The following are equivalent \begin{enumerate} \item $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, E \otimes^\mathbf{L} K) < \infty$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_{\mathbf{P}^n_k}(E, K) < \infty$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, K \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_k}(d)) < \infty$ for $d = 0, 1, \ldots, n$. \end{enumerate}


equiv-lemma-finiteness

Let $X$ be a scheme proper over a field $k$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and let $E$ in $D(\mathcal{O}_X)$ be perfect. Then $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$.


equiv-lemma-characterize-dbcoh-projective

\begin{reference} \cite[Lemma 7.46]{Rouquier-dimensions} and implicit in \cite[Theorem A.1]{BvdB} \end{reference} Let $X$ be a projective scheme over a field $k$. Let $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. The following are equivalent \begin{enumerate} \item $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$, and \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$ for all perfect $E$ in $D(\mathcal{O}_X)$. \end{enumerate}


equiv-lemma-maps-from-compact-filtered

Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. Let $X \in \Ob(\mathcal{D})$. The category of arrows $E \to X$ with $E \in \Ob(\mathcal{D}')$ is filtered.


equiv-lemma-van-den-bergh

\begin{reference} \cite[Lemma 2.14]{CKN} \end{reference} Let $k$ be a field. Let $\mathcal{D}$ be a $k$-linear triangulated category which has direct sums and is compactly generated. Denote $\mathcal{D}_c$ the full subcategory of compact objects. Let $H : \mathcal{D}_c^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $\dim_k H(X) < \infty$ for all $X \in \Ob(\mathcal{D}_c)$. Then $H$ is isomorphic to the functor $X \mapsto \Hom(X, Y)$ for some $Y \in \Ob(\mathcal{D})$.


equiv-lemma-characterize-dbcoh-proper-regular

Let $X$ be a proper scheme over a field $k$ which is regular. Let $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. The following are equivalent \begin{enumerate} \item $K \in D^b_{\textit{Coh}}(\mathcal{O}_X) = D_{perf}(\mathcal{O}_X)$, and \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$ for all perfect $E$ in $D(\mathcal{O}_X)$. \end{enumerate}


equiv-lemma-bondal-van-den-bergh

Let $X$ be a proper scheme over a field $k$ which is regular. \begin{enumerate} \item Let $F : D_{perf}(\mathcal{O}_X)^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k F(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \mapsto \Hom_X(E, K)$ for some $K \in D_{perf}(\mathcal{O}_X)$. \item Let $G : D_{perf}(\mathcal{O}_X) \to \text{Vect}_k$ be a $k$-linear homological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k G(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $G$ is isomorphic to a functor of the form $E \mapsto \Hom_X(K, E)$ for some $K \in D_{perf}(\mathcal{O}_X)$. \end{enumerate}


equiv-lemma-always-right-adjoints

Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. If $X$ is regular, then $k$-linear any exact functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ has an exact right adjoint and an exact left adjoint.


equiv-lemma-fourier-Mukai-QCoh

Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$ if $K$ is in $D_\QCoh(\mathcal{O}_{X \times_S Y})$ and $X \to S$ is quasi-compact and quasi-separated.


equiv-lemma-compose-fourier-mukai

Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume $X \to S$, $Y \to S$, and $Z \to S$ are quasi-compact and quasi-separated. Let $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Let $K' \in D_\QCoh(\mathcal{O}_{Y \times_S Z})$. Consider the Fourier-Mukai functors $\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ and $\Phi_{K'} : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Z)$. If $X$ and $Z$ are tor independent over $S$ and $Y \to S$ is flat, then $$ \Phi_{K'} \circ \Phi_K = \Phi_{K''} : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_Z) $$ where $$ K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K') $$ in $D_\QCoh(\mathcal{O}_{X \times_S Z})$.


equiv-lemma-fourier-mukai

Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D_{perf}(\mathcal{O}_X)$ into $D_{perf}(\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \begin{enumerate} \item $S$ is Noetherian, $X \to S$ and $Y \to S$ are of finite type, $K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})$, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_2^{-1}\mathcal{O}_Y)$, \item $X \to S$ is of finite presentation and $K$ can be represented by a bounded complex $\mathcal{K}^\bullet$ of finitely presented $\mathcal{O}_{X \times_S Y}$-modules, flat over $Y$, with support proper over $Y$, \item $X \to S$ is a proper flat morphism of finite presentation and $K$ is perfect, \item $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is perfect \item $X \to S$ is a proper flat morphism of finite presentation and $K$ is $Y$-perfect, \item $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is $Y$-perfect. \end{enumerate}


equiv-lemma-fourier-mukai-Coh

Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type over $S$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \begin{enumerate} \item the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$, \item $K$ can be represented by a bounded complex $\mathcal{K}^\bullet$ of coherent $\mathcal{O}_{X \times_S Y}$-modules, flat over $X$, with support proper over $Y$, \item the support of $H^i(K)$ is proper over $Y$ for all $i$ and $X$ is a regular scheme, \item $K$ is perfect, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $Y \to S$ is flat. \end{enumerate} Furthermore in each case the support condition is automatic if $X \to S$ is proper.


equiv-lemma-fourier-mukai-right-adjoint

\begin{reference} Compare with discussion in \cite{Rizzardo}. \end{reference} Let $X \to S$ and $Y \to S$ be morphisms of quasi-compact and quasi-separated schemes. Let $\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ be a Fourier-Mukai functor with pseudo-coherent kernel $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Let $a : D_\QCoh(\mathcal{O}_Y) \to  D_\QCoh(\mathcal{O}_{X \times_S Y})$ be the right adjoint to $R\text{pr}_{2, *}$, see Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image}. Denote $$ K' = (Y \times_S X \to X \times_S Y)^* R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \in D_\QCoh(\mathcal{O}_{Y \times_S X}) $$ and denote $\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical map $$ \Hom_X(M, \Phi'(N)) \longrightarrow \Hom_Y(\Phi(M), N) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$ which is an isomorphism if \begin{enumerate} \item $N$ is perfect, or \item $K$ is perfect and $X \to S$ is proper flat and of finite presentation. \end{enumerate}


equiv-lemma-fourier-mukai-left-adjoint

\begin{reference} Compare with discussion in \cite{Rizzardo}. \end{reference} Let $S$ be a Noetherian scheme. Let $Y \to S$ be a flat proper Gorenstein morphism and let $X \to S$ be a finite type morphism. Denote $\omega^\bullet_{Y/S}$ the relative dualizing complex of $Y$ over $S$. Let $\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ be a Fourier-Mukai functor with perfect kernel $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Denote $$ K' = (Y \times_S X \to X \times_S Y)^*(K^\vee \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} L\text{pr}_2^*\omega^\bullet_{Y/S}) \in D_\QCoh(\mathcal{O}_{Y \times_S X}) $$ and denote $\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical isomorphism $$ \Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$.


equiv-lemma-fourier-mukai-flat-proper-over-noetherian

Let $S$ be a Noetherian scheme. \begin{enumerate} \item For $X$, $Y$ proper and flat over $S$ and $K$ in $D_{perf}(\mathcal{O}_{X \times_S Y})$ we obtain a Fourier-Mukai functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. \item For $X$, $Y$, $Z$ proper and flat over $S$, $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$, $K' \in D_{perf}(\mathcal{O}_{Y \times_S Z})$ the composition $\Phi_{K'} \circ \Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Z)$ is equal to $\Phi_{K''}$ with $K'' \in D_{perf}(\mathcal{O}_{X \times_S Z})$ computed as in Lemma \ref{lemma-compose-fourier-mukai}, \item For $X$, $Y$, $K$, $\Phi_K$ as in (1) if $X \to S$ is Gorenstein, then $\Phi_{K'} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)$ is a right adjoint to $\Phi_K$ where $K' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ is the pullback of $L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee$ by $Y \times_S X \to X \times_S Y$. \item For $X$, $Y$, $K$, $\Phi_K$ as in (1) if $Y \to S$ is Gorenstein, then $\Phi_{K''} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)$ is a left adjoint to $\Phi_K$ where $K'' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ is the pullback of $L\text{pr}_2^*\omega_{Y/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee$ by $Y \times_S X \to X \times_S Y$. \end{enumerate}


equiv-lemma-on-product

Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$ having the resolution property. For any coherent $\mathcal{O}_{X \times_R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_Y$-module.


equiv-lemma-diagonal-resolution

Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme over $R$ which has the resolution property. Set $\mathcal{O}_\Delta = \Delta_*(\mathcal{O}_X)$ where $\Delta : X \to X \times_R X$ is the diagonal of $X/k$. There exists a resolution $$ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 $$ where each $\mathcal{E}_i$ and $\mathcal{G}_i$ is a finite locally free $\mathcal{O}_X$-module.


equiv-lemma-Ext-0-regular

Let $X$ be a regular Noetherian scheme of dimension $d < \infty$. Then \begin{enumerate} \item for $\mathcal{F}$, $\mathcal{G}$ coherent $\mathcal{O}_X$-modules we have $\Ext^n_X(\mathcal{F}, \mathcal{G}) = 0$ for $n > d$, and \item for $K, L \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and $a \in \mathbf{Z}$ if $H^i(K) = 0$ for $i < a + d$ and $H^i(L) = 0$ for $i \geq a$ then $\Hom_X(K, L) = 0$. \end{enumerate}


equiv-lemma-split-complex-regular

Let $X$ be a regular Noetherian scheme of dimension $d < \infty$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and $a \in \mathbf{Z}$. If $H^i(K) = 0$ for $a < i < a + d$, then $K = \tau_{\leq a}K \oplus \tau_{\geq a + d}K$.


equiv-lemma-diagonal-trick

Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$. There exist finite locally free $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $$ \mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle $$ in $D(\mathcal{O}_{X \times X})$ where the notation is as in Derived Categories, Section \ref{derived-section-generators}.


equiv-lemma-smooth-proper-strong-generator

Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$. Then $D_{perf}(\mathcal{O}_X)$ has a strong generator.


equiv-lemma-diagonal-trick-proper

Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$. There exists integers $m, n \geq 1$ and a finite locally free $\mathcal{O}_X$-module $\mathcal{G}$ such that every coherent $\mathcal{O}_X$-module is contained in $smd(add(\mathcal{G}[-m, m])^{\star n})$ with notation as in Derived Categories, Section \ref{derived-section-operate-on-full}.


equiv-lemma-functor-quasi-coherent-from-affine

Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ affine. There is an equivalence of categories between \begin{enumerate} \item the category of $R$-linear functors $F : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ which are right exact and commute with arbitrary direct sums, and \item the category $\QCoh(\mathcal{O}_{X \times_R Y})$ \end{enumerate} given by sending $\mathcal{K}$ to the functor $F$ in (\ref{equation-FM-QCoh}).


