Math 256B. Answers to Problem Set #7

1. (20 points) Do III Ex. 4.2. Hint: For part (a), consider subsheaves of the constant sheaf K(X) .

(a). If Spec A is a nonempty open affine in Y and Spec B = f^{−1}(Spec A) is its inverse image, then B is finitely generated and integral over A, so K(X) = Frac B is finite over K(Y) = Frac A. Let x_{1}, …, x_{r} be a basis, and let ℳ be the 𝒪_{X}-submodule of the constant sheaf K(X) on X generated by x_{1}, …, x_{r}. It is coherent because if Spec B is an open affine in X, then ℳ |_{Spec B} = ; ∑ Bx_{i}. Let η be the generic point of Y . Then (f_{∗}ℳ)_{η} is a vector space over K(Y) with basis x_{1}, …, x_{r}. Since x_{1}, …, x_{r} are global sections of ℳ , they are also global sections of f_{∗}ℳ , so they determine a sheaf morphism α: 𝒪_{Y}^{r} → f_{∗}ℳ . This is an isomorphism at η because of the known structure of (f_{∗}ℳ)_{η}. (b). Following the hint, applying ℋ om (·, ℱ) to α gives a morphism

β: ℋ om _{𝒪_{Y}} (f_{∗}ℳ , ℱ) ⟶ ℋ om (𝒪_{Y}^{r} , ℱ) = ℱ^{r}

inducing an isomorphism on the stalks at η. Let 𝒜 = f_{∗}𝒪_{X} ; it is a sheaf of 𝒪_{Y} - algebras on Y . We can turn ℋ om _{𝒪_{Y}} (f_{∗}ℳ , ℱ) into an 𝒜 -module (in the sense of (II Ex. 5.17e)) as follows. Let U ⊆ Y be an open subset, let a ∈ 𝒜 (U) = 𝒪_{X}(f^{−1}(U)) , and let ϕ ∈ ℋ om _{𝒪_{Y}} (f_{∗}ℳ , ℱ) . Then ϕ is an 𝒪_{Y} (U)-module homomorphism

ℳ (f^{−1}(U)) = (f_{∗}ℳ)(U) → ℱ (U) .

We define aϕ: ℳ (f^{−1}(U)) → ℱ (U) by (aϕ)(s) = ϕ(as) (recall that this same trick was used in showing that the category of A-modules has enough injectives). By Ex. 5.17e, there is a quasi-coherent sheaf 𝒢 on X such that f_{∗}𝒢 ≅ ℋ om _{𝒪_{Y}} (f_{∗}ℳ , ℱ) . Since (f_{∗}𝒢)_{η} is finitely generated, we may use (II Ex. 5.15e) to replace 𝒢 with a coherent subsheaf which still satisfies the condition at η. (c). Following the hint again, we use noetherian induction on closed subsets of Y (closed subschemes would also work). If Y = ∅, then it is affine, so the theorem is true in that case.

Let f: X → Y be a finite surjective morphism of noetherian separated schemes, with X affine.

First note that finiteness and surjectivity (of f ) are preserved by base change. For finiteness this is easy to see from the definition; for surjectivity this was proved in the solution of (II Ex. 4.4). Also, if we change base to a closed subscheme of Y , then X and Y remain noetherian and separated, since X too is replaced by a closed subscheme of itself.

We now reduce to Y reduced, by base change to Y_{red} . If X is affine then so is X ×_{Y} Y_{red} , since the latter is a closed subscheme of X (II Ex. 3.11b). If we can show that Y_{red} is affine, this will imply that Y is affine, by Ex. 3.1. So, we may assume that Y is reduced.

Next reduce to Y integral. Let Y′ be an irreducible component of Y , with reduced induced subscheme structure, let X′ = X ×_{Y} Y′, and let f′: X′ → Y′ be the projection map. Then X′ will still be affine (II Ex. 3.11b again), and f′ will still satisfy all the conditions of the theorem, so if we can show that Y′ is affine then it will follow by Ex. 3.2 that Y is affine. After doing this for all irreducible components Y′ of Y , we may assume that Y is integral.

Finally, reduce to X integral. Since f is closed (II Ex. 4.1) and Y is irreducible, some irreducible component of X maps onto Y . We may replace X with that irreducible component (with reduced induced subscheme structure) and it will still be affine and will still satisfy the hypotheses. Thus, we may assume that X and Y are integral.

Now let ℱ be a coherent sheaf on Y , and let β: f_{∗}𝒢 → ℱ^{r} be as in part (b). We have an exact sequence

0 ⟶ ker β ⟶ f_{∗}𝒢 ⟶ ℱ^{r} ⟶ coker β ⟶ 0  (*)

of coherent sheaves on Y , with ker β and coker β supported on proper subsets of Y .

Lemma. Let 𝒦 be a coherent sheaf on a noetherian scheme W . Then there is a closed subscheme i: Z → W corresponding to the closed subset Supp 𝒦 , and a coherent sheaf 𝒦 ′ on Z, such that 𝒦 ≅ i_{∗}𝒦 ′.

