Math 256B. Homework 12

Due Wednesday 24 April

1. State an analogue of Theorem 7.1 for n = 0 . Prove it directly, using as few results from Section 7 as possible. (You may use the two definitions of dualizing sheaf and the fact that they are equivalent.)

2(nc). Let 𝔘 = (U_{i})_{i∈I} be an open covering of a topological space X, and let I′ be a subset of I such that 𝔘′ := (U_{i})_{i∈I′} also covers X. Construct natural maps

𝒞^{p}(𝔘, ℱ) → 𝒞^{p}(𝔘′, ℱ)

for all p ∈ ℕ and all sheaves ℱ of abelian groups on X, functorially in ℱ , such that (i). 𝒞 ·(𝔘, ℱ) → 𝒞 ·(𝔘′, ℱ) is a map of complexes for all ℱ , and (ii). if X is a scheme and if ℱ , 𝔘, and 𝔘′ satisfy the hypotheses of (III, Thm. 4.5), then the maps Ȟ^{p}(𝔘, ℱ) → Ȟ^{p}(𝔘′, ℱ) induced by this map of complexes are compatible with the maps of (III, Thm. 4.5).

3. Let k be a field. Compute the dualizing sheaf of the (reduced) scheme

X = V (z) ∪ V (x, y) ⊆ ℙ_{k}^{2} .

(This is the disjoint union of a line and a point.)