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AG for NT Week 6 AG for NT Week 6 Recap 1. A ring, X = Spec A, M an A-module which we associated a sheaf Mf of OX -modules. Have shown: Any quasi-coherent sheafs on Spec A has this form. 2. S graded ring, X = PrjS, M graded S-module, associated to it Mf a quasi-coherent sheaf. Mf(D+(f)) = fhomogeneous elements of degree 0 in Mf g. Want to show: Given any quasi-coherent sheaf on X = ProjS, we have f =∼ Mf for some graded S-module M. We won't (can't?) say this in general, but Proposition 0.1. S graded ring, nitely generated by S1 as an S0-algebra (S = ⊕a≥0Sa). Let X = ProjS and f a quasi coherent sheaf on X. Then there is a graded S-module M such that f =∼ Mf. Remark. 1. Applies to for some ring , i.e., to n S = A[x0; : : : ; xn] A PA 2. Don't have an equivalence of categories between graded S-modules and quasi-coherent sheafs on ProjS. Indeed, 2 dierent graded modules can give the same sheaf. Example: and for some ⊕n≥0Mn ⊕n≥n0 Mn n0 ≥ 0 3. How to dene M given f? Cannot just set M = Γ(X; f) as in the ane case. Solution is to use twists. Example. , Proj r . S = A[x0; : : : ; xr] X = S = PA Claim. Γ(X; OX (n)) = Sn = fhomogenous polynomials of degree n}. In ∼ particular, Γ(X; OX ) = A. Proof. The sets D+(xi) cover X. A section t 2 Γ(X; OX (n)) is the same as an (r + 1)-tuple of sections (t0; : : : ; tr) with ti 2 OX (n)(D+(xi)) com- patible on D+(xi) \ D+(xj) = D+(xixj) Now, homogeneous elements in and OX (n)(D+(xi)) = fdeg n Sxi g OX (n)(D+(xixj)) = homogeneous elements in . We are considering fdeg n Sxixj g OX (n)(D+(x0 : : : xr)) = 1 in homogeneous elements in . We have , fdeg n Sx0:::xr g 8i; j S,! Sxi ,! where the maps are dened respectively , Sxixj ,! Sx0:::xr f 7! f=1 k k k and k k k. f=xi 7! fxj =(xixj) f=(xixj) 7! f(x0 : : : xi−1xi+1 : : : xj−1xj+1 : : : xr) =(x0 : : : xr) These preserve gradings, injective as no xi is a zero divisor in S and induce the relevant restriction maps. So to give a section t 2 Γ(X; OX (n)) is to give a homogeneous elements of in , lying in deg n Sx0:::xr \iSxi = S So we have ∼ . S = ⊕n2ZΓ(X; OX (n)) = ⊕n2ZΓ(X; Se(n)) Denition. S a graded ring, generated by S1 as S0-algebra. X = ProjS, f sheaf of OX -modules. Dene the graded S-module associated to f to be . Γ∗(f) = ⊕n2ZΓ(X; f(n)) Module Structure: If s 2 Sd, view this as an element of Γ(X; OX (d)) (glue the in . Then ∼ . So get homomor- S=1 OX (d)(D+(f)) f(n)⊗OX OX (d) = f(n+d) phism so for Γ(X; f(n))⊗Γ(X;OX ) Γ(X; OX (d)) ! Γ(X; f(n+d)) t 2 Γ(X; f(n)) dene s · t to be the image of t ⊗ s in Γ(X; f(n + d)). ∼ Remark. For S = A[x0; : : : ; xm] have shown S = Γ∗(OX ) as graded S-modules. Proposition. 0.1 again. Let S be a graded ring, nitely generated by S1 as ∼ S0-algebra, X = ProjS, f a quasi coherent sheaf on X. Then f = Γ^∗(f) Proof. See Hartshorne Closed subschemes Recall: A closed immersion is a morphism f : Y ! Xof schemes such that f induces a homeomorphism from Y to a closed subset of X, and such that # f : OX ! f∗OY is surjective. Denition. A closed subscheme of a scheme X is an equivalence class of closed immersion where f : Y ! X, f 0 : Y 0 ! X are equivalent if there is an isomor- phism i : Y 0 ! Y such that f 0 = f · i. Example. A ring, aCA, then homomorphism A ! A=a induces closed immer- sion Spec(A=a) ! Spec A image v(a). # Aim: To show that the kernel of f , that is, the sheaf u 7! ker(OX (U) ! (f∗OY )(U)) is quasi-coherent when f is closed immersion. Denition. A topological space X is quasi-separated if the intersection of any two quasi-compact open subsets is again quasi-compact. Exercise. For any ring A, Spec A is quasi-separated. Denition 0.2. A morphism of schemes f : X ! Y is quasi-compact (respec- tively quasi-separated) if for every ane open U ⊃ Y , f −1(U) is quasi-compact (respectively quasi-seperated) 2 Remark. If X is a Noetherian scheme, then