equiv-lemma-functor-quasi-coherent-from-affine-compose

In Lemma \ref{lemma-functor-quasi-coherent-from-affine} let $F$ correspond to $\mathcal{K}$ in $\QCoh(\mathcal{O}_{X \times_R Y})$. We have \begin{enumerate} \item If $f : X' \to X$ is an affine morphism, then $F \circ f_*$ corresponds to $(f \times \text{id}_Y)^*\mathcal{K}$. \item If $g : Y' \to Y$ is a quasi-compact and quasi-separated flat morphism, then $g^* \circ F$ corresponds to $(\text{id}_X \times g)^*\mathcal{K}$. \item If $j : V \to Y$ is an open immersion, then $j^* \circ F$ corresponds to $\mathcal{K}|_{X \times_R V}$. \end{enumerate}


equiv-lemma-coh-noetherian-from-affine-flat

In Lemma \ref{lemma-functor-quasi-coherent-from-affine} if $F$ is an exact functor, then the corresponding object $\mathcal{K}$ of $\QCoh(\mathcal{O}_{X \times_R Y})$ is flat over $X$.


equiv-lemma-functor-quasi-coherent-from-affine-diagonal

Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is quasi-compact and that the diagonal morphism of $X$ is affine. There is an equivalence of categories between \begin{enumerate} \item the category of $R$-linear exact functors $F : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ which commute with arbitrary direct sums, and \item the full subcategory of $\QCoh(\mathcal{O}_{X \times_R Y})$ consisting of $\mathcal{K}$ such that \begin{enumerate} \item $\mathcal{K}$ is flat over $X$, \item for $\mathcal{F} \in \QCoh(\mathcal{O}_X)$ we have $R^q\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) = 0$ for $q > 0$. \end{enumerate} \end{enumerate} given by sending $\mathcal{K}$ to the functor $F$ in (\ref{equation-FM-QCoh}).


equiv-lemma-persistence-exactness

Let $R$, $X$, $Y$, and $\mathcal{K}$ be as in Lemma \ref{lemma-functor-quasi-coherent-from-affine-diagonal} part (2). Then for any scheme $T$ over $R$ we have $$ R^q\text{pr}_{13, *}(\text{pr}_{12}^*\mathcal{F} \otimes_{\mathcal{O}_{T \times_R X \times_R Y}} \text{pr}_{23}^*\mathcal{K}) = 0 $$ for $\mathcal{F}$ quasi-coherent on $T \times_R X$ and $q > 0$.


equiv-lemma-functor-quasi-coherent-from-separated

In Lemma \ref{lemma-functor-quasi-coherent-from-affine-diagonal} let $F$ and $\mathcal{K}$ correspond. If $X$ is separated and flat over $R$, then there is a surjection $\mathcal{O}_X \boxtimes F(\mathcal{O}_X) \to \mathcal{K}$.


equiv-lemma-functor-coherent

Let $X$ and $Y$ be Noetherian schemes. Let $F : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_Y)$ be a functor. Then $F$ extends uniquely to a functor $\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ which commutes with filtered colimits. If $F$ is additive, then its extension commutes with arbitrary direct sums. If $F$ is exact, left exact, or right exact, so is its extension.


equiv-lemma-characterize-finite

Let $f : V \to X$ be a quasi-finite separated morphism of Noetherian schemes. If there exists a coherent $\mathcal{O}_V$-module $\mathcal{K}$ whose support is $V$ such that $f_*\mathcal{K}$ is coherent and $R^qf_*\mathcal{K} = 0$, then $f$ is finite.


equiv-lemma-functor-coherent-over-field

Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. There is an equivalence of categories between \begin{enumerate} \item the category of $k$-linear exact functors $F : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_Y)$, and \item the category of coherent $\mathcal{O}_{X \times Y}$-modules $\mathcal{K}$ which are flat over $X$ and have support finite over $Y$ \end{enumerate} given by sending $\mathcal{K}$ to the restriction of the functor (\ref{equation-FM-QCoh}) to $\textit{Coh}(\mathcal{O}_X)$.


equiv-lemma-pushforward-invertible-pre

Let $f : X \to Y$ be a finite type separated morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent module on $X$ with support finite over $Y$ and with $\mathcal{L} = f_*\mathcal{F}$ an invertible $\mathcal{O}_X$-module. Then there exists a section $s : Y \to X$ such that $\mathcal{F} \cong s_*\mathcal{L}$.


equiv-lemma-sibling-fully-faithful

Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D}$ be exact functors of triangulated categories. Assume \begin{enumerate} \item the functors $F \circ i$ and $F' \circ i$ are isomorphic where $i : \mathcal{A} \to D^b(\mathcal{A})$ is the inclusion functor, and \item for all $X, Y \in \Ob(\mathcal{A})$ we have $\Ext^q_\mathcal{D}(F(X), F(Y)) = 0$ for $q < 0$ (for example if $F$ is fully faithful). \end{enumerate} Then $F$ and $F'$ are siblings.


equiv-lemma-sibling-faithful

Let $F$ and $F'$ be siblings as in Definition \ref{definition-siblings}. Then \begin{enumerate} \item if $F$ is essentially surjective, then $F'$ is essentially surjective, \item if $F$ is fully faithful, then $F'$ is fully faithful. \end{enumerate}


equiv-lemma-get-fully-faithful

\begin{reference} Variant of \cite[Lemma 2.15]{Orlov-K3} \end{reference} Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of triangulated categories. Let $S \subset \Ob(\mathcal{D})$ be a set of objects. Assume \begin{enumerate} \item $F$ has both right and left adjoints, \item for $K \in \mathcal{D}$ if $\Hom(E, K[i]) = 0$ for all $E \in S$ and $i \in \mathbf{Z}$ then $K = 0$, \item for $K \in \mathcal{D}$ if $\Hom(K, E[i]) = 0$ for all $E \in S$ and $i \in \mathbf{Z}$ then $K = 0$, \item the map $\Hom(E, E'[i]) \to \Hom(F(E), F(E')[i])$ induced by $F$ is bijective for all $E, E' \in S$ and $i \in \mathbf{Z}$. \end{enumerate} Then $F$ is fully faithful.


equiv-lemma-duality-at-point

Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \in X$ be a closed point. For a coherent $\mathcal{O}_X$-module $\mathcal{F}$ supported at $x$ choose a coherent $\mathcal{O}_X$-module $\mathcal{F}'$ supported at $x$ such that $\mathcal{F}_x$ and $\mathcal{F}'_x$ are Matlis dual. Then there is an isomorphism $$ \Hom_X(\mathcal{F}, M) = H^0(X, M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}'[-d_x]) $$ where $d_x = \dim(\mathcal{O}_{X, x})$ functorial in $M$ in $D_{perf}(\mathcal{O}_X)$.


equiv-lemma-orthogonal-point-sheaf

Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \in X$ be a closed point and denote $\mathcal{O}_x$ the skyscraper sheaf at $x$ with value $\kappa(x)$. Let $K$ in $D_{perf}(\mathcal{O}_X)$. \begin{enumerate} \item If $\Ext^i_X(\mathcal{O}_x, K) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^{i - d_x}(K)|_U = 0$ where $d_x = \dim(\mathcal{O}_{X, x})$. \item If $\Hom_X(\mathcal{O}_x, K[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \item If $\Ext^i_X(K, \mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K^\vee)|_U = 0$. \item If $\Hom_X(K, \mathcal{O}_x[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \item If $H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K)|_U = 0$. \item If $H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0$ for $i \in \mathbf{Z}$ then $K$ is zero in an open neighbourhood of $x$. \end{enumerate}


equiv-lemma-get-fully-faithful-geometric

Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. Assume $X$ is regular. Then a $k$-linear exact functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is fully faithful if and only if for any closed points $x, x' \in X$ the maps $$ F : \Ext^i_X(\mathcal{O}_x, \mathcal{O}_{x'}) \longrightarrow \Ext^i_Y(F(\mathcal{O}_x), F(\mathcal{O}_{x'})) $$ are isomorphisms for all $i \in \mathbf{Z}$. Here $\mathcal{O}_x$ is the skyscraper sheaf at $x$ with value $\kappa(x)$.


equiv-lemma-noah-pre

\begin{reference} Email from Noah Olander of Jun 9, 2020 \end{reference} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then $F$ is fully faithful.


equiv-lemma-exact-functor-preserving-Coh

Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ be a $k$-linear exact functor sending $\textit{Coh}(\mathcal{O}_X) \subset D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $\textit{Coh}(\mathcal{O}_Y) \subset D^b_{\textit{Coh}}(\mathcal{O}_Y)$. Then there exists a Fourier-Mukai functor $F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ whose kernel is a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$.


equiv-lemma-preserves-Coh

Let $k$ be a field. Let $X$ be a separated scheme of finite type over $k$ which is regular. Let $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism of $k$-vector spaces $$ \Hom_X(\mathcal{F}, M) = \Hom_X(\mathcal{F}, F(M)) $$ functorial in $M$ in $D_{perf}(\mathcal{O}_X)$. Then there exists an automorphism $f : X \to X$ over $k$ which induces the identity on the underlying topological space\footnote{This often forces $f$ to be the identity, see Lemma \ref{lemma-automorphism}.} and an invertible $\mathcal{O}_X$-module $\mathcal{L}$ such that $F$ and $F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}$ are siblings.


equiv-lemma-automorphism

Let $X$ be a reduced scheme of finite type over a field $k$. Let $f : X \to X$ be an automorphism over $k$ which induces the identity map on the underlying topological space of $X$. Then \begin{enumerate} \item $f^*\mathcal{F} \cong \mathcal{F}$ for every coherent $\mathcal{O}_X$-module, and \item if $\dim(Z) > 0$ for every irreducible component $Z \subset X$, then $f$ is the identity. \end{enumerate}


equiv-lemma-noah

\begin{reference} Email from Noah Olander of Jun 8, 2020 \end{reference} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then there exists an automorphism $f : X \to X$ over $k$ which induces the identity on the underlying topological space\footnote{This often forces $f$ to be the identity, see Lemma \ref{lemma-automorphism}.} and an invertible $\mathcal{O}_X$-module $\mathcal{L}$ such that $F$ and $F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}$ are siblings.


equiv-lemma-two-functors

Let $k$ be a field. Let $X$, $Y$ be smooth proper schemes over $k$. Let $F, G : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ be $k$-linear exact functors such that \begin{enumerate} \item $F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$, \item $F$ is fully faithful, and \item $G$ is a Fourier-Mukai functor whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$. \end{enumerate} Then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ such that $F$ and $F'$ are siblings.


equiv-lemma-fully-faithful

Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. Given a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is a sibling to $F$.


equiv-lemma-uniqueness

Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $K \in D_{perf}(\mathcal{O}_{X \times X})$. If the Fourier-Mukai functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ is isomorphic to the identity functor, then $K \cong \Delta_*\mathcal{O}_X$ in $_{perf}(\mathcal{O}_{X \times X})$.


equiv-lemma-base-change-is-functor

Let $S' \to S$ be a morphism of schemes. The rule which sends \begin{enumerate} \item a smooth proper scheme $X$ over $S$ to $X' =  S' \times_S X$, and \item the isomorphism class of an object $K$ of $D_{perf}(\mathcal{O}_{X \times_S Y})$ to the isomorphism class of $L(X' \times_{S'} Y' \to X \times_S Y)^*K$ in $D_{perf}(\mathcal{O}_{X' \times_{S'} Y'})$ \end{enumerate} is a functor from the category defined for $S$ to the category defined for $S'$.


