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)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BDI. Beware of the difference between the letter 'O' and the digit '0'.