Bryden Cais.  Algebraic Geometry, HW 3.  March 5, 2003

1. Given a smooth curve C and an open subset U ⊂ C with a morphism f : U → ℙ_{k}^{n} we must show that f extends uniquely to a morphism C → ℙ_{k}^{n}. Since C has dimension 1 and U is open, U is the complement of finitely many points. Thus, we can reduce to the case that C − U = P consists of a single point. Working locally, we can replace U by Spec k(C) and C by O_{C,P}. Now we have a morphism f : Spec k(C) → ℙ_{k}^{n} given by f = (f_{0}, f_{1}, …, f_{n}) for some f_{i} ∈ k(C) with V (f_{0}, f_{1}, …, f_{n}) = ∅. Since O_{C,P} is a DVR, there is a valuation v on k(C) = frac O_{C,P} such that O_{C,P} = {f ∈ k(C) : v(f) ≥ 0}. Now let i be such that v(f_{i}) ≤ v(f_{j}) for all j ≠ i. Such a minimum exists since there are only finitely many f_{j}. Then observe that v(f_{j}/f_{i}) = v(f_{j}) − v(f_{i}) ≥ 0 so that f_{j}/f_{i} ∈ O_{C,P} for each j. Thus we define

f̃ = (f_{0}/f_{i}, f_{1}/f_{i}, …, f_{n}/f_{i}).

Clearly, since the i th coordinate is 1, this gives a well defined morphism O_{C,P} → ℙ_{k}^{n}. Moreover, we have f_{i}f̃ = f so that f̃ extends f.

2. Let X = Spec A and F = M̃ for some finite A module M. We claim that Supp F = V (I) where I = Ann M. Indeed, by definition Supp F = {P ∈ X : F_{P} ≠ 0}. Moreover, we have seen that F_{P} ≃ M_{P}. Thus, we are reduced to showing that M_{P} = 0 if and only if Ann M ⊈ P. Indeed, suppose that Ann M ⊈ P and let f ∈ Ann M − P. Then for any m/g ∈ M_{P} with m ∈ M and g ∉ P we have fm/g = 0 and f ∉ P so that m/g ∼ 0 in M_{P}. Thus, M_{P} = 0.

Conversely, suppose that M_{P} = 0. Then for all m ∈ M we have m ∼ 0 in M_{P} (since, in particular, M ⊂ M_{P}). That is, there exists f_{m} ∉ P such that f_{m}(m − 0) = f_{m}m = 0 for all m ∈ M. Now since M is a finitely generated A module, we can find an A basis m_{1}, …, m_{s} for M. Now let f = f_{m_{1}}f_{m_{2}} ⋯ f_{m_{s}}. By construction, f ∈ Ann M. Moreover, since f_{m_{i}} ∉ P, we have f ∉ P. It follows that Ann M ⊈ P.

3. Let X be a scheme and F a sheaf of O_{X} modules on X. We show that F is coherent if and only if there exists an open cover {U_{i}} of X such that, for each i, the sheaf F|_{U_{i}} is the cokernel of a morphism of free O_{X} modules of finite rank.

One direction is trivial: if there exists an open cover {U_{i}} of X such that, for each i, the sheaf F|_{U_{i}} is the cokernel of a morphism of free O_{X} modules of finite rank then certainly F|_{U_{i}} is coherent. If {V_{i,j}} is an open affine cover of U_{i}, for each i then F|_{U_{i}}(V_{i,j}) is a finite A_{i,j} module, where V_{i,j} = Spec A_{i,j}. Since the V_{i,j} cover X, we see that F is coherent.