equiv-lemma-base-change-rek

With notation as in Definition \ref{definition-relative-equivalence-kernel} let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Let $S_1 \to S$ be a morphism of schemes. Let $X_1 = S_1 \times_S X$ and $Y_1 = S_1 \times_S Y$. Then the pullback $K_1 = L(X_1 \times_{S_1} Y_1 \to X \times_S Y)^*K$ is the Fourier-Mukai kernel of a relative equivalence from $X_1$ to $Y_1$ over $S_1$.


equiv-lemma-descend-rek

Let $S = \lim_{i \in I} S_i$ be a limit of a directed system of schemes with affine transition morphisms $g_{i'i} : S_{i'} \to S_i$. We assume that $S_i$ is quasi-compact and quasi-separated for all $i \in I$. Let $0 \in I$. Let $X_0 \to S_0$ and $Y_0 \to S_0$ be smooth proper morphisms. We set $X_i = S_i \times_{S_0} X_0$ for $i \geq 0$ and $X = S \times_{S_0} X_0$ and similarly for $Y_0$. If $K$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$ then for some $i \geq 0$ there exists a Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$.


equiv-lemma-deform-koszul

Let $(R, \mathfrak m, \kappa) \to (A, \mathfrak n, \lambda)$ be a flat local ring homorphism of local rings which is essentially of finite presentation. Let $\overline{f}_1, \ldots, \overline{f}_r \in \mathfrak n/\mathfrak m A \subset A/\mathfrak m A$ be a regular sequence. Let $K \in D(A)$. Assume \begin{enumerate} \item $K$ is perfect, \item $K \otimes_A^\mathbf{L} A/\mathfrak m A$ is isomorphic in $D(A/\mathfrak m A)$ to the Koszul complex on $\overline{f}_1, \ldots, \overline{f}_r$. \end{enumerate} Then $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence $f_1, \ldots, f_r \in A$ lifting the given elements $\overline{f}_1, \ldots, \overline{f}_r$. Moreover, $A/(f_1, \ldots, f_r)$ is flat over $R$.


equiv-lemma-limit-arguments

Let $R \to S$ be a finite type flat ring map of Noetherian rings. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak p \subset R$. Let $K \in D(S)$ be perfect. Let $f_1, \ldots, f_r \in \mathfrak q S_\mathfrak q$ be a regular sequence such that $S_\mathfrak q/(f_1, \ldots, f_r)$ is flat over $R$ and such that $K \otimes_S^\mathbf{L} S_\mathfrak q$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$. Then there exists a $g \in S$, $g \not \in \mathfrak q$ such that \begin{enumerate} \item $f_1, \ldots, f_r$ are the images of $f'_1, \ldots, f'_r \in S_g$, \item $f'_1, \ldots, f'_r$ form a regular sequence in $S_g$, \item $S_g/(f'_1, \ldots, f'_r)$ is flat over $R$, \item $K \otimes_S^\mathbf{L} S_g$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$. \end{enumerate}


equiv-lemma-isomorphism-in-neighbourhood

Let $S$ be a Noetherian scheme. Let $s \in S$. Let $p : X \to Y$ be a morphism of schemes over $S$. Assume \begin{enumerate} \item $Y \to S$ and $X \to S$ proper, \item $X$ is flat over $S$, \item $X_s \to Y_s$ an isomorphism. \end{enumerate} Then there exists an open neighbourhood $U \subset S$ of $s$ such that the base change $X_U \to Y_U$ is an isomorphism.


equiv-lemma-no-deformations

Let $k$ be a field. Let $S$ be a finite type scheme over $k$ with $k$-rational point $s$. Let $Y \to S$ be a smooth proper morphism. Let $X = Y_s \times S \to S$ be the constant family with fibre $Y_s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Assume the restriction $$ L(Y_s \times_S Y_s \to X \times_S Y)^*K \cong  \Delta_{Y_s/k, *} \mathcal{O}_{Y_s} $$ in $D(\mathcal{O}_{Y_s \times Y_s})$. Then there is an open neighbourhood $s \in U \subset S$ such that $Y|_U$ is isomorphic to $Y_s \times U$ over $U$.


equiv-lemma-no-deformations-better

Let $k$ be an algebraically closed field. Let $X$ be a smooth proper scheme over $k$. Let $f : Y \to S$ be a smooth proper morphism with $S$ of finite type over $k$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X \times S$ to $Y$ over $S$. Then $S$ can be covered by open subschemes $U$ such that there is a $U$-isomorphism $f^{-1}(U) \cong Y_0 \times U$ for some $Y_0$ proper and smooth over $k$.


equiv-lemma-countable-finite-type

Let $R$ be a countable Noetherian ring. Then the category of schemes of finite type over $R$ is countable.


equiv-lemma-countable-abelian

Let $\mathcal{A}$ be a countable abelian category. Then $D^b(\mathcal{A})$ is countable.


equiv-lemma-countable-perfect

Let $X$ be a scheme of finite type over a countable Noetherian ring. Then the categories $D_{perf}(\mathcal{O}_X)$ and $D^b_{\textit{Coh}}(\mathcal{O}_X)$ are countable.


equiv-lemma-countable-isos

Let $K$ be an algebraically closed field. Let $S$ be a finite type scheme over $K$. Let $X \to S$ and $Y \to S$ be finite type morphisms. There exists a countable set $I$ and for $i \in I$ a pair $(S_i \to S, h_i)$ with the following properties \begin{enumerate} \item $S_i \to S$ is a morphism of finite type, set $X_i = X \times_S S_i$ and $Y_i = Y \times_S S_i$, \item $h_i : X_i \to Y_i$ is an isomorphism over $S_i$, and \item for any closed point $s \in S(K)$ if $X_s \cong Y_s$ over $K = \kappa(s)$ then $s$ is in the image of $S_i \to S$ for some $i$. \end{enumerate}


equiv-lemma-countable-equivs

Let $K$ be an algebraically closed field. There exists a countable set $I$ and for $i \in I$ a pair $(S_i/K, X_i \to S_i, Y_i \to S_i, M_i)$ with the following properties \begin{enumerate} \item $S_i$ is a scheme of finite type over $K$, \item $X_i \to S_i$ and $Y_i \to S_i$ are proper smooth morphisms of schemes, \item $M_i \in D_{perf}(\mathcal{O}_{X_i \times_{S_i} Y_i})$ is the Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$, and \item for any smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ there exists an $i \in I$ and a $s \in S_i(K)$ such that $X \cong (X_i)_s$ and $Y \cong (Y_i)_s$. \end{enumerate}


equiv-proposition-siblings-isomorphic

\begin{reference} \cite[Proposition 2.16]{Orlov-K3} \end{reference} Let $F$ and $F'$ be siblings as in Definition \ref{definition-siblings}. Assume that $F$ is fully faithful and that $\mathcal{A}$ has enough negative objects (see above). Then $F$ and $F'$ are isomorphic functors.


equiv-proposition-equivalence

Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. If $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is a $k$-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is an equivalence and a sibling of $F$.


stacks-more-morphisms-theorem-chow-finite-type

\begin{reference} This is a result due to Ofer Gabber, see \cite[Theorem 1.1]{olsson_proper} \end{reference} Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Assume \begin{enumerate} \item $Y$ is quasi-compact and quasi-separated, \item $f$ is separated of finite type. \end{enumerate} Then there exists a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[rd] & X \ar[l] \ar[d] \ar[r] & \overline{X} \ar[ld] \\ & Y } $$ where $X \to \mathcal{X}$ is proper surjective, $X \to \overline{X}$ is an open immersion, and $\overline{X} \to Y$ is proper morphism of algebraic spaces.


stacks-more-morphisms-theorem-keel-mori

Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is finite. Then there exists a uniform categorical moduli space $$ f : \mathcal{X} \longrightarrow M $$ and $f$ is separated, quasi-compact, and a universal homeomorphism.


stacks-more-morphisms-lemma-thickening

Let $i : \mathcal{X} \to \mathcal{X}'$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $i$ is a thickening of algebraic stacks (abuse of language as above), and \item $i$ is representable by algebraic spaces and is a thickening in the sense of Properties of Stacks, Section \ref{stacks-properties-section-properties-morphisms}. \end{enumerate} In this case $i$ is a closed immersion and a universal homeomorphism.


stacks-more-morphisms-lemma-base-change-thickening

Let $\mathcal{Y} \subset \mathcal{Y}'$ be a thickening of algebraic stacks. Let $\mathcal{X}' \to \mathcal{Y}'$ be a morphism of algebraic stacks and set $\mathcal{X} = \mathcal{Y} \times_{\mathcal{Y}'} \mathcal{X}'$. Then $(\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ is a morphism of thickenings. If $\mathcal{Y} \subset \mathcal{Y}'$ is a first order thickening, then $\mathcal{X} \subset \mathcal{X}'$ is a first order thickening.


stacks-more-morphisms-lemma-composition-thickening

If $\mathcal{X} \subset \mathcal{X}'$ and $\mathcal{X}' \subset \mathcal{X}''$ are thickenings of algebraic stacks, then so is $\mathcal{X} \subset \mathcal{X}''$.


stacks-more-morphisms-lemma-reduced-diagonal

Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Then $\mathcal{X} \times_\mathcal{Y} \mathcal{X} \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{X}'$ is a thickening and the canonical diagram $$ \xymatrix{ \mathcal{X} \ar[r]_-\Delta \ar[d] & \mathcal{X} \times_\mathcal{Y} \mathcal{X} \ar[d] \\ \mathcal{X}' \ar[r]^-{\Delta'} & \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{X}' } $$ is cartesian.


stacks-more-morphisms-lemma-thickening-diagonals

Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Let $\Delta : \mathcal{X} \to \mathcal{X} \times_\mathcal{Y} \mathcal{X}$ and $\Delta' : \mathcal{X}' \to \mathcal{X}' \times_{\mathcal{Y}'} \mathcal{X}'$ be the corresponding diagonal morphisms. Then each property from the following list is satisfied by $\Delta$ if and only if it is satisfied by $\Delta'$: (a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact, (e) universally closed, (f) integral, (g) quasi-separated, (h) separated, (i) universally injective, (j) universally open, (k) locally quasi-finite, (l) finite, (m) unramified, (n) monomorphism, (o) immersion, (p) closed immersion, and (q) proper.