Proof. Let ℐ be the coherent sheaf of ideals on W defined by ℐ |_{Spec A} = (Ann M)

for all open affines Spec A, where M = 𝒦 (Spec A) , so that 𝒦 |_{Spec A} ≅ M̃. To show that this definition is consistent, it suffices to show that Ann M_{f} = (Ann M)_{f} for all f ∈ A, by Problem 3 in Problem Set #3, since the intersection of any two open affines in W can be covered by simultaneously principal open subsets. The inclusion “ ⊇ ” is trivial. To show “ ⊆,” let a/f^{r} ∈ Ann M_{f} with a ∈ A and r ∈ ℕ and let m_{1}, …, m_{s} be a finite generating set for M . Then there are integers n_{1}, …, n_{s} such that f^{n_{i}}am_{i} = 0 for all i. Taking n = max{n_{1}, …, n_{s}}, we find that f^{n}a ∈ Ann M , so a/f^{r} ∈ (Ann M)_{f} , as was to be shown.

Let Z be the closed subscheme corresponding to ℐ ; then the lemma is true by (II Prop. 5.2d). □

By the lemma, there is a proper closed subscheme i: Z → Y and a sheaf 𝒦 ′ on Z such that i_{∗}𝒦 ′ ≅ ker β ⊕ coker β. Since X ×_{Y} Z is a closed subscheme of X, it is affine. Applying Chevalley’s theorem to X ×_{Y} Z → Z by noetherian induction, it follows that Z is affine. Thus H^{i}(Z, 𝒦 ′) = 0 for all i > 0 by Theorem 3.5, so Lemma 2.10 and compatibility of cohomology with direct sums gives

H^{i}(Y, ker β) = H^{i}(Y, coker β) = 0

for all i > 0 .

Now split up the exact sequence (*) into short exact sequences

0 ⟶ ker β ⟶ f_{∗}𝒢 ⟶ im β ⟶ 0

and

0 ⟶ im β ⟶ ℱ^{r} ⟶ coker β ⟶ 0 .

The long exact sequence in cohomology associated to the first of these, combined with Ex. 4.1, gives an exact sequence

H^{i}(X, 𝒢) ⟶ H^{i}(Y, im β) ⟶ H^{i+1}(Y, ker β)

for all i > 0 . But the first of these groups vanishes by Thm. 3.5, and the last one also was shown to be zero, so H^{i}(Y, im β) = 0 for all i > 0 . The long exact sequence associated to the second of the above two short exact sequences gives an exact sequence

H^{i}(Y, im β) ⟶ H^{i}(Y, ℱ^{r}) ⟶ H^{i}(Y, coker β)

for all i > 0 . Again, the two ends vanish, so H^{i}(Y, ℱ^{r}) = 0 for all i > 0 .

Since H^{i}(Y, ℱ^{r}) ≅ H^{i}(Y, ℱ)^{r}, it follows that H^{i}(Y, ℱ) = 0 for all i > 0 . Since this holds for all coherent sheaves ℱ on Y , it follows by Thm. 3.7 that Y is affine, as was to be shown.

2. (10 points) Do III Ex. 5.3.

(a). If X is integral and k is algebraically closed, then X is a variety, and by (I Thm. 3.4a) H^{0}(X, 𝒪_{X}) = 𝒪_{X}(X) = k. Thus h^{0}(X, 𝒪_{X}) = 1 ,

χ(X) − 1 = ∑_{i=1}^{r}(−1)^{i}h^{i}(X, 𝒪_{X}) ,

and

p_{a}(X) = ∑_{i=1}^{r}(−1)^{r−i}h^{i}(X, 𝒪_{X}) = ∑_{i=0}^{r−1}(−1)^{i}h^{r−i}(X, 𝒪_{X}) .

If X is a curve, then r = 1 and we just get p_{a}(X) = h^{1}(X, 𝒪_{X}) .

(b). Let S_{X} be the homogeneous coordinate ring of X associated to its embedding in ℙ_{k}^{r}. Then 𝒪_{X} = S̃_{X} and therefore S_{X} ≈ Γ_{∗}(𝒪_{X}) by (II Ex. 5.9b), so the Hilbert polynomial P_{X} of X equals the Hilbert polynomial of 𝒪_{X} defined in Ex. 5.2. Thus P_{X}(0) = χ(𝒪_{X}(0)) = χ(𝒪_{X}) . This suffices to show that the two definitions of arithmetic genus are the same in this case.

(c). For nonsingular projective curves, birational means isomorphic (I Cor. 6.12), so the first sentence is obvious. If X is a nonsingular plane curve of degree d ≥ 3 , then by (I Ex. 7.2), p_{a}(X) ≠ 0 . Since p_{a}(ℙ^{1}) = h^{1}(ℙ^{1}, 𝒪_{ℙ^{1}}) = h^{0}(ℙ^{1}, 𝒪(−2)) = 0 , this leads to a contradiction if X is rational. Therefore X is not rational.

3. (10 points) Do III Ex. 5.10.

