The Stacks project

36.31 Perfect complexes

We first talk about jumping loci for betti numbers of perfect complexes. Given a complex $E$ on a scheme $X$ and a point $x$ of $X$ we often write $E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x)$ instead of the more correct $Li_ x^*E$, where $i_ x : x \to X$ is the canonical morphism.

Lemma 36.31.1. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function

\[ \beta _ i : X \longrightarrow \{ 0, 1, 2, \ldots \} ,\quad x \longmapsto \dim _{\kappa (x)} H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x)) \]

Then we have

  1. formation of $\beta _ i$ commutes with arbitrary base change,

  2. the functions $\beta _ i$ are upper semi-continuous, and

  3. the level sets of $\beta _ i$ are locally constructible in $X$.

Proof. Consider a morphism of schemes $f : Y \to X$ and a point $y \in Y$. Let $x$ be the image of $y$ and consider the commutative diagram

\[ \xymatrix{ y \ar[r]_ j \ar[d]_ g & Y \ar[d]^ f \\ x \ar[r]^ i & X } \]

Then we see that $Lg^* \circ Li^* = Lj^* \circ Lf^*$. This implies that the function $\beta '_ i$ associated to the pseudo-coherent complex $Lf^*E$ is the pullback of the function $\beta _ i$, in a formula: $\beta '_ i = \beta _ i \circ f$. This is the meaning of (1).

Fix $i$ and let $x \in X$. It is enough to prove (2) and (3) holds in an open neighbourhood of $x$, hence we may assume $X$ affine. Then we can represent $E$ by a bounded above complex $\mathcal{F}^\bullet $ of finite free modules (Lemma 36.13.3). Then $P = \sigma _{\geq i - 1}\mathcal{F}^\bullet $ is a perfect object and $P \to E$ induces an isomorphism

\[ H^ i(P \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x')) \to H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x')) \]

for all $x' \in X$. Thus we may assume $E$ is perfect. In this case by More on Algebra, Lemma 15.74.6 there exists an affine open neighbourhood $U$ of $x$ and $a \leq b$ such that $E|_ U$ is represented by a complex

\[ \ldots \to 0 \to \mathcal{O}_ U^{\oplus \beta _ a(x)} \to \mathcal{O}_ U^{\oplus \beta _{a + 1}(x)} \to \ldots \to \mathcal{O}_ U^{\oplus \beta _{b - 1}(x)} \to \mathcal{O}_ U^{\oplus \beta _ b(x)} \to 0 \to \ldots \]

(This also uses earlier results to turn the problem into algebra, for example Lemmas 36.3.5 and 36.10.7.) It follows immediately that $\beta _ i(x') \leq \beta _ i(x)$ for all $x' \in U$. This proves that $\beta _ i$ is upper semi-continuous.

To prove (3) we may assume that $X$ is affine and $E$ is given by a complex of finite free $\mathcal{O}_ X$-modules (for example by arguing as in the previous paragraph, or by using Cohomology, Lemma 20.46.3). Thus we have to show that given a complex

\[ \mathcal{O}_ X^{\oplus a} \to \mathcal{O}_ X^{\oplus b} \to \mathcal{O}_ X^{\oplus c} \]

the function associated to a point $x \in X$ the dimension of the cohomology of $\kappa _ x^{\oplus a} \to \kappa _ x^{\oplus b} \to \kappa _ x^{\oplus c}$ in the middle has constructible level sets. Let $A \in \text{Mat}(a \times b, \Gamma (X, \mathcal{O}_ X))$ be the matrix of the first arrow. The rank of the image of $A$ in $\text{Mat}(a \times b, \kappa (x))$ is equal to $r$ if all $(r + 1) \times (r + 1)$-minors of $A$ vanish at $x$ and there is some $r \times r$-minor of $A$ which does not vanish at $x$. Thus the set of points where the rank is $r$ is a constructible locally closed set. Arguing similarly for the second arrow and putting everything together we obtain the desired result. $\square$

Lemma 36.31.2. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. The function

\[ \chi _ E : X \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^ i \dim _{\kappa (x)} H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x)) \]

is locally constant on $X$.

Proof. By Cohomology, Lemma 20.46.3 we see that we can, locally on $X$, represent $E$ by a finite complex $\mathcal{E}^\bullet $ of finite free $\mathcal{O}_ X$-modules. On such an open the function $\chi _ E$ is constant with value $\sum (-1)^ i \text{rank}(\mathcal{E}^ i)$. $\square$

Lemma 36.31.3. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. Given $i, r \in \mathbf{Z}$, there exists an open subscheme $U \subset X$ characterized by the following

  1. $E|_ U \cong H^ i(E|_ U)[-i]$ and $H^ i(E|_ U)$ is a locally free $\mathcal{O}_ U$-module of rank $r$,

  2. a morphism $f : Y \to X$ factors through $U$ if and only if $Lf^*E$ is isomorphic to a locally free module of rank $r$ placed in degree $i$.