stacks-more-morphisms-lemma-thickening-properties

\begin{reference} \cite[Theorem 2.2.5]{Conrad-moduli} \end{reference} Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Then \begin{enumerate} \item $\mathcal{X}$ is an algebraic space if and only if $\mathcal{X}'$ is an algebraic space, \item $\mathcal{X}$ is a scheme if and only if $\mathcal{X}'$ is a scheme, \item $\mathcal{X}$ is DM if and only if $\mathcal{X}'$ is DM, \item $\mathcal{X}$ is quasi-DM if and only if $\mathcal{X}'$ is quasi-DM, \item $\mathcal{X}$ is separated if and only if $\mathcal{X}'$ is separated, \item $\mathcal{X}$ is quasi-separated if and only if $\mathcal{X}'$ is quasi-separated, and \item add more here. \end{enumerate}


stacks-more-morphisms-lemma-thicken-property-morphisms

Let $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ be a morphism of thickenings of algebraic stacks. Then \begin{enumerate} \item $f$ is an affine morphism if and only if $f'$ is an affine morphism, \item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism, \item $f$ is quasi-compact if and only if $f'$ quasi-compact, \item $f$ is universally closed if and only if $f'$ is universally closed, \item $f$ is integral if and only if $f'$ is integral, \item $f$ is universally injective if and only if $f'$ is universally injective, \item $f$ is universally open if and only if $f'$ is universally open, \item $f$ is quasi-DM if and only if $f'$ is quasi-DM, \item $f$ is DM if and only if $f'$ is DM, \item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated, \item $f$ is representable if and only if $f'$ is representable, \item $f$ is representable by algebraic spaces if and only if $f'$ is representable by algebraic spaces, \item add more here. \end{enumerate}


stacks-more-morphisms-lemma-flatness-morphism-thickenings-fp-over-ft

Consider a commutative diagram $$ \xymatrix{ (\mathcal{X} \subset \mathcal{X}') \ar[rr]_{(f, f')} \ar[rd] & & (\mathcal{Y} \subset \mathcal{Y}') \ar[ld] \\ & (\mathcal{B} \subset \mathcal{B}') } $$ of thickenings of algebraic stacks. Assume \begin{enumerate} \item $\mathcal{Y}' \to \mathcal{B}'$ is locally of finite type, \item $\mathcal{X}' \to \mathcal{B}'$ is flat and locally of finite presentation, \item $f$ is flat, and \item $\mathcal{X} = \mathcal{B} \times_{\mathcal{B}'} \mathcal{X}'$ and $\mathcal{Y} = \mathcal{B} \times_{\mathcal{B}'} \mathcal{Y}'$. \end{enumerate} Then $f'$ is flat and for all $y' \in |\mathcal{Y}'|$ in the image of $|f'|$ the morphism $\mathcal{Y}' \to \mathcal{B}'$ is flat at $y'$.


stacks-more-morphisms-lemma-deform-property-fp-over-ft

Consider a commutative diagram $$ \xymatrix{ (\mathcal{X} \subset \mathcal{X}') \ar[rr]_{(f, f')} \ar[rd] & & (\mathcal{Y} \subset \mathcal{Y}') \ar[ld] \\ & (\mathcal{B} \subset \mathcal{B}') } $$ of thickenings of algebraic stacks. Assume $\mathcal{Y}' \to \mathcal{B}'$ locally of finite type, $\mathcal{X}' \to \mathcal{B}'$ flat and locally of finite presentation, $\mathcal{X} = \mathcal{B} \times_{\mathcal{B}'} \mathcal{X}'$, and $\mathcal{Y} = \mathcal{B} \times_{\mathcal{B}'} \mathcal{Y}'$. Then \begin{enumerate} \item $f$ is flat if and only if $f'$ is flat, \label{item-flat-fp-over-ft} \item $f$ is an isomorphism if and only if $f'$ is an isomorphism, \label{item-isomorphism-fp-over-ft} \item $f$ is an open immersion if and only if $f'$ is an open immersion, \label{item-open-immersion-fp-over-ft} \item $f$ is a monomorphism if and only if $f'$ is a monomorphism, \label{item-monomorphism-fp-over-ft} \item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite, \label{item-quasi-finite-fp-over-ft} \item $f$ is syntomic if and only if $f'$ is syntomic, \label{item-syntomic-fp-over-ft} \item $f$ is smooth if and only if $f'$ is smooth, \label{item-smooth-fp-over-ft} \item $f$ is unramified if and only if $f'$ is unramified, \label{item-unramified-fp-over-ft} \item $f$ is \'etale if and only if $f'$ is \'etale, \label{item-etale-fp-over-ft} \item $f$ is finite if and only if $f'$ is finite, and \label{item-finite-fp-over-ft} \item add more here. \end{enumerate}


stacks-more-morphisms-lemma-morphisms-lifts-etale

For any morphism (\ref{equation-morphism}) the map $f' : V' \to U'$ is \'etale.


stacks-more-morphisms-lemma-gerbe-of-lifts-fibred

The category $p : \mathcal{C} \to W_{spaces, \etale}$ constructed in Remark \ref{remark-gerbe-of-lifts} is fibred in groupoids.


stacks-more-morphisms-lemma-gerbe-of-lifts-stack

The category $p : \mathcal{C} \to W_{spaces, \etale}$ constructed in Remark \ref{remark-gerbe-of-lifts} is a stack in groupoids.


stacks-more-morphisms-lemma-etale-local-lifts

Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Let $W$ be an algebraic space and let $W \to \mathcal{X}$ be a smooth morphism. There exists an \'etale covering $\{W_i \to W\}_{i \in I}$ and for each $i$ a cartesian diagram $$ \xymatrix{ W_i \ar[r] \ar[d] & W_i' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } $$ with $W_i' \to \mathcal{X}'$ smooth.


stacks-more-morphisms-lemma-etale-local-lifts-isomorphic

Let $\mathcal{X} \subset \mathcal{X}'$ be a thickening of algebraic stacks. Consider a commutative diagram $$ \xymatrix{ W'' \ar[d]_{x''} & W \ar[l] \ar[r] \ar[d]_x & W' \ar[d]^{x'} \\ \mathcal{X}' & \mathcal{X} \ar[l] \ar[r] & \mathcal{X}' } $$ with cartesian squares where $W', W, W''$ are algebraic spaces and the vertical arrows are smooth. Then there exist \begin{enumerate} \item an \'etale covering $\{f'_k : W'_k \to W'\}_{k \in K}$, \item \'etale morphisms $f''_k : W'_k \to W''$, and \item $2$-morphisms $\gamma_k : x'' \circ f''_k \to x' \circ f'_k$ \end{enumerate} such that (a) $(f'_k)^{-1}(W) = (f''_k)^{-1}(W)$, (b) $f'_k|_{(f'_k)^{-1}(W)} = f''_k|_{(f''_k)^{-1}(W)}$, and (c) pulling back $\gamma_k$ to the closed subscheme of (a) agrees with the $2$-morphism given by the commutativity of the initial diagram over $W$.


stacks-more-morphisms-lemma-gerbe-of-lifts

The category $p : \mathcal{C} \to W_{spaces, \etale}$ constructed in Remark \ref{remark-gerbe-of-lifts} is a gerbe.


stacks-more-morphisms-lemma-gerbe-of-lifts-first-order

In Remark \ref{remark-gerbe-of-lifts} assume $\mathcal{X} \subset \mathcal{X}'$ is a first order thickening. Then \begin{enumerate} \item the automorphism sheaves of objects of the gerbe $p : \mathcal{C} \to W_{spaces, \etale}$ constructed in Remark \ref{remark-gerbe-of-lifts} are abelian, and \item the sheaf of groups $\mathcal{G}$ constructed in Stacks, Lemma \ref{stacks-lemma-gerbe-abelian-auts} is a quasi-coherent $\mathcal{O}_W$-module. \end{enumerate}


stacks-more-morphisms-lemma-inf-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack over a scheme $S$. Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally of finite presentation. Let $A \to B$ be a flat $S$-algebra homomorphism. Let $x$ be an object of $\mathcal{X}$ over $A$ and set $y = x|_B$. Then $\text{Inf}_x(M) \otimes_A B = \text{Inf}_y(M \otimes_A B)$.


stacks-more-morphisms-lemma-sheaf-of-infinitesimal-lifts

Let $\mathcal{X}$ be an algebraic stack over a base scheme $S$. Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is locally of finite presentation. Let $(A' \to A, x)$ be a deformation situation. Then the functor $$ F : B' \longmapsto \{\text{lifts of }x|_{B' \otimes_{A'} A}\text{ to } B'\}/\text{isomorphisms} $$ is a sheaf on the site $(\textit{Aff}/\Spec(A'))_{fppf}$ of Topologies, Definition \ref{topologies-definition-big-small-fppf}.


stacks-more-morphisms-lemma-T-quasi-coherent

Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose structure morphism $\mathcal{X} \to S$ is locally of finite presentation. Let $A \to B$ be a flat $S$-algebra homomorphism. Let $x$ be an object of $\mathcal{X}$ over $A$. Then $T_x(M) \otimes_A B = T_y(M \otimes_A B)$.


stacks-more-morphisms-lemma-local-lift-enough

Let $\mathcal{X}$ be an algebraic stack over a scheme $S$ whose structure morphism $\mathcal{X} \to S$ is locally of finite presentation. Let $(A' \to A, x)$ be a deformation situation. If there exists a faithfully flat finitely presented $A'$-algebra $B'$ and an object $y'$ of $\mathcal{X}$ over $B'$ lifting $x|_{B' \otimes_{A'} A}$, then there exists an object $x'$ over $A'$ lifting $x$.


stacks-more-morphisms-lemma-reformulate-formal-smoothness

A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is formally smooth (Definition \ref{definition-formally-smooth}) if and only if for every diagram (\ref{equation-diagram}) and $\gamma$ the category of dotted arrows is nonempty.


stacks-more-morphisms-lemma-lift-to-smooth

Let $T \to T'$ be a first order thickening of affine schemes. Let $\mathcal{X}'$ be an algebraic stack over $T'$ whose structure morphism $\mathcal{X}' \to T'$ is smooth. Let $x : T \to \mathcal{X}'$ be a morphism over $T'$. Then there exists a morphsm $x' : T' \to \mathcal{X}'$ over $T'$ with $x'|_T = x$.


stacks-more-morphisms-lemma-smooth-formally-smooth

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent: \begin{enumerate} \item The morphism $f$ is smooth. \item The morphism $f$ is locally of finite presentation and formally smooth. \end{enumerate}


stacks-more-morphisms-lemma-flatten-stack

Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Let $V \subset Y$ be an open subspace. Assume \begin{enumerate} \item $Y$ is quasi-compact and quasi-separated, \item $f$ is of finite type and quasi-separated, \item $V$ is quasi-compact, and \item $\mathcal{X}_V$ is flat and locally of finite presentation over $V$. \end{enumerate} Then there exists a $V$-admissible blowup $Y' \to Y$ and a closed substack $\mathcal{X}' \subset \mathcal{X}_{Y'}$ with $\mathcal{X}'_V = \mathcal{X}_V$ such that $\mathcal{X}' \to Y'$ is flat and of finite presentation.


stacks-more-morphisms-lemma-finite-cover-factor

Let $Y$ be a quasi-compact and quasi-separated algebraic space. Let $V \subset Y$ be a quasi-compact open. Let $f : \mathcal{X} \to V$ be surjective, flat, and locally of finite presentation. Then there exists a finite surjective morphism $g : Y' \to Y$ such that $V' = g^{-1}(V) \to Y$ factors Zariski locally through $f$.