for any scheme Y and morphism f : X ! Y , f is both quasi-compact and quasi-separated. If f : X ! Y is separated then it's quasi-separated. Lemma. Let i : Y ! X be a close immersion. Then i is both quasi-compact and quasi-separated. Proof. Let V = i(Y ) and take U ⊂ X ane open (Say U = Spec A). Then i−1(U) is homeomorphic to U \V . As V is closed in X, U \V is closed in U. So −1 U \ V is of the form V (a) for some ideal a C A. Then i (U) is homeomorphic to Spec(A=a) hence both quasi-separated and quasi-compact. Lemma. X a scheme. Then the kernel of any morphism of quasi-compact sheaves φ : F!G on X is also quasi-coherent. Proof. The question is local, so we may assume X = Spec A is ane. Then there are A-modules, M; N such that F =∼ Mf and G =∼ Ne and φ is induced from a homomorphism : M ! N. It is easy to show that ker φ =∼ ker] : Lemma. Let f : X ! Y be a morphism of schemes. Suppose f is quasi- compact and quasi-separated. Let F be a quasi-coherent sheaf on X. Then f∗F is a quasi-coherent sheaf on Y . Proof. We may assume Y is ane (exercise). Then by assumption, X = f −1(Y ) is both quasi-compact and quasi-seperated. So we may cover X by nitely many open ane Ui and for each i; j we may cover Ui \Uj by nitely many open ane Uijk. −1 Now let V ⊂ Y open. To give a section, s 2 (f∗F)(V ) = F(f (V )) is the −1 −1 same as giving sections si over each f (V )\Ui agreeing on each f (V )\Uijk. So f∗F ts into an exact sequence 0 ! f∗F ! ⊕if∗(FjUi) ! ⊕i;j;kf∗(FjUijk) of sheaves on . As each is ane, both and Y Ui;Uijk ⊕if∗(FjUi ) ⊕i;j;kf∗(FjUijk ) are quasi-coherent (1st lemma last week). Thus, as the kernel of a morphism of quasi-coherent sheaves, f∗F is also quasi-coherent. Corollary. Let i : Y ! X closed immersion of schemes. Then i∗OY is quasi- coherent. Denition. (X; OX ) be a scheme. A sheaf of ideals on X is a sheaf F on X such that for each U ⊂ X, F(U) is an ideal of OX (U). These are clearly OX -modules. Denition. Let Y closed subscheme of X and i : Y ! X corresponding closed # immersion. Dene the ideal sheaf of Y , IY to be the kernel of i : OX ! i∗OY . It depends only on the equivalence class of the closed immersion. Note. IY , as dened above is quasi-coherent. Example. Let X = Spec A ane, Y = Spec(A=a), i : Y ! X obvious closed immersion, then . In particular ∼ . Γ(X; IY ) = a IY = ea 3 Proposition. X = Spec A ane scheme. Then any closed subscheme U of X is of the form (Spec A=a; i) where a C A and i : Spec A=a ! Spec A is induced from A ! A=a. This gives a one to one correspondence between ideals of A and closed subschemes of Spec A. In particular, any closed subscheme of an ane scheme is ane. Proof. Note that distinct ideals give non-equivalent closed subschemes as can recover a as Γ(X; IY ) as in example above. Now need a lemma Lemma. X a topological space, i : Y ! X continuous and induces homeomor- −1 phism from Y to a closed subset of X. Let F be any sheaf on Y . Then i i∗f is canonically isomorphic to F Proof. Exercise And a denition Denition. X a topological space, F a sheaf on X. Dene the support of F, supp(F) to be the set fx 2 Xjfx 6= 0g Let i : Y ! X = Spec A closed immersion. IY be the corresponding ideal sheaf. Then is quasi-coherent so ∼ where . Now IY IY = ea a = Γ(X; IY ) C A # i : OX ! i∗OY is surjective, so have a short exact sequence of sheaves. , but is isomorphic to canonically, so # 0 ! ea !OX ! i∗OY ! 0 OX =ea (^A=a) i ∼ induces isomorphism (^A=a) ! i∗OY . Now for p 2 Spec A we have ( ∼ 0 p2 = i(Y ) (i∗OY )p = : OY;i−1(p) 6= 0 p 2 i(Y ) So i(Y ) = supp(i∗OY ) = Supp((^A=a)) = V (a). Moreover, have an isomorphism ∼ −1 ∼ −1 of sheaves of rings on Y , i (A=ag ) ! i i∗OY ! OY :It is an exercise to use lemma this to dene a morphism Y ! Spec A=a such that i Y / X = Spec A O $ Spec A=a commutes Proposition. Let X be any scheme.
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