The Stacks project

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 15.75.6 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 20.49.3). 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. $\square$

Comments (2)

Comment #5399 by Davis Lazowski on

I think whenever you write , you mean . You write in three places:

  1. , at the start of the proof of 1);
  2. , in the proof of 2);
  3. at the start of the proof of 3)

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