stacks-more-morphisms-lemma-refined-valuative-criterion-proper

Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ and $h$ are of finite type, that $f$ is separated, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram $$ \xymatrix{ \Spec(K) \ar[r]_-u \ar[d]_j & \mathcal{U} \ar[r]_h & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[rr]^-y & & \mathcal{Y} } $$ where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$ there exist an extension $K'/K$ of fields, a valuation ring $A' \subset K'$ dominating $A$ such that the category of dotted arrows for the induced diagram $$ \xymatrix{ \Spec(K') \ar[r]_-{x'} \ar[d]_{j'} & \mathcal{X} \ar[d]^f \\ \Spec(A') \ar[r]^-{y'} \ar@{..>}[ru] & \mathcal{Y} } $$ with induced $2$-arrow $\gamma' : y' \circ j' \to f \circ x'$ is nonempty (Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-fill-in-diagram}), then $f$ is proper.


stacks-more-morphisms-lemma-refined-valuative-criterion-separated

Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram $$ \xymatrix{ \Spec(K) \ar[r]_-u \ar[d]_j & \mathcal{U} \ar[r]_h & \mathcal{X} \ar[d]^f \\ \Spec(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{Y} } $$ where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then $f$ is separated.


stacks-more-morphisms-lemma-quotient-compare

Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with $s, t : R \to U$ flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$. Given an algebraic space $Y$ there is a $1$-to-$1$ correspondence between morphisms $f : \mathcal{X} \to Y$ and $R$-invariant morphisms $\phi : U \to Y$.


stacks-more-morphisms-lemma-categorical-quotient-compare

With assumption and notation as in Lemma \ref{lemma-quotient-compare}. Then $f$ is a (uniform) categorical moduli space if and only if $\phi$ is a (uniform) categorical quotient. Similarly for moduli spaces in a full subcategory.


stacks-more-morphisms-lemma-check-uniform-categorical-quotient-on-affines

Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. If for every affine scheme $Y'$ and flat morphism $Y' \to Y$ the base change $f' : Y' \times_Y \mathcal{X} \to Y'$ is a categorical moduli space, then $f$ is a uniform categorical moduli space.


stacks-more-morphisms-lemma-well-nigh-affine

Let $\mathcal{X}$ be an algebraic stack. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is well-nigh affine, and \item there exists a groupoid scheme $(U, R, s, t, c)$ with $U$ and $R$ affine and $s, t : R \to U$ finite locally free such that $\mathcal{X} = [U/R]$. \end{enumerate} If true then $\mathcal{X}$ is quasi-compact, quasi-DM, and separated.


stacks-more-morphisms-lemma-affine-over-well-nigh-affine

Let the algebraic stack $\mathcal{X}$ be well-nigh affine. \begin{enumerate} \item If $\mathcal{X}$ is an algebraic space, then it is affine. \item If $\mathcal{X}' \to \mathcal{X}$ is an affine morphism of algebraic stacks, then $\mathcal{X}'$ is well-nigh affine. \end{enumerate}


stacks-more-morphisms-lemma-well-nigh-affine-moduli-space

Let the algebraic stack $\mathcal{X}$ be well-nigh affine. There exists a uniform categorical moduli space $$ f : \mathcal{X} \longrightarrow M $$ in the category of affine schemes. Moreover $f$ is separated, quasi-compact, and a universal homeomorphism.


stacks-more-morphisms-lemma-well-nigh-affine-moduli-space-etale

Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $\mathcal{X}'$ and $\mathcal{X}$ are well-nigh affine, $h$ is \'etale, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \ref{stacks-morphisms-remark-identify-automorphism-groups}). Then there exists a cartesian diagram $$ \xymatrix{ \mathcal{X}' \ar[d] \ar[r] & \mathcal{X} \ar[d] \\ M' \ar[r] & M } $$ where $M' \to M$ is \'etale and the vertical arrows are the moduli spaces constructed in Lemma \ref{lemma-well-nigh-affine-moduli-space}.


stacks-more-morphisms-lemma-moduli-space-finite-affine

Let the algebraic stack $\mathcal{X}$ be well-nigh affine. The morphism $$ f : \mathcal{X} \longrightarrow M $$ of Lemma \ref{lemma-well-nigh-affine-moduli-space} is a uniform categorical moduli space.


stacks-more-morphisms-lemma-etale-separated-over-well-nigh-affine

Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $\mathcal{X}$ is well-nigh affine, $h$ is \'etale, $h$ is separated, and $h$ induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \ref{stacks-morphisms-remark-identify-automorphism-groups}). Then there exists a cartesian diagram $$ \xymatrix{ \mathcal{X}' \ar[d] \ar[r] & \mathcal{X} \ar[d] \\ M' \ar[r] & M } $$ where $M' \to M$ is a separated \'etale morphism of schemes and $\mathcal{X} \to M$ is the moduli space constructed in Lemma \ref{lemma-well-nigh-affine-moduli-space}.


stacks-more-morphisms-lemma-etale-local-finite-inertia

Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is finite. Then there exist a set $I$ and for $i \in I$ a morphism of algebraic stacks $$ g_i : \mathcal{X}_i \longrightarrow \mathcal{X} $$ with the following properties \begin{enumerate} \item $|\mathcal{X}| = \bigcup |g_i|(|\mathcal{X}_i|)$, \item $\mathcal{X}_i$ is well-nigh affine, \item $\mathcal{I}_{\mathcal{X}_i} \to \mathcal{X}_i \times_\mathcal{X} \mathcal{I}_\mathcal{X}$ is an isomorphism, and \item $g_i : \mathcal{X}_i \to \mathcal{X}$ is representable by algebraic spaces, separated, and \'etale, \end{enumerate}


stacks-more-morphisms-lemma-etale-separated-over-keel-mori

Let $h : \mathcal{X}' \to \mathcal{X}$ be a morphism of algebraic stacks. Assume \begin{enumerate} \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is finite, \item $h$ is \'etale, separated, and induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark \ref{stacks-morphisms-remark-identify-automorphism-groups}). \end{enumerate} Then there exists a cartesian diagram $$ \xymatrix{ \mathcal{X}' \ar[d] \ar[r] & \mathcal{X} \ar[d] \\ M' \ar[r] & M } $$ where $M' \to M$ is a separated \'etale morphism of algebraic spaces and the vertical arrows are the moduli spaces constructed in Theorem \ref{theorem-keel-mori}.


stacks-more-morphisms-lemma-keel-mori-finite-type

Let $p : \mathcal{X} \to Y$ be a morphism of an algebraic stack to an algebraic space. Assume \begin{enumerate} \item $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is finite, \item $Y$ is locally Noetherian, and \item $p$ is locally of finite type. \end{enumerate} Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem \ref{theorem-keel-mori}. Then $M \to Y$ is locally of finite type.


stacks-more-morphisms-lemma-keel-mori-diagonal

Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is finite. Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem \ref{theorem-keel-mori}. \begin{enumerate} \item If $\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated. \item If $\mathcal{X}$ is separated, then $M$ is separated. \item Add more here, for example relative versions of the above. \end{enumerate}


perfect-theorem-approximation

Let $X$ be a quasi-compact and quasi-separated scheme. Then approximation by perfect complexes holds on $X$.


perfect-theorem-bondal-van-den-Bergh

Let $X$ be a quasi-compact and quasi-separated scheme. The category $D_\QCoh(\mathcal{O}_X)$ can be generated by a single perfect object. More precisely, there exists a perfect object $P$ of $D(\mathcal{O}_X)$ such that for  $E \in D_\QCoh(\mathcal{O}_X)$ the following are equivalent \begin{enumerate} \item $E = 0$, and \item $\Hom_{D(\mathcal{O}_X)}(P[n], E) = 0$ for all $n \in \mathbf{Z}$. \end{enumerate}


perfect-theorem-DQCoh-is-Ddga

Let $X$ be a quasi-compact and quasi-separated scheme. Then there exist a differential graded algebra $(E, \text{d})$ with only a finite number of nonzero cohomology groups $H^i(E)$ such that $D_\QCoh(\mathcal{O}_X)$ is equivalent to $D(E, \text{d})$.


perfect-lemma-quasi-coherence-direct-sums

Let $X$ be a scheme. Then $D_\QCoh(\mathcal{O}_X)$ has direct sums.


perfect-lemma-Rlim-quasi-coherent

Let $X$ be a scheme. Let $(K_n)$ be an inverse system of $D_\QCoh(\mathcal{O}_X)$ with derived limit $K = R\lim K_n$ in $D(\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \to H^q(K_n)$ is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$. Then \begin{enumerate} \item $H^q(K) = \lim H^q(K_n)$, \item $R\lim H^q(K_n) = \lim H^q(K_n)$, and \item for every affine open $U \subset X$ we have $H^p(U, \lim H^q(K_n)) = 0$ for $p > 0$. \end{enumerate}


perfect-lemma-nice-K-injective

Let $X$ be a scheme. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Then the canonical map $E \to R\lim \tau_{\geq -n}E$ is an isomorphism\footnote{In particular, $E$ has a K-injective representative as in Cohomology, Lemma \ref{cohomology-lemma-K-injective}.}.


perfect-lemma-application-nice-K-injective

Let $X$ be a scheme. Let $F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}$ be an additive functor and $N \geq 0$ an integer. Assume that \begin{enumerate} \item $F$ commutes with countable direct products, \item $R^pF(\mathcal{F}) = 0$ for all $p \geq N$ and $\mathcal{F}$ quasi-coherent. \end{enumerate} Then for $E \in D_\QCoh(\mathcal{O}_X)$ \begin{enumerate} \item $H^i(RF(\tau_{\leq a}E)) \to H^i(RF(E))$ is an isomorphism for $i \leq a$, \item $H^i(RF(E)) \to H^i(RF(\tau_{\geq b - N + 1}E))$ is an isomorphism for $i \geq b$, \item if $H^i(E) = 0$ for $i \not \in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^i(RF(E)) = 0$ for $i \not \in [a, b + N - 1]$. \end{enumerate}


perfect-lemma-affine-compare-bounded

Let $X = \Spec(A)$ be an affine scheme. All the functors in the diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[rr]_{(\ref{equation-compare})} & & D_\QCoh(\mathcal{O}_X) \ar[ld]^{R\Gamma(X, -)} \\ & D(A) \ar[lu]^{\widetilde{\ \ }} } $$ are equivalences of triangulated categories. Moreover, for $E$ in $D_\QCoh(\mathcal{O}_X)$ we have $H^0(X, E) = H^0(X, H^0(E))$.


perfect-lemma-affine-K-flat

Let $X = \Spec(A)$ be an affine scheme. If $K^\bullet$ is a K-flat complex of $A$-modules, then $\widetilde{K^\bullet}$ is a K-flat complex of $\mathcal{O}_X$-modules.


perfect-lemma-quasi-coherence-pushforward

If $f : X \to Y$ is a morphism of affine schemes given by the ring map $A \to B$, then the diagram $$ \xymatrix{ D(B) \ar[d] \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(A) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ commutes.


