Lecture 13: Invertible Sheaves

Lecture  18.725 Algebraic Geometry I Lecture 13: Invertible Sheaves Last time we showed that when X = Spec A is an affine scheme, we have the equivalence QCoh(X) ∼ = Mod(A) given by the Γ and the Loc functors. In particular, these functors are exact, and we have Γ(F) = 0 =⇒ F = 0. This in particular implies that Γ ◦ Loc = 1 (We know this holds for A, now check the general case by choosing a presentation.). We need to check the other direction: Loc ◦ Γ(F) = F. Definition 1. A functor F : C1 → C2 is called conservative if for every g ∈ Hom(C1 ), F(g) is an isomorphism implies that g is an isomorphism. Note that this does not say that F(A) ∼ = F(B) =⇒ A ∼ = B. Example 1. Let C1 , C2 be abelian categories, and F an exact functor. Then ker(F(f )) = F(ker(f )), and the same holds for cokernels. Lemma 1. Let L, R be adjoint functors, L fully faithful (i.e. R ◦ L ∼ = Id), R is conservative, then the two functors are inverse pairs in an categorical equivalence. Proof. We need RL ∼ = Id, which follows from RLR ∼ = R by conservative property, which in turns follows from the fully faithfulness of F. Now back to the discussion on Loc and Γ. We already know that Loc is fully faithful, and it is sufficient to f. The image of M f are the functors show it is essentially surjective, i.e. every F has some M such that F = M ⊕I ⊕J that have presentations, i.e. O → O → F → 0, so it suffices to check that every F has a presentation. We check that for every F, there exists a surjection O⊕J  F. To see so, consider Γ(F) = Hom(O, F) (structure sheaf is the terminal object in the category of sheaves). So if we take a set of generators mj , j ∈ J of F, we obtain an onto map Γ(O⊕J ) → Γ(F), so O⊕J → F is surjective. Remark 1. Results of this type are generally referred to as Morita theories. Now suppose A contains arbitrary direct sums and that Hom(P, •) commutes with the direct sum. We say P ∈ A is a projective generator if the P -projection functor, X 7→ Hom(P, X), is an exact functor, and that Hom(P, X) = 0 ⇔ X = 0. In this case, one can show that A ∼ = Mod(End P )opp , and, in particular, as ∼ a corollary, we have Mod(A)f.g. = Coh(X). Lemma 2. f : X → Y is an affine morphism if and only if for every open U ⊆ U , f −1 (U ) is affine. f : X → Y is a finite morphism if and only if it is affine and, for every open U ⊆ Y such that U = Spec A, if f −1 (U ) = Spec B then B is a finite A-algebra. [ Proof. Let U be affine. By definition, there exists some affine cover U = Ui such that f −1 (Ui ) is affine. Write V = f −1 (U ), then we want to have V = Spec A. Note that k[Ui ] = f∗ (O)(Ufi ) = f∗ (O)(U )fi = A(fi ) , and each A(fi ) is finitely generated. Take all those rings together as an algebra over B = k[U ], we obtain a finitely generated ring A. The check that V = Spec A is routine. For the second part, suppose f : X → Y finite (in the old definition), then f∗ OX is a coherent sheaf on Y , i.e. f∗ OX (U ) is finite over OY for some open set U . Proposition 1. For any fixed Y , the category of X that has an affine morphism to Y corresponds to the opposite category of quasicoherent sheaves of OY -algebra (which is finitely generated and reduced). To see this, given any map f : X → [Y we obviously obtain a sheaf f∗ OX . Conversely, given a sheaf A of OY algebra, pick an affine cover Y = Ui , glue together all the Spec A[Ui ] by identifying Spec A[Ui ∩ Uj ] i that sits in two copies (here we assume seperatedness). Proposition 2. Suppose X → Y is affine. Let A = f∗ OX , then Qcoh(X) = {Qcoh(Y ) with an A action}, where the map is F 7→ f∗ F. Let i : Z ,→ X be an embedding of a closed subvariety, then i∗ is a full embedding of a subcategory, with one-sided inverse i∗ . It is easy to see that the image of i∗ consists of those F such that F|X−Z = 0. On the other hand, for every Z ⊆ X we have a subsheaf IZ ⊆ OX consisting of those f that vanish on Z. It is obviously an ideal sheaf, and we in fact have a correspondence between closed subvarieties and radical ideal sheaves. 1 Lecture  18.725 Algebraic Geometry I Proposition 3. i∗ : Qcoh(Z) → Qcoh(X) (or coherent to coherent) is a full embedding and the image are the Fs such that IZ F = 0. For example, consider X = Spec A, and let Z = Spec A/I, then A/I modules are the A modules that are killed by I. Let U = X − Z, then i∗ F|U = 0. Note the converse doesn’t hold: there might be F that restricts to U to be trivial, but does not come from i∗ M for any M . For instance, let X = A1 , Z = {0}, let f, and let i : k[t] → k that sends t to 0. There does exist a weaker property: if F|U = 0, M = k[t]/t2 , F = M σ is a section of F, then there exists some n such that IZn σ = 0. In addition, if F is coherent, then we actually have ssome n such that IZn F = 0. Locally free sheaves of rank 1 are called invertible sheaves. ˜ ]d = {p/q | deg p − deg q = d, q | 6= 0} is an invertible Example 2. Let X = Pn , then OPn (d)(U ) = k[U Ũ ˜ sheaf on X, where U ,→ U is the projection compatible with An+1 − {0} ,→ An+1 . We would like to understand maps X → Pn , by which we mean the similar knowledge as the fact that T.F.A.E.: • Maps X → An ; • Homs k[x1 , . . . , xn ] → k[X]; • n-tuple elements in k[X]. And our claim is that T.F.A.E.: • Maps X → Pn ; • Invertible sheaves L on X with (n + 1) elements s0 , . . . , sn in Γ(L) such that they generate L. Here to a map f : X → Pn we assign f ∗ O(1) with sections t0 , . . . , tn . Conversely, given L generated by s0 , . . . , sn set f = (s0 : . . . : sn ), locally we can identify L with O so s0 , . . . , sn give functions on U with no common zeroes. If f0 , . . . , fn are these functions, then x 7→ (f0 (x) : . . . : fn (x)) is a map U 7→ Pn indepdendent of choice that gives an isomorphism L ∼ = O. 2