Schemes

Exercises 6

Schemes – Examples again

Feel free to quote results from Hartshorne or EGA.

1. Čech cohomology. Here k is a field.

(a) Let X be a scheme with an open covering U : X = U_{1}∪U_{2}, with U_{1} = Spec k[x], U_{2} = Spec k[y] with U_{1} ∩ U_{2} = Spec k[z, 1/z] and with open immersions U_{1} ∩ U_{2} → U_{1} resp. U_{1} ∩ U_{2} → U_{2} determined by x ↦ z resp. y ↦ z (and I really mean this). (We’ve seen in the lectures that such an X exists; it is the affine line zith zero doubled.) Compute Ȟ^{1}(U, O); eg. give a basis for it as a k-vectorspace.

(b) For each element in Ȟ^{1}(U, O) construct an exact sequence of sheaves of O_{X}-modules

0 → O_{X} → E → O_{X} → 0

such that the boundary δ(1) ∈ Ȟ^{1}(U, O) equals the given element. (Part of the problem is to make sense of this. It is also OK to show abstractly such a thing has to exist.)

Definition of delta. Suppose that

0 → F_{1} → F_{2} → F_{3} → 0

is a short exact sequence of abelian sheaves on any topological space X. The boundary map δ : H^{0}(X, F_{3}) → Ȟ^{1}(X, F_{1}) is defined as follows. Take an element τ ∈ H^{0}(X, F_{3}). Choose an open covering U : X = ⋃_{i∈I} U_{i} such that for each i there exists a section τ̃_{i} ∈ F_{2} lifting the restriction of τ to U_{i}. Then consider the assignment

(i_{0}, i_{1}) ⟼ τ̃_{i_{0}}|_{U_{i_{0}i_{1}}} − τ̃_{i_{1}}|_{U_{i_{0}i_{1}}} .

This is clearly a 1-coboundary in the Čech complex Č^{∗}(U, F_{2}). But we observe that (thinking of F_{1} as a subsheaf of F_{2}) the RHS always is a section of F_{1} over U_{i_{0}i_{1}}. Hence we see that the assignment defines a 1-cochain in the complex Č^{∗}(U, F_{2}). The cohomology class of this 1-cochain is by definition δ(τ).

2. Algebra. (Silly and should be easy.)

(a) Give an example of a ring A and a nonsplit short exact sequence of A-modules

0 → M_{1} → M_{2} → M_{3} → 0.

(b) Give an example of a nonsplit sequence of A-modules as above and a faithfully flat A → B such

that

0 → M_{1} ⊗_{A} B → M_{2} ⊗_{A} B → M_{3} ⊗_{A} B → 0.

is split as a sequence of B-modules.

3. Maps of Proj. Let R and S be graded rings. So R = ⊕_{d≥0}R_{d} and R_{a} · R_{b} ⊂ R_{a+b}. Suppose we have a

ring map

φ : R → S

such that there exists an integer e ≥ 1 such that φ(R_{d}) ⊂ S_{de}.

(a) For which elements 𝔭 ∈ Proj(S) is there a well-defined corresponding point in Proj(R)? In other words, find a suitable open U ⊂ Proj(S) such that φ defines a continuous map Proj(φ) : U → Proj(R).

(b) Give an example where U ≠ Proj(S).

(c) Give an example where U = Proj(S).

(d) (Do not write this down.) Convince yourself that the continuous map U → Proj(R) comes canonically with a map on sheaves so that Proj(φ) is a morphism of schemes:

Proj(S) ⊃ U ⟶ Proj(R)

Notation. Let R be a graded ring as above and let n ≥ 0 be an integer. Let X = Proj(R). Then there is a unique quasi-coherent O_{X}-module O_{X}(n) on X such that for every homogeneous element f ∈ R of positive degree we have O_{X}|_{D_{+}(f)} is the quasi-coherent sheaf associated to the R_{(f)} = (R_{f})_{0}-module (R_{f})_{n} (=elements homogenous of degree n in R_{f} = R[1/f]). See Hartshorne, page 116+. Note that there are natural maps

O_{X}(n_{1}) ⊗_{O_{X}} O_{X}(n_{2}) ⟶ O_{X}(n_{1} + n_{2})

4. Pathologies in Proj. Give examples of R as above such that (a) O_{X}(1) is not an invertible O_{X}-module.

(b) O_{X}(1) is invertible, but the natural map O_{X}(1) ⊗_{O_{X}} O_{X}(1) → O_{X}(2) is NOT an isomorphism.