perfect-lemma-quasi-coherence-pullback

Let $f : Y \to X$ be a morphism of schemes. \begin{enumerate} \item The functor $Lf^*$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$. \item If $X$ and $Y$ are affine and $f$ is given by the ring map $A \to B$, then the diagram $$ \xymatrix{ D(B) \ar[r] & D_\QCoh(\mathcal{O}_Y) \\ D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} & D_\QCoh(\mathcal{O}_X) \ar[u]_{Lf^*} } $$ commutes. \end{enumerate}


perfect-lemma-quasi-coherence-tensor-product

Let $X$ be a scheme. \begin{enumerate} \item For objects $K, L$ of $D_\QCoh(\mathcal{O}_X)$ the derived tensor product $K \otimes^\mathbf{L}_{\mathcal{O}_X} L$ is in $D_\QCoh(\mathcal{O}_X)$. \item If $X = \Spec(A)$ is affine then $$ \widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{K^\bullet} = \widetilde{M^\bullet \otimes_A^\mathbf{L} K^\bullet} $$ for any pair of complexes of $A$-modules $K^\bullet$, $M^\bullet$. \end{enumerate}


perfect-lemma-quasi-coherence-direct-image

Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. \begin{enumerate} \item The functor $Rf_*$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_S)$. \item If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$ such that for an object $E$ of $D_\QCoh(\mathcal{O}_X)$ with $H^m(E) = 0$ for $m > 0$ we have $H^m(Rf_*E) = 0$ for $m \geq N$. \item In fact, if $S$ is quasi-compact we can find $N = N(X, S, f)$ such that for every morphism of schemes $S' \to S$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \to S'$ is the base change of $f$. \end{enumerate}


perfect-lemma-acyclicity-lemma

Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of schemes. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules each of which is right acyclic for $f_*$. Then $f_*\mathcal{F}^\bullet$ represents $Rf_*\mathcal{F}^\bullet$ in $D(\mathcal{O}_S)$.


perfect-lemma-acyclicity-lemma-global

Let $X$ be a quasi-separated and quasi-compact scheme. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules each of which is right acyclic for $\Gamma(X, -)$. Then $\Gamma(X, \mathcal{F}^\bullet)$ represents $R\Gamma(X, \mathcal{F}^\bullet)$ in $D(\Gamma(X, \mathcal{O}_X)$.


perfect-lemma-spectral-sequence

Let $X$ be a quasi-separated and quasi-compact scheme. For any object $K$ of $D_\QCoh(\mathcal{O}_X)$ the spectral sequence $$ E_2^{i, j} = H^i(X, H^j(K)) \Rightarrow H^{i + j}(X, K) $$ of Cohomology, Example \ref{cohomology-example-spectral-sequence} is bounded and converges.


perfect-lemma-quasi-coherence-pushforward-direct-sums

Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of schemes. Then $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$ commutes with direct sums.


perfect-lemma-pushforward-affine-morphism

Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Then $f_*\mathcal{F}^\bullet = Rf_*\mathcal{F}^\bullet$.


perfect-lemma-affine-morphism

Let $f : X \to S$ be an affine morphism of schemes. Then $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$ reflects isomorphisms.


perfect-lemma-affine-morphism-pull-push

Let $f : X \to S$ be an affine morphism of schemes. For $E$ in $D_\QCoh(\mathcal{O}_S)$ we have $Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_S} f_*\mathcal{O}_X$.


perfect-lemma-affine-morphism-equivalence

Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ induces an equivalence $$ \Phi : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(f_*\mathcal{O}_X) $$ whose composition with $D_\QCoh(f_*\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$.


perfect-lemma-extended-alternating-zero

With $X$, $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$, and $\mathcal{F}$ as in Remark \ref{remark-support-c-equations} the complex (\ref{equation-extended-alternating}) restricts to an acyclic complex over $X \setminus Z$.


perfect-lemma-extended-alternating-represented

With $X$, $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$, and $\mathcal{F}$ as in Remark \ref{remark-support-c-equations}. If $\mathcal{F}$ is quasi-coherent, then the complex (\ref{equation-extended-alternating}) represents $i_* R\mathcal{H}_Z(\mathcal{F})$ in $D_Z(\mathcal{O}_X)$.


perfect-lemma-supported-trivial-vanishing

Let $X$ be a scheme. Let $T \subset X$ be a closed subset which can locally be cut out by at most $c$ elements of the structure sheaf. Then $\mathcal{H}^i_Z(\mathcal{F}) = 0$ for $i > c$ and any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$.


perfect-lemma-supported-vanishing

Let $X$ be a scheme. Let $T \subset X$ be a closed subset which can locally be cut out by a Koszul regular sequence having $c$ elements. Then $\mathcal{H}^i_Z(\mathcal{F}) = 0$ for $i \not = c$ for every flat, quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$.


perfect-lemma-supported-map-determinant

With $X$, $f_1, \ldots, f_c \in \Gamma(X, \mathcal{O}_X)$, and $\mathcal{F}$ as in Remark \ref{remark-support-c-equations}. Let $a_{ji} \in \Gamma(X, \mathcal{O}_X)$ for $1 \leq i, j \leq c$ and set $g_j = \sum_{i = 1, \ldots, c} a_{ji}f_i$. Assume $g_1, \ldots, g_c$ scheme theoretically cut out $Z$. If $\mathcal{F}$ is quasi-coherent, then $$ c_{f_1, \ldots, f_c} = \det(a_{ji}) c_{g_1, \ldots, g_c} $$ where $c_{f_1, \ldots, f_c}$ and $c_{g_1, \ldots, g_c}$ are as in Remark \ref{remark-supported-map-c-equations}.


perfect-lemma-affine-pushforward

Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ defines a derived functor $f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$. This functor has the property that $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ commutes.


perfect-lemma-flat-pushforward-coherator

Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is quasi-compact, quasi-separated, and flat. Then, denoting $$ \Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y)) $$ the right derived functor of $f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$ we have $RQ_Y \circ Rf_* = \Phi \circ RQ_X$.


perfect-lemma-affine-coherator

Let $X = \Spec(A)$ be an affine scheme. Then \begin{enumerate} \item $Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)$ is the functor which sends $\mathcal{F}$ to the quasi-coherent $\mathcal{O}_X$-module associated to the $A$-module $\Gamma(X, \mathcal{F})$, \item $RQ_X : D(\mathcal{O}_X) \to D(\QCoh(\mathcal{O}_X))$ is the functor which sends $E$ to the complex of quasi-coherent $\mathcal{O}_X$-modules associated to the object $R\Gamma(X, E)$ of $D(A)$, \item restricted to $D_\QCoh(\mathcal{O}_X)$ the functor $RQ_X$ defines a quasi-inverse to (\ref{equation-compare}). \end{enumerate}


perfect-lemma-argument-proves

Let $X$ be a quasi-compact and quasi-separated scheme. Suppose that for every affine open $U \subset X$ the right derived functor $$ \Phi : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_X)) $$ of the left exact functor $j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)$ fits into a commutative diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_U)) \ar[d]_\Phi \ar[r]_{i_U} & D_\QCoh(\mathcal{O}_U) \ar[d]^{Rj_*} \\ D(\QCoh(\mathcal{O}_X)) \ar[r]^{i_X} & D_\QCoh(\mathcal{O}_X) } $$ 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-lemma-injective-quasi-coherent-sheaf-Noetherian

Let $X$ be a Noetherian scheme. Let $\mathcal{J}$ be an injective object of $\QCoh(\mathcal{O}_X)$. Then $\mathcal{J}$ is a flasque sheaf of $\mathcal{O}_X$-modules.


perfect-lemma-Noetherian-pushforward

Let $f : X \to Y$ be a morphism of Noetherian schemes. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$ such that the diagram $$ \xymatrix{ D(\QCoh(\mathcal{O}_X)) \ar[d]_{\Phi} \ar[r] & D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\ D(\QCoh(\mathcal{O}_Y)) \ar[r] & D_\QCoh(\mathcal{O}_Y) } $$ commutes.


perfect-lemma-alternating-cech-complex

In Situation \ref{situation-complex}. Let $M$ be an $A$-module and denote $\mathcal{F}$ the associated $\mathcal{O}_X$-module. Then there is a canonical isomorphism of complexes $$ \colim_e \Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M) \longrightarrow \check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) $$ functorial in $M$.


perfect-lemma-alternating-cech-complex-complex

In Situation \ref{situation-complex}. Let $M^\bullet$ be a complex of $A$-modules and denote $\mathcal{F}^\bullet$ the associated complex of $\mathcal{O}_X$-modules. Then there is a canonical isomorphism of complexes $$ \colim_e \text{Tot}(\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet)) \longrightarrow \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) $$ functorial in $M^\bullet$.


perfect-lemma-alternating-cech-complex-complex-computes-cohomology

In Situation \ref{situation-complex}. Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent $\mathcal{O}_X$-modules. Then there is a canonical isomorphism $$ \text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet)) \longrightarrow R\Gamma(U, \mathcal{F}^\bullet) $$ in $D(A)$ functorial in $\mathcal{F}^\bullet$.


perfect-lemma-represent-cohomology-class-on-closed

In Situation \ref{situation-complex}. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Assume that $H^i(E)|_U = 0$ for $i = - r + 1, \ldots, 0$. Then given $s \in H^0(X, E)$ there exists an $e \geq 0$ and a morphism $K_e \to E$ such that $s$ is in the image of $H^0(X, K_e) \to H^0(X, E)$.


perfect-lemma-pseudo-coherent

Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent object of $D(\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent $\mathcal{O}_X$-module for $i > m$ and $H^m(E)$ is a quotient of a quasi-coherent $\mathcal{O}_X$-module. If $E$ is pseudo-coherent, then $E$ is an object of $D_\QCoh(\mathcal{O}_X)$.


perfect-lemma-pseudo-coherent-affine

Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_X)$. Then $E$ is an $m$-pseudo-coherent (resp.\ pseudo-coherent) as an object of $D(\mathcal{O}_X)$ if and only if $M^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) as a complex of $A$-modules.


perfect-lemma-identify-pseudo-coherent-noetherian

Let $X$ be a Noetherian scheme. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. For $m \in \mathbf{Z}$ the following are equivalent \begin{enumerate} \item $H^i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$, and \item $E$ is $m$-pseudo-coherent. \end{enumerate} In particular, $E$ is pseudo-coherent if and only if $E$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_X)$.


perfect-lemma-tor-dimension-affine

Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item $E$ has tor amplitude in $[a, b]$ if and only if $M^\bullet$ has tor amplitude in $[a, b]$. \item $E$ has finite tor dimension if and only if $M^\bullet$ has finite tor dimension. \end{enumerate}


perfect-lemma-tor-dimension-rel-affine

Let $f : X \to S$ be a morphism of affine schemes corresponding to the ring map $R \to A$. Let $M^\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item $E$ as an object of $D(f^{-1}\mathcal{O}_S)$ has tor amplitude in $[a, b]$ if and only if $M^\bullet$ has tor amplitude in $[a, b]$ as an object of $D(R)$. \item $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_S)$ if and only if $M^\bullet$ has finite tor dimension as an object of $D(R)$. \end{enumerate}


perfect-lemma-tor-qc-qs

Let $X$ be a quasi-separated scheme. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Let $a \leq b$. The following are equivalent \begin{enumerate} \item $E$ has tor amplitude in $[a, b]$, and \item for all $\mathcal{F}$ in $\QCoh(\mathcal{O}_X)$ we have $H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0$ for $i \not \in [a, b]$. \end{enumerate}


perfect-lemma-perfect-affine

Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a complex of $A$-modules and let $E$ be the corresponding object of $D(\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\mathcal{O}_X)$ if and only if $M^\bullet$ is perfect as an object of $D(A)$.


