The Stacks project

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20.40 Strictly perfect complexes

Strictly perfect complexes of modules are used to define the notions of pseudo-coherent and perfect complexes later on. They are defined as follows.

Definition 20.40.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We say $\mathcal{E}^\bullet $ is strictly perfect if $\mathcal{E}^ i$ is zero for all but finitely many $i$ and $\mathcal{E}^ i$ is a direct summand of a finite free $\mathcal{O}_ X$-module for all $i$.

Warning: Since we do not assume that $X$ is a locally ringed space, it may not be true that a direct summand of a finite free $\mathcal{O}_ X$-module is finite locally free.

Lemma 20.40.2. The cone on a morphism of strictly perfect complexes is strictly perfect.

Proof. This is immediate from the definitions. $\square$

Lemma 20.40.3. The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect.

Proof. Omitted. $\square$

Lemma 20.40.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. If $\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_ Y$-modules, then $f^*\mathcal{F}^\bullet $ is a strictly perfect complex of $\mathcal{O}_ X$-modules.

Proof. The pullback of a finite free module is finite free. The functor $f^*$ is additive functor hence preserves direct summands. The lemma follows. $\square$

Lemma 20.40.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given a solid diagram of $\mathcal{O}_ X$-modules

\[ \xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_ p } \]

with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_ X$-module and $p$ surjective, then a dotted arrow making the diagram commute exists locally on $X$.

Proof. We may assume $\mathcal{E} = \mathcal{O}_ X^{\oplus n}$ for some $n$. In this case finding the dotted arrow is equivalent to lifting the images of the basis elements in $\Gamma (X, \mathcal{F})$. This is locally possible by the characterization of surjective maps of sheaves (Sheaves, Section 6.16). $\square$

Lemma 20.40.6. Let $(X, \mathcal{O}_ X)$ be a ringed space.

  1. Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet $ be a morphism of complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect and $\mathcal{F}^\bullet $ acyclic. Then $\alpha $ is locally on $X$ homotopic to zero.

  2. Let $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet $ be a morphism of complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect, $\mathcal{E}^ i = 0$ for $i < a$, and $H^ i(\mathcal{F}^\bullet ) = 0$ for $i \geq a$. Then $\alpha $ is locally on $X$ homotopic to zero.

Proof. The first statement follows from the second, hence we only prove (2). We will prove this by induction on the length of the complex $\mathcal{E}^\bullet $. If $\mathcal{E}^\bullet \cong \mathcal{E}[-n]$ for some direct summand $\mathcal{E}$ of a finite free $\mathcal{O}_ X$-module and integer $n \geq a$, then the result follows from Lemma 20.40.5 and the fact that $\mathcal{F}^{n - 1} \to \mathop{\mathrm{Ker}}(\mathcal{F}^ n \to \mathcal{F}^{n + 1})$ is surjective by the assumed vanishing of $H^ n(\mathcal{F}^\bullet )$. If $\mathcal{E}^ i$ is zero except for $i \in [a, b]$, then we have a split exact sequence of complexes

\[ 0 \to \mathcal{E}^ b[-b] \to \mathcal{E}^\bullet \to \sigma _{\leq b - 1}\mathcal{E}^\bullet \to 0 \]

which determines a distinguished triangle in $K(\mathcal{O}_ X)$. Hence an exact sequence

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}( \sigma _{\leq b - 1}\mathcal{E}^\bullet , \mathcal{F}^\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(\mathcal{E}^ b[-b], \mathcal{F}^\bullet ) \]

by the axioms of triangulated categories. The composition $\mathcal{E}^ b[-b] \to \mathcal{F}^\bullet $ is locally homotopic to zero, whence we may assume our map comes from an element in the left hand side of the displayed exact sequence above. This element is locally zero by induction hypothesis. $\square$

Lemma 20.40.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given a solid diagram of complexes of $\mathcal{O}_ X$-modules

\[ \xymatrix{ \mathcal{E}^\bullet \ar@{..>}[dr] \ar[r]_\alpha & \mathcal{F}^\bullet \\ & \mathcal{G}^\bullet \ar[u]_ f } \]

with $\mathcal{E}^\bullet $ strictly perfect, $\mathcal{E}^ j = 0$ for $j < a$ and $H^ j(f)$ an isomorphism for $j > a$ and surjective for $j = a$, then a dotted arrow making the diagram commute up to homotopy exists locally on $X$.