Proof. Let $\beta _ j : X \to \{ 0, 1, 2, \ldots \} $ for $j \in \mathbf{Z}$ be the functions of Lemma 36.31.1. Then the set

\[ W = \{ x \in X \mid \beta _ j(x) \leq 0\text{ for all }j \not= i\} \]

is open in $X$ and its formation commutes with pullback to any $Y$ over $X$. This follows from the lemma using that apriori in a neighbourhood of any point only a finite number of the $\beta _ j$ are nonzero. Thus we may replace $X$ by $W$ and assume that $\beta _ j(x) = 0$ for all $x \in X$ and all $j \not= i$. In this case $H^ i(E)$ is a finite locally free module and $E \cong H^ i(E)[-i]$, see for example More on Algebra, Lemma 15.74.6. Thus $X$ is the disjoint union of the open subschemes where the rank of $H^ i(E)$ is fixed and we win. $\square$

Lemma 36.31.4. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect of tor-amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $r \geq 0$. Then there exists a locally closed subscheme $j : Z \to X$ characterized by the following

  1. $H^ a(Lj^*E)$ is a locally free $\mathcal{O}_ Z$-module of rank $r$, and

  2. a morphism $f : Y \to X$ factors through $Z$ if and only if for all morphisms $g : Y' \to Y$ the $\mathcal{O}_{Y'}$-module $H^ a(L(f \circ g)^*E)$ is locally free of rank $r$.

Moreover, $j : Z \to X$ is of finite presentation and we have

  1. if $f : Y \to X$ factors as $Y \xrightarrow {g} Z \to X$, then $H^ a(Lf^*E) = g^*H^ a(Lj^*E)$,

  2. if $\beta _ a(x) \leq r$ for all $x \in X$, then $j$ is a closed immersion and given $f : Y \to X$ the following are equivalent

    1. $f : Y \to X$ factors through $Z$,

    2. $H^0(Lf^*E)$ is a locally free $\mathcal{O}_ Y$-module of rank $r$,

    and if $r = 1$ these are also equivalent to

    1. $\mathcal{O}_ Y \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(H^0(Lf^*E), H^0(Lf^*E))$ is injective.

Proof. First, let $U \subset X$ be the locally constructible open subscheme where the function $\beta _ a$ of Lemma 36.31.1 has values $\leq r$. Let $f : Y \to X$ be as in (2). Then for any $y \in Y$ we have $\beta _ a(Lf^*E) = r$ hence $y$ maps into $U$ by Lemma 36.31.1. Hence $f$ as in (2) factors through $U$. Thus we may replace $X$ by $U$ and assume that $\beta _ a(x) \in \{ 0, 1, \ldots , r\} $ for all $x \in X$. We will show that in this case there is a closed subscheme $Z \subset X$ cut out by a finite type quasi-coherent ideal characterized by the equivalence of (4) (a), (b) and (4)(c) if $r = 1$ and that (3) holds. This will finish the proof because it will a fortiori show that morphisms as in (2) factor through $Z$.

If $x \in X$ and $\beta _ a(x) < r$, then there is an open neighbourhood of $x$ where $\beta _ a < r$ (Lemma 36.31.1). In this way we see that set theoretically at least $Z$ is a closed subset.

To get a scheme theoretic structure, consider a point $x \in X$ with $\beta _ a(x) = r$. Set $\beta = \beta _{a + 1}(x)$. By More on Algebra, Lemma 15.74.6 there exists an affine open neighbourhood $U$ of $x$ such that $K|_ U$ is represented by a complex

\[ \ldots \to 0 \to \mathcal{O}_ U^{\oplus r} \xrightarrow {(f_{ij})} \mathcal{O}_ U^{\oplus \beta } \to \ldots \to \mathcal{O}_ U^{\oplus \beta _{b - 1}(x)} \to \mathcal{O}_ U^{\oplus \beta _ b(x)} \to 0 \to \ldots \]

(This also uses earlier results to turn the problem into algebra, for example Lemmas 36.3.5 and 36.10.7.) Now, if $g : Y \to U$ is any morphism of schemes such that $g^\sharp (f_{ij})$ is nonzero for some pair $i, j$, then $H^0(Lg^*E)$ is not a locally free $\mathcal{O}_ Y$-module of rank $r$. See More on Algebra, Lemma 15.15.7. Trivially $H^0(Lg^*E)$ is a locally free $\mathcal{O}_ Y$-module if $g^\sharp (f_{ij}) = 0$ for all $i, j$. Thus we see that over $U$ the closed subscheme cut out by all $f_{ij}$ satisfies (3) and we have the equivalence of (4)(a) and (b). The characterization of $Z$ shows that the locally constructed patches glue (details omitted). Finally, if $r = 1$ then (4)(c) is equivalent to (4)(b) because in this case locally $H^0(Lg^*E) \subset \mathcal{O}_ Y$ is the annihilator of the ideal generated by the elements $g^\sharp (f_{ij})$. $\square$


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