perfect-lemma-quasi-coherence-internal-hom

Let $X$ be a scheme. \begin{enumerate} \item If $L$ is in $D^+_\QCoh(\mathcal{O}_X)$ and $K$ in $D(\mathcal{O}_X)$ is pseudo-coherent, then $R\SheafHom(K, L)$ is in $D_\QCoh(\mathcal{O}_X)$ and locally bounded below. \item If $L$ is in $D_\QCoh(\mathcal{O}_X)$ and $K$ in $D(\mathcal{O}_X)$ is perfect, then $R\SheafHom(K, L)$ is in $D_\QCoh(\mathcal{O}_X)$. \item If $X = \Spec(A)$ is affine and $K, L \in D(A)$ then $$ R\SheafHom(\widetilde{K}, \widetilde{L}) = \widetilde{R\Hom_A(K, L)} $$ in the following two cases \begin{enumerate} \item $K$ is pseudo-coherent and $L$ is bounded below, \item $K$ is perfect and $L$ arbitrary. \end{enumerate} \item If $X = \Spec(A)$ and $K, L$ are in $D(A)$, then the $n$th cohomology sheaf of $R\SheafHom(\widetilde{K}, \widetilde{L})$ is the sheaf associated to the presheaf $$ X \supset D(f) \longmapsto \Ext^n_{A_f}(K \otimes_A A_f, L \otimes_A A_f) $$ for $f \in A$. \end{enumerate}


perfect-lemma-internal-hom-evaluate-tensor-isomorphism

Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\QCoh(\mathcal{O}_X)$. The map $$ K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L) \longrightarrow R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L) $$ of Cohomology, Lemma \ref{cohomology-lemma-internal-hom-diagonal-better} is an isomorphism in the following cases \begin{enumerate} \item $M$ perfect, or \item $K$ is perfect, or \item $M$ is pseudo-coherent, $L \in D^+(\mathcal{O}_X)$, and $K$ has finite tor dimension. \end{enumerate}


perfect-lemma-coh-to-qcoh

Let $X$ be a Noetherian scheme. Then the functor $$ D^-(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^-_{\textit{Coh}(\mathcal{O}_X)}(\QCoh(\mathcal{O}_X)) $$ is an equivalence.


perfect-lemma-direct-image-coherent

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_S)$.


perfect-lemma-direct-image-coherent-bdd-below

Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^+_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$ is proper over $S$ for all $i$. Then $Rf_*E$ is an object of $D^+_{\textit{Coh}}(\mathcal{O}_S)$.


perfect-lemma-coherent-internal-hom

Let $X$ be a locally Noetherian scheme. If $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_X)$, then $R\SheafHom(K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$.


perfect-lemma-perfect-on-noetherian

Let $X$ be a Noetherian scheme. Let $E$ in $D(\mathcal{O}_X)$ be perfect. Then \begin{enumerate} \item $E$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$, \item if $L$ is in $D_{\textit{Coh}}(\mathcal{O}_X)$ then $E \otimes_{\mathcal{O}_X}^\mathbf{L} L$ and $R\SheafHom_{\mathcal{O}_X}(E, L)$ are in $D_{\textit{Coh}}(\mathcal{O}_X)$, \item if $L$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ then $E \otimes_{\mathcal{O}_X}^\mathbf{L} L$ and $R\SheafHom_{\mathcal{O}_X}(E, L)$ are in $D^b_{\textit{Coh}}(\mathcal{O}_X)$, \item if $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$ then $E \otimes_{\mathcal{O}_X}^\mathbf{L} L$ and $R\SheafHom_{\mathcal{O}_X}(E, L)$ are in $D^+_{\textit{Coh}}(\mathcal{O}_X)$, \item if $L$ is in $D^-_{\textit{Coh}}(\mathcal{O}_X)$ then $E \otimes_{\mathcal{O}_X}^\mathbf{L} L$ and $R\SheafHom_{\mathcal{O}_X}(E, L)$ are in $D^-_{\textit{Coh}}(\mathcal{O}_X)$. \end{enumerate}


perfect-lemma-ext-finite

Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$. For $L$ in $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_X)$, the $A$-modules $\Ext_{\mathcal{O}_X}^n(K, L)$ are finite.


perfect-lemma-perfect-on-regular

Let $X$ be a Noetherian regular scheme of finite dimension. Then every object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ is perfect and conversely every perfect object of $D(\mathcal{O}_X)$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$.


perfect-lemma-tor-amplitude-descends

Let $f : X \to Y$ be a surjective flat morphism of schemes (or more generally locally ringed spaces). Let $E \in D(\mathcal{O}_Y)$. Let $a, b \in \mathbf{Z}$. Then $E$ has tor-amplitude in $[a, b]$ if and only if $Lf^*E$ has tor-amplitude in $[a, b]$.


perfect-lemma-pseudo-coherent-descends-fpqc

Let $\{f_i : X_i \to X\}$ be an fpqc covering of schemes. Let $E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each $Lf_i^*E$ is $m$-pseudo-coherent.


perfect-lemma-closed-push-pseudo-coherent

Let $i : Z \to X$ be a morphism of ringed spaces such that $i$ is a closed immersion of underlying topological spaces and such that $i_*\mathcal{O}_Z$ is pseudo-coherent as an $\mathcal{O}_X$-module. Let $E \in D(\mathcal{O}_Z)$. Then $E$ is $m$-pseudo-coherent if and only if $Ri_*E$ is $m$-pseudo-coherent.


perfect-lemma-lift-quasi-coherent

Let $X$ be a scheme and let $j : U \to X$ be a quasi-compact open immersion. The functors $$ D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_U) \quad\text{and}\quad D^+_\QCoh(\mathcal{O}_X) \to D^+_\QCoh(\mathcal{O}_U) $$ are essentially surjective. If $X$ is quasi-compact, then the functors $$ D^-_\QCoh(\mathcal{O}_X) \to D^-_\QCoh(\mathcal{O}_U) \quad\text{and}\quad D^b_\QCoh(\mathcal{O}_X) \to D^b_\QCoh(\mathcal{O}_U) $$ are essentially surjective.


perfect-lemma-lift-coherent

Let $X$ be a Noetherian scheme and let $j : U \to X$ be an open immersion. The functor $D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_U)$ is essentially surjective.


perfect-lemma-lift-pseudo-coherent

Let $X$ be an affine scheme and let $U \subset X$ be a quasi-compact open subscheme. For any pseudo-coherent object $E$ of $D(\mathcal{O}_U)$ there exists a bounded above complex of finite free $\mathcal{O}_X$-modules  whose restriction to $U$ is isomorphic to $E$.


perfect-lemma-vanishing-ext

Let $X$ be a quasi-compact and quasi-separated scheme. Let $E \in D^b_\QCoh(\mathcal{O}_X)$. There exists an integer $n_0 > 0$ such that $\Ext^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) = 0$ for every finite locally free $\mathcal{O}_X$-module $\mathcal{E}$ and every $n \geq n_0$.


perfect-lemma-lift-perfect-complex-plus-locally-free

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $E$ of $D(\mathcal{O}_U)$ there exists an integer $r$ and a finite locally free sheaf $\mathcal{F}$ on $U$ such that $\mathcal{F}[-r] \oplus E$ is the restriction of a perfect object of $D(\mathcal{O}_X)$.


perfect-lemma-lift-map

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. Let $E, E'$ be objects of $D_\QCoh(\mathcal{O}_X)$ with $E$ perfect. For every map $\alpha : E|_U \to E'|_U$ there exist maps $$ E \xleftarrow{\beta} E_1 \xrightarrow{\gamma} E' $$ of perfect complexes on $X$ such that $\beta : E_1 \to E$ restricts to an isomorphism on $U$ and such that $\alpha = \gamma|_U \circ \beta|_U^{-1}$. Moreover we can assume $E_1 = E \otimes_{\mathcal{O}_X}^\mathbf{L} I$ for some perfect complex $I$ on $X$.


perfect-lemma-lift-perfect-complex-plus-shift

Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_U)$ the object $F \oplus F[1]$ is the restriction of a perfect object of $D(\mathcal{O}_X)$.


perfect-lemma-perfect-into-support-on-T

Let $X$ be a quasi-compact and quasi-separated scheme. Let $f \in \Gamma(X, \mathcal{O}_X)$. For any morphism $\alpha : E \to E'$ in $D_\QCoh(\mathcal{O}_X)$ such that \begin{enumerate} \item $E$ is perfect, and \item $E'$ is supported on $T = V(f)$ \end{enumerate} there exists an $n \geq 0$ such that $f^n \alpha  = 0$.


perfect-lemma-lift-perfect-complex-plus-shift-support

Let $X$ be an affine scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $U \subset X$ be a quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_U)$ supported on $T \cap U$ the object $F \oplus F[1]$ is the restriction of a perfect object $E$ of $D(\mathcal{O}_X)$ supported in $T$.


perfect-lemma-lift-map-from-perfect-complex-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open. Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact in $X$. Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. Let $\alpha : P \to E|_U$ be a map where $P$ is a perfect object of $D(\mathcal{O}_U)$ supported on $T \cap U$. Then there exists a map $\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_X)$ supported on $T$ such that $P$ is a direct summand of $R|_U$ in $D(\mathcal{O}_U)$ compatible $\alpha$ and $\beta|_U$.


perfect-lemma-open

Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. Let $(T, E, m)$ be a triple as in Definition \ref{definition-approximation-holds}. If \begin{enumerate} \item $T \subset U$, \item approximation holds for $(T, E|_U, m)$, and \item the sheaves $H^i(E)$ for $i \geq m$ are supported on $T$, \end{enumerate} then approximation holds for $(T, E, m)$.


perfect-lemma-approximation-affine

Let $X$ be an affine scheme. Then approximation holds for every triple $(T, E, m)$ as in Definition \ref{definition-approximation-holds} such that there exists an integer $r \geq 0$ with \begin{enumerate} \item $E$ is $m$-pseudo-coherent, \item $H^i(E)$ is supported on $T$ for $i \geq m - r + 1$, \item $X \setminus T$ is the union of $r$ affine opens. \end{enumerate} In particular, approximation by perfect complexes holds for affine schemes.


perfect-lemma-induction-step

Let $X$ be a scheme. Let $X = U \cup V$ be an open covering with $U$ quasi-compact, $V$ affine, and $U \cap V$ quasi-compact. If approximation by perfect complexes holds on $U$, then approximation holds on $X$.


perfect-lemma-direct-summand-of-a-restriction

Let $X$ be a quasi-compact and quasi-separated scheme. Let $U$ be a quasi-compact open subscheme. Let $P$ be a perfect object of $D(\mathcal{O}_U)$. Then $P$ is a direct summand of the restriction of a perfect object of $D(\mathcal{O}_X)$.