It will suffice to prove this when r = 3 . Pick a very ample (or just ample) line sheaf 𝒪(1) on X, and let

ℱ^{1} ⟶^{ϕ} ℱ^{2} ⟶^{ψ} ℱ^{3}

be an exact sequence of coherent sheaves on X. Let 𝒢 = im ϕ = ker ψ; it is a subsheaf of ℱ^{2} . Pick n_{0} such that H^{1}(X, (ker ϕ)(n)) = 0 for all n ≥ n_{0} . For such n the long exact sequence associated to the short exact sequence

0 ⟶ (ker ϕ)(n) ⟶ ℱ^{1}(n) ⟶ 𝒢 (n) ⟶ 0

gives

H^{0}(X, ℱ^{1}(n)) ⟶ H^{0}(X, 𝒢 (n)) ⟶ 0 ;  (1)

in addition, the exact sequence

0 ⟶ 𝒢 (n) ⟶ ℱ^{2}(n) ⟶ ℱ^{3}(n)

gives an exact sequence

0 ⟶ H^{0}(X, 𝒢 (n)) ⟶ H^{0}(X, ℱ^{2}(n)) ⟶ H^{0}(X, ℱ^{3}(n))  (2)

since the global section functor is left exact. Combining (1) and (2) then gives the exact sequence

H^{0}(X, ℱ^{1}(n)) ⟶ H^{0}(X, ℱ^{2}(n)) ⟶ H^{0}(X, ℱ^{3}(n)) ,

as was to be shown.

4. (10 points) Do II Ex. 8.3.

(a). Applying (8.11) to the sequence X ×_{S} Y ⟶^{p_{1}} X ⟶ S, where p_{i} denotes projection to the i^{th} factor, gives an exact sequence

p_{1}^{∗}Ω_{X/S} ⟶ Ω_{X×_{S}Y/S} ⟶ Ω_{X×_{S}Y/X} ⟶ 0 .

By (8.10), Ω_{X×_{S}Y/X} ≅ p_{2}^{∗}Ω_{Y/S} , so this exact sequence becomes

p_{1}^{∗}Ω_{X/S} ⟶^{ϕ} Ω_{X×_{S}Y/S} ⟶ p_{2}^{∗}Ω_{Y/S} ⟶ 0 .  (*)

Interchanging the roles of the factors X and Y gives a similar exact sequence

0 ⟵ p_{1}^{∗}Ω_{X/S} ⟵^{ψ} Ω_{X×_{S}Y/S} ⟵ p_{2}^{∗}Ω_{Y/S} .

Now let U = Spec A and V = Spec B be open affines in X and Y , respectively, both lying over an open affine W = Spec C of S. Then U ×_{W} V = Spec(A ⊗_{C} B) is an open affine of X ×_{S} Y . Over this open affine, p_{1}^{∗}Ω_{X/S} = (Ω_{A/C} ⊗_{A} (A ⊗_{C} B)) , Ω_{X×_{S}Y/S} = (Ω_{A⊗_{C}B/C}) , and Ω_{X×_{S}Y/Y} = (Ω_{A⊗_{C}B/B}) .

On this open affine, the map ϕ: (Ω_{A/C} ⊗_{A} (A ⊗_{C} B)) → (Ω_{A⊗_{C}B/C}) is given by da ⊗ (a′ ⊗ b) ↦ (a′ ⊗ b)d(a ⊗ 1) for all a, a′ ∈ A and all b ∈ B. Recalling that ψ is a composite

(Ω_{A⊗_{C}B/C}) ⟶ (Ω_{A⊗_{C}B/B}) ⟶ (Ω_{A/C} ⊗_{A} (A ⊗_{C} B)) ;

it is given by

d(a ⊗ b) ↦ d(a ⊗ b) = (1 ⊗ b)d(a ⊗ 1) ↦ da ⊗ (1 ⊗ b)

for all a ∈ A and all b ∈ B. Thus ψ ∘ ϕ is given by

da ⊗ (a′ ⊗ b) ↦ (a′ ⊗ b)da ⊗ (1 ⊗ 1) = da ⊗ (a′ ⊗ b) ;

i.e., it is the identity on p_{1}^{∗}Ω_{X/S} .

This implies that ϕ is injective; moreover, ψ splits the short exact sequence derived from (*), giving Ω_{X×_{S}Y/S} ≅ p_{1}^{∗}Ω_{X/S} ⊕ p_{2}^{∗}Ω_{Y/S} , as was to be shown. (b). This is immediate from the definition of canonical sheaf (p. 180), part (a), (I Ex. 3.15d), and (II Ex. 5.16d). (c). By Example 8.20.3, ω_{Y} = 𝒪(3−2−1) is the trivial line sheaf, so by (b) ω_{X} ≅ 𝒪_{X} and therefore p_{g}(X) = 1 . By (I Ex. 7.2b) we have p_{a}(Y) = 1 , so by (I Ex. 7.2e)

p_{a}(X) = 1^{2} − 1 − 1 = −1 ,

as was to be shown.