The relevant theorem supplies, locally on

S = Spec A

, a bounded complex

K^{∙}

of finite free

A

-modules such that

H^{p} (X, F \otimes_{A} M) ≅ H^{p} (K^{∙} \otimes_{A} M)

naturally for every

A

-module

M

. In particular the fiber cohomology at

s

is computed by

K^{∙} \otimes_{A} κ (s)

The Euler characteristic of a finite-dimensional complex equals the alternating sum of the dimensions of its terms. Hence

χ (X_{s}, F_{s}) = \sum_{i} (- 1)^{i} rank K^{i},

which is constant wherever one complex works. It is therefore locally constant on

S

, and is constant when

S

is connected.

Solved by gpt-5.6-sol high.

b

Solution

A

. The condition that such a matrix have rank at most

r

is closed, being defined by its

(r + 1) \times (r + 1)

minors. Since

dim H^{p} (K^{∙} \otimes κ (s)) = dim K^{p} - rank d_{s}^{p - 1} - rank d_{s}^{p},

the condition that this dimension be at least

n

is a finite union of intersections of closed rank loci. It is therefore closed. This is the Semicontinuity theorem for coherent cohomology.

Solved by gpt-5.6-sol high.

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