perfect-lemma-orthogonal-koszul-complex

\begin{reference} \cite[Proposition 6.1]{Bokstedt-Neeman} \end{reference} 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 $A$. For $E \in D_\QCoh(\mathcal{O}_X)$ the following are equivalent \begin{enumerate} \item $E = Rj_*(E|_U)$, and \item $\Hom_{D(\mathcal{O}_X)}(K[n], E) = 0$ for all $n \in \mathbf{Z}$. \end{enumerate}


perfect-lemma-generator-with-support

\begin{reference} \cite[Theorem 6.8]{Rouquier-dimensions} \end{reference} Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. With notation as above, the category $D_{\QCoh, T}(\mathcal{O}_X)$ is generated by a single perfect object.


perfect-lemma-nonzero-some-cohomology

Let $X$ be a scheme and $\mathcal{L}$ an ample invertible $\mathcal{O}_X$-module. If $K$ is a nonzero object of $D_\QCoh(\mathcal{O}_X)$, then for some $n \geq 0$ and $p \in \mathbf{Z}$ the cohomology group $H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes n})$ is nonzero.


perfect-lemma-construct-the-next-one

Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. For every $a \in \mathbf{Z}$ there exists an exact complex $$ 0 \to \mathcal{O}_X(a) \to \ldots \to \mathcal{O}_X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_X(a + n + 1) \to 0 $$ of vector bundles on $X$.


perfect-lemma-generator-P1

Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. Then $$ E = \mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n) $$ is a generator (Derived Categories, Definition \ref{derived-definition-generators}) of $D_\QCoh(X)$.


perfect-lemma-compact-is-perfect-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. An object of $D_{\QCoh, T}(\mathcal{O}_X)$ is compact if and only if it is perfect as an object of $D(\mathcal{O}_X)$.


perfect-lemma-map-from-pseudo-coherent-to-complex-with-support

Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $U = X \setminus T$ is quasi-compact. Let $\alpha : P \to E$ be a morphism of $D_\QCoh(\mathcal{O}_X)$ with either \begin{enumerate} \item $P$ is perfect and $E$ supported on $T$, or \item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below. \end{enumerate} Then there exists a perfect complex of $\mathcal{O}_X$-modules $I$ and a map $I \to \mathcal{O}_X[0]$ such that $I \otimes^\mathbf{L} P \to E$ is zero and such that $I|_U \to \mathcal{O}_U[0]$ is an isomorphism.


perfect-lemma-tensor-with-QCoh-complex

Let $X$ be a scheme. Let $K^\bullet$ be a complex of $\mathcal{O}_X$-modules whose cohomology sheaves are quasi-coherent. Let $(E, d) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)$ be the endomorphism differential graded algebra. Then the functor $$ - \otimes_E^\mathbf{L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}_X) $$ of Differential Graded Algebra, Lemma \ref{dga-lemma-tensor-with-complex-derived} has image contained in $D_\QCoh(\mathcal{O}_X)$.


perfect-lemma-ext-from-perfect-into-bounded-QCoh

Let $X$ be a quasi-compact and quasi-separated scheme. Let $K$ be a perfect object of $D(\mathcal{O}_X)$. Then \begin{enumerate} \item there exist integers $a \leq b$ such that $\Hom_{D(\mathcal{O}_X)}(K, L) = 0$ for $L \in D_\QCoh(\mathcal{O}_X)$ with $H^i(L) = 0$ for $i \in [a, b]$, and \item if $L$ is bounded, then $\Ext^n_{D(\mathcal{O}_X)}(K, L)$ is zero for all but finitely many $n$. \end{enumerate}


perfect-lemma-pseudo-coherent-hocolim

Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \in D(\mathcal{O}_X)$. The following are equivalent \begin{enumerate} \item $K$ is pseudo-coherent, and \item $K = \text{hocolim} K_n$ where $K_n$ is perfect and $\tau_{\geq -n}K_n \to \tau_{\geq -n}K$ is an isomorphism for all $n$. \end{enumerate}


perfect-lemma-Pn-module-category

\begin{reference} \cite{Beilinson} \end{reference} Let $A$ be a ring. Let $X = \mathbf{P}^n_A = \text{Proj}(S)$ where $S = A[X_0, \ldots, X_n]$. With $P$ as in (\ref{equation-generator-Pn}) and $R$ as in (\ref{equation-algebra-for-Pn}) the functor $$ - \otimes_R^\mathbf{L} P : D(R) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an $A$-linear equivalence of triangulated categories sending $R$ to $P$.


perfect-lemma-better-coherator

Let $X$ be a quasi-compact and quasi-separated scheme. The inclusion functor $D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)$ has a right adjoint $DQ_X$.


perfect-lemma-pushforward-better-coherator

Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. If the right adjoints $DQ_X$ and $DQ_Y$ of the inclusion functors $D_\QCoh \to D$ exist for $X$ and $Y$, then $$ Rf_* \circ DQ_X = DQ_Y \circ Rf_* $$


perfect-lemma-boundedness-better-coherator

Let $X$ be a quasi-compact and quasi-separated scheme. The functor $DQ_X$ of Lemma \ref{lemma-better-coherator} has the following boundedness property: there exists an integer $N = N(X)$ such that, if $K$ in $D(\mathcal{O}_X)$ with $H^i(U, K) = 0$ for $U$ affine open in $X$ and $i \not \in [a, b]$, then the cohomology sheaves $H^i(DQ_X(K))$ are zero for $i \not \in [a, b + N]$.


perfect-lemma-cohomology-base-change

Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. For $E$ in $D_\QCoh(\mathcal{O}_X)$ and $K$ in $D_\QCoh(\mathcal{O}_Y)$ the map $$ Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K \longrightarrow Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K) $$ defined in Cohomology, Equation (\ref{cohomology-equation-projection-formula-map}) is an isomorphism.


perfect-lemma-tor-independent

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes. The following are equivalent \begin{enumerate} \item $X$ and $Y$ are tor independent over $S$, and \item for every affine opens $U \subset X$, $V \subset Y$, $W \subset S$ with $f(U) \subset W$ and $g(V) \subset W$ the rings $\mathcal{O}_X(U)$ and $\mathcal{O}_Y(V)$ are tor independent over $\mathcal{O}_S(W)$. \item there exists an affine open overing $S = \bigcup W_i$ and for each $i$ affine open coverings $f^{-1}(W_i) = \bigcup U_{ij}$ and $g^{-1}(W_i) = \bigcup V_{ik}$ such that the rings $\mathcal{O}_X(U_{ij})$ and $\mathcal{O}_Y(V_{ik})$ are tor independent over $\mathcal{O}_S(W_i)$ for all $i, j, k$. \end{enumerate}


perfect-lemma-flat-base-change-tor-independent

Let $X \to S$ and $Y \to S$ be morphisms of schemes. Let $S' \to S$ be a morphism of schemes and denote $X' = X \times_S S'$ and $Y' = Y \times_S S'$. If $X$ and $Y$ are tor independent over $S$ and $S' \to S$ is flat, then $X'$ and $Y'$ are tor independent over $S'$.


perfect-lemma-compare-base-change

Let $g : S' \to S$ be a morphism of schemes. Let $f : X \to S$ be quasi-compact and quasi-separated. Consider the base change diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ If $X$ and $S'$ are Tor independent over $S$, then for all $E \in D_\QCoh(\mathcal{O}_X)$ we have $Rf'_*L(g')^*E = Lg^*Rf_*E$.


perfect-lemma-affine-morphism-and-hom-out-of-perfect

Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ of quasi-compact and quasi-separated schemes. Assume $g$ and $f$ Tor independent and $S = \Spec(R)$, $S' = \Spec(R')$ affine. For $M, K \in D(\mathcal{O}_X)$ the canonical map $$ R\Hom_X(M, K) \otimes^\mathbf{L}_R R' \longrightarrow R\Hom_{X'}(L(g')^*M, L(g')^*K) $$ in $D(R')$ is an isomorphism in the following two cases \begin{enumerate} \item $M \in D(\mathcal{O}_X)$ is perfect and $K \in D_\QCoh(X)$, or \item $M \in D(\mathcal{O}_X)$ is pseudo-coherent, $K \in D_\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$. \end{enumerate}


perfect-lemma-tor-independence-and-tor-amplitude

Consider a cartesian square of schemes $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ Assume $g$ and $f$ Tor independent. \begin{enumerate} \item If $E \in D(\mathcal{O}_X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_S$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_{S'}$-modules. \item If $\mathcal{G}$ is an $\mathcal{O}_X$-module flat over $S$, then $L(g')^*\mathcal{G} = (g')^*\mathcal{G}$. \end{enumerate}


perfect-lemma-compare-base-change-closed-immersion

Consider a cartesian diagram of schemes $$ \xymatrix{ Z' \ar[r]_{i'} \ar[d]_g & X' \ar[d]^f \\ Z \ar[r]^i & X } $$ where $i$ is a closed immersion. If $Z$ and $X'$ are tor independent over $X$, then $Ri'_* \circ Lg^* = Lf^* \circ Ri_*$ as functors $D(\mathcal{O}_Z) \to D(\mathcal{O}_{X'})$.


perfect-lemma-kunneth

In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor-independent over $S$, then (\ref{equation-kunneth}) is an isomorphism for $K \in D_\QCoh(\mathcal{O}_X)$ and $M \in D_\QCoh(\mathcal{O}_Y)$. If in addition $S = \Spec(A)$ is affine, then the map (\ref{equation-kunneth-global}) is an isomorphism.


perfect-lemma-cohomology-de-rham-base-change

Let $a : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}^\bullet$ be a locally bounded complex of $a^{-1}\mathcal{O}_S$-modules. Assume for all $n \in \mathbf{Z}$ the sheaf $\mathcal{F}^n$ is a flat $a^{-1}\mathcal{O}_S$-module and $\mathcal{F}^n$ has the structure of a quasi-coherent $\mathcal{O}_X$-module compatible with the given $a^{-1}\mathcal{O}_S$-module structure (but the differentials in the complex $\mathcal{F}^\bullet$ need not be $\mathcal{O}_X$-linear). Then the following hold \begin{enumerate} \item $Ra_*\mathcal{F}^\bullet$ is locally bounded, \item $Ra_*\mathcal{F}^\bullet$ is in $D_\QCoh(\mathcal{O}_S)$, \item $Ra_*\mathcal{F}^\bullet$ locally has finite tor dimension, \item $\mathcal{G} \otimes_{\mathcal{O}_S}^\mathbf{L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}\mathcal{G} \otimes_{a^{-1}\mathcal{O}_S} \mathcal{F}^\bullet)$ for $\mathcal{G} \in \QCoh(\mathcal{O}_S)$, and \item $K \otimes_{\mathcal{O}_S}^\mathbf{L} Ra_*\mathcal{F}^\bullet = Ra_*(a^{-1}K \otimes_{a^{-1}\mathcal{O}_S}^\mathbf{L} \mathcal{F}^\bullet)$ for $K \in D_\QCoh(\mathcal{O}_S)$. \end{enumerate}