Conversely, suppose that F is coherent. Let {U_{i}} be an affine open cover of X with U_{i} = Spec A_{i} such that F|_{U_{i}} ≃ M̃_{i}, with M_{i} a finite A_{i} module. Let m_{1,i} …, m_{s(i),i} be a basis of M_{i} over A_{i}. Define a homomorphism ϕ_{i} : A_{i}^{s(i)} → M_{i} by (0, 0, …, 0, 1, 0, …, 0) ↦ m_{j,i}, where the 1 is in the j th position. Now ker ϕ_{i} is a submodule of M and hence finitely generated. So let n_{1,i}, …, n_{r(i),i} be an A_{i} basis of ϕ_{i}. Define ψ_{i} : A_{i}^{r(i)} → ker ϕ_{i} as above. Then ψ_{i}(A_{i}^{r(i)}) ≃ ker ϕ_{i} so that coker ψ_{i} = A_{i}^{s(i)}/ ker ϕ_{i} ≃ M_{i}. It follows that the sheaf F|_{U_{i}} is the cokernel of a morphism of free O_{X} modules of finite rank.

4. Let X be a reduced connected scheme and F a coherent sheaf on X. Define ϕ : X → Z by

ϕ(x) = dim_{k(x)} F_{x} ⊗_{O_{X,x}} k(x)

where k(x) = O_{X,x}/𝔪_{X,x}.

(a) We show that the set {x ∈ X : ϕ(x) < n} is open for any integer n. To do this, it suffices to produce an open neighborhood U of any point x such that for all y ∈ U we have ϕ(y) ≤ ϕ(x). Let V = Spec A be any affine open set containing x with F|_{V} = M̃ and let P ⊂ A correspond to x. Then we have F_{x} = M_{P} and O_{X,x} = A_{P} so that ϕ(x) = dim_{A_{P} /P A_{P}} M_{P} /P M_{P} = s, say. Let α_{1}, …, α_{s} ∈ M_{P} be elements whose reductions modulo P M_{P} form a basis for M_{P} /P M_{P} over A_{P} /P A_{P}. Without loss of generality, we may take α_{i} ∈ M since we can just clear denominators. By Nakayama’s Lemma (Atiya Macdonald, p. 22) α_{i} generate M_{P} as an A_{P} module.

Since M is a finite A module, we let m_{1}, …, m_{r} generate M over A. Since m_{i} ∈ M_{P}, we can write

m_{i} = ∑_{j} \frac{a_{i,j}}{b_{i,j}} α_{j}

for a_{i,j}, b_{i,j} ∈ A and b_{i,j} ∉ P. Let t_{i} = ∏_{j} b_{i,j}. Then t_{i}m_{i} is in the A span of α_{i} and t_{i} ∉ P. Now set f = ∏_{i} t_{i}. Then f ∉ P so that P ∈ D(f) and D(f) is open. Moreover, if Q ∈ D(f) then t_{i} ∉ Q so that t_{i} is a unit in A_{Q}. Since c_{i}m_{i} is in the A span of α_{i}, it follows that m_{i} is in the A_{Q} span of α_{i}. Since m_{i} span M over A, we have m_{i} span M_{Q} over A_{Q} so that the α_{i} span M_{Q} over A_{Q}. It follows that the images of α_{i} in M_{Q}/QM_{Q} span M_{Q}/QM_{Q} as a A_{Q}/QA_{Q} vector space; that is, ϕ(Q) ≤ s = ϕ(P).

(b) Suppose that F is locally free and let U_{i} be an open cover of X on which F|_{U_{i}} = M̃_{i} with F_{U_{i}} a free O_{U_{i}} module. Suppose that for some i, U_{i} = Spec A_{i} and M_{i} has rank n as an A_{i} module. Now let W be the union of all U_{j} such that M_{j} if free or rank n and let V be the union of all U_{j} with M_{j} free of rank m ≠ n. By construction, W, V are open and W is nonempty. But if x ∈ W ∩ V then F_{x} ≃ M_{P} has rank n and rank m ≠ n, and this is absurd. Hence we conclude that W ∩ V = ∅. Then since we clearly have X = W ∪ V and since X is connected by hypothesis, we must have V = ∅. It follows that ϕ(x) = n is constant on X.

