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Cohomology of shifted Nisnevich units

PDF p. 4: Property (2)

Statement

Assume the Axiom of Choice and the size bounds in Derived global sections on a scheme site. For a scheme $X$ , put $A = R Γ_{N i s} (X, G_{m}) [- 1]$ , using the small Nisnevich site and the sheaf of units of Units and Picard groups in Nisnevich degree zero and one. Then $H^{i} (A) = H_{N i s}^{i - 1} (X, G_{m})$ for every integer $i$ ; these groups are zero for $i \leq 0$ , the global units for $i = 1$ , and $Pic (X)$ for $i = 2$ .

Facts & Assumptions

Given: $X$ , the stated site, Choice, and the size bounds.

Goal: Compute the indicated cohomology groups.

[F1]

Derived sections of an abelian sheaf concentrated in degree zero have no negative cohomology (Derived global sections on a scheme site).

[F2]

Degree-zero Nisnevich cohomology of $G_{m}$ is $Γ (X, O_{X})^{\times}$ , and degree one is $Pic (X)$ , the group of line bundles under tensor product (Units and Picard groups in Nisnevich degree zero and one).

Proof

1.1givenalgebra

For the cohomological shift, a complex $C [- 1]$ has $C [- 1]^{i} = C^{i - 1}$ , with the corresponding signed differential. Its cycles and boundaries in degree $i$ are those of $C$ in degree $i - 1$ , so $H^{i} (C [- 1]) = H^{i - 1} (C)$ . Apply this to the derived-sections complex.

2.1F1F2step 1.1∎

If $i \leq 0$ , then $i - 1 < 0$ , so [F1] gives zero. For $i = 1, 2$ , [F2] gives global units and the Picard group respectively. These identifications use the specified small site and retain the Choice and size hypotheses.

Depends on

Used by · 1

Low-degree consequences of a weight-one comparison

Dependency tree · 3 levels

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