Proof. Our assumptions on $f$ imply the cone $C(f)^\bullet $ has vanishing cohomology sheaves in degrees $\geq a$. Hence Lemma 20.40.6 guarantees there is an open covering $X = \bigcup U_ i$ such that the composition $\mathcal{E}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet $ is homotopic to zero over $U_ i$. Since

\[ \mathcal{G}^\bullet \to \mathcal{F}^\bullet \to C(f)^\bullet \to \mathcal{G}^\bullet [1] \]

restricts to a distinguished triangle in $K(\mathcal{O}_{U_ i})$ we see that we can lift $\alpha |_{U_ i}$ up to homotopy to a map $\alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{G}^\bullet |_{U_ i}$ as desired. $\square$

Lemma 20.40.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect.

  1. For any element $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is given by a morphism of complexes $\alpha _ i : \mathcal{E}^\bullet |_{U_ i} \to \mathcal{F}^\bullet |_{U_ i}$.

  2. Given a morphism of complexes $\alpha : \mathcal{E}^\bullet \to \mathcal{F}^\bullet $ whose image in the group $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is zero, there exists an open covering $X = \bigcup U_ i$ such that $\alpha |_{U_ i}$ is homotopic to zero.

Proof. Proof of (1). By the construction of the derived category we can find a quasi-isomorphism $f : \mathcal{F}^\bullet \to \mathcal{G}^\bullet $ and a map of complexes $\beta : \mathcal{E}^\bullet \to \mathcal{G}^\bullet $ such that $\alpha = f^{-1}\beta $. Thus the result follows from Lemma 20.40.7. We omit the proof of (2). $\square$

Lemma 20.40.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with $\mathcal{E}^\bullet $ strictly perfect. Then the internal hom $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $\mathcal{H}^\bullet $ with terms

\[ \mathcal{H}^ n = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p) \]

and differential as described in Section 20.36.

Proof. Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ into a K-injective complex. Let $(\mathcal{H}')^\bullet $ be the complex with terms

\[ (\mathcal{H}')^ n = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{L}^{-q}, \mathcal{I}^ p) \]

which represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ by the construction in Section 20.36. It suffices to show that the map

\[ \mathcal{H}^\bullet \longrightarrow (\mathcal{H}')^\bullet \]

is a quasi-isomorphism. Given an open $U \subset X$ we have by inspection

\[ H^0(\mathcal{H}^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{K}^\bullet |_ U) \to H^0((\mathcal{H}')^\bullet (U)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet |_ U, \mathcal{K}^\bullet |_ U) \]

By Lemma 20.40.8 the sheafification of $U \mapsto H^0(\mathcal{H}^\bullet (U))$ is equal to the sheafification of $U \mapsto H^0((\mathcal{H}')^\bullet (U))$. A similar argument can be given for the other cohomology sheaves. Thus $\mathcal{H}^\bullet $ is quasi-isomorphic to $(\mathcal{H}')^\bullet $ which proves the lemma. $\square$

Lemma 20.40.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $, $\mathcal{F}^\bullet $ be complexes of $\mathcal{O}_ X$-modules with

  1. $\mathcal{F}^ n = 0$ for $n \ll 0$,

  2. $\mathcal{E}^ n = 0$ for $n \gg 0$, and

  3. $\mathcal{E}^ n$ isomorphic to a direct summand of a finite free $\mathcal{O}_ X$-module.

Then the internal hom $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $\mathcal{H}^\bullet $ with terms

\[ \mathcal{H}^ n = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p) \]

and differential as described in Section 20.36.

Proof. Choose a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ where $\mathcal{I}^\bullet $ is a bounded below complex of injectives. Note that $\mathcal{I}^\bullet $ is K-injective (Derived Categories, Lemma 13.29.4). Hence the construction in Section 20.36 shows that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{E}^\bullet , \mathcal{F}^\bullet )$ is represented by the complex $(\mathcal{H}')^\bullet $ with terms

\[ (\mathcal{H}')^ n = \prod \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{I}^ p) = \bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{I}^ p) \]

(equality because there are only finitely many nonzero terms). Note that $\mathcal{H}^\bullet $ is the total complex associated to the double complex with terms $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^ p)$ and similarly for $(\mathcal{H}')^\bullet $. The natural map $(\mathcal{H}')^\bullet \to \mathcal{H}^\bullet $ comes from a map of double complexes. Thus to show this map is a quasi-isomorphism, we may use the spectral sequence of a double complex (Homology, Lemma 12.22.6)

\[ {}'E_1^{p, q} = H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}^{-q}, \mathcal{F}^\bullet )) \]

converging to $H^{p + q}(\mathcal{H}^\bullet )$ and similarly for $(\mathcal{H}')^\bullet $. To finish the proof of the lemma it suffices to show that $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ induces an isomorphism

\[ H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{F}^\bullet )) \longrightarrow H^ p(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{E}, \mathcal{I}^\bullet )) \]

on cohomology sheaves whenever $\mathcal{E}$ is a direct summand of a finite free $\mathcal{O}_ X$-module. Since this is clear when $\mathcal{E}$ is finite free the result follows. $\square$


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