Conversely, suppose that ϕ(x) = n is constant for all x ∈ X. We show that the stalk F_{x} is free. Let U = Spec A be an open affine set containing x with F_{x} = M_{P} where P ⊂ A corresponds to x. Let α_{1}, …, α_{n} ∈ M have images forming a basis of M_{P} /P M_{P} over A_{P} /P A_{P}. By Nakayama’s lemma, as above, the α_{i} generate M_{P} as an A_{P} module; we must show they are linearly independent. So suppose we have a relation ∑_{i=1}^{n} \frac{a_{i}}{b_{i}} α_{i} = 0 in M_{P}. We can clear denominators to suppose that we have a relation

∑_{i=1}^{n} a_{i}α_{i} = 0

in M_{P} with a_{i} ∈ A; that is, there is some c ∉ P with c ∑ a_{i}α_{i} = 0 in M. Certainly we then have a linear relation in M_{P} /P_{M}, and since the α_{i} are linearly independent over A_{P} /P A_{P} we have, for each i, some c_{i} ∉ P with c_{i}a_{i} = 0 in M_{P} /P M_{P}; that is, c_{i}a_{i} ∈ P. It follows that a_{i} ∈ P for each i. By our arguments above, there is an open set D(f) with P ∈ D(f) and for all Q ∈ D(f) the α_{i} generate M_{Q} over A_{Q}. Now if Q ∈ D(cf) then since c, g ∉ Q we have

∑_{i=1}^{n} a_{i}α_{i} = 0

in M_{Q}. Since ϕ is constant, the images of α_{i} form a basis of M_{Q}/QM_{Q} over A_{Q}/QA_{Q} from which it follows, as above, that a_{i} ∈ Q. We have thus shown that for all Q ∈ D(cf), we have a_{i} ∈ Q. By Atiyah Macdonald p. 5, this implies that a_{i} lies in the nilradical of A_{cf} (since the prime ideals of A_{cf} are exactly the elements of D(cf)). But since X is reduced, we have A reduced and hence A_{cf} reduced. Thus, a_{i} = 0 in A_{cf}. Finally, since cf ∉ P we have A_{cf} ⊂ A_{P} whence a_{i} = 0 in A_{P}. It follows that the α_{i} are linearly independent over A_{P} and hence F_{x} is free for each x. We conclude that F is locally free.

5. Let X/k be a scheme and L a locally free sheaf of rank 1. Construct a scheme π : L → X as follows: let {U_{i}} be an open affine cover of X and ϕ_{i} : L|_{U_{i}} ˜→O_{U_{i}} a trivialization of L|_{U_{i}}. Set U_{ij} = U_{i} ∩ U_{j} and let g_{ij} ∈ O_{X}^{×}(U_{ij}) be the transition functions. Set V_{i} = U_{i} × A_{t_{i}}^{1} and glue the natural maps V_{i} → U_{i} and V_{j} → U_{j} to form π : L → X by

U_{ij} × 𝔸_{t_{i}}^{1} → U_{ij} × 𝔸_{t_{j}}^{1}, (u, t_{i}) ↦ (u, g_{ij}t_{i}).

We claim that Γ(U, L) is identified with sections σ : U → L|_{U} with π ∘ σ = id. Indeed, since U_{i} is affine, for each i we have U_{i} = Spec A_{i} and V_{i} = π^{−1}(U_{i}) = Spec A_{i}[t_{i}]. Now any section σ : U_{i} → V_{i} induces an A-algebra ring homomorphism θ_{i,σ} : A_{i}[t_{i}] → A_{i}. We thus have the correspondence

{σ} ↦ {s ∈ Γ(U, L) : s|_{U_{i}} = θ_{i,σ}(t_{i}) ∈ A_{i}}.

In the reverse direction, we have

{s ∈ Γ(U, L) : s|_{U_{i}} = a_{i} ∈ A_{i}} ↦ {σ : θ_{i,σ}|_{A_{i}} = id, θ_{i,σ}(t_{i}) = a_{i}}.

It is not difficult to see that σ defined in this way is a section (follows from the requirement that θ_{i,σ}|_{A_{i}} = id) and that the local pieces defining σ glue together compatibly (indeed, we have g_{ij}a_{j} = a_{i}). Moreover, we see that these are inverse correspondences.