Relative Affine Schemes

Size: px
Start display at page:

Transcription

1 Relative Affine Schemes Daniel Murfet October 5, 2006 The fundamental Spec( ) construction associates an affine scheme to any ring. In this note we study the relative version of this construction, which associates a scheme to any sheaf of algebras. The contents of this note are roughly EGA II 1.2, 1.3. Contents 1 Affine Morphisms 1 2 The Spec Construction 4 3 The Sheaf Associated to a Module 8 1 Affine Morphisms Definition 1. Let f : X Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y, if there is a nonempty open cover {V α } α Λ of Y by open affine subsets V α such that for every α, f 1 V α is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the affine property is stable under composition with isomorphisms on either end. Example 1. Any closed immersion X Y is an affine morphism by our solution to (Ex 4.3). Remark 1. A scheme X affine over S is not necessarily affine (for example X = S) and if an affine scheme X is an S-scheme, it is not necessarily affine over S. However, if S is separated then an S-scheme X which is affine is affine over S. Lemma 1. An affine morphism is quasi-compact and separated. Any finite morphism is affine. Proof. Let f : X Y be affine. Then f is separated since any morphism of affine schemes is separated, and the separatedness condition is local. Since any affine scheme is quasi-compact it is clear that f is quasi-compact. It follows from (Ex. 3.4) that a finite morphism is an affine morphism of a very special type. The next two result show that being affine is a property local on the base: Lemma 2. If f : X Y is affine and V Y is open then f 1 V V is also affine. Proof. Let {Y α } α Λ be a nonempty affine open cover of Y such that f 1 Y α is affine for every α Λ. For every point y V find α such that y V Y α. If Y α = SpecBα then there is g B α with y D(g) V Y α. If f 1 Y α = SpecAα and ϕ : B α A α corresponds to f 1 Y α Y α then D(ϕ(g)) = f 1 D(g) is an affine open subset of f 1 V. Lemma 3. If f : X Y is a morphism of schemes and {Y α } α Λ is a nonempty open cover of Y such that f 1 Y α Y α is affine for every α, then f is affine. Proof. We can take a cover of Y consisting of affine open sets contained in some Y α whose inverse image is affine. This shows that f is affine. 1

2 Proposition 4. A morphism f : X Y is affine if and only if for every open affine subset U Y the open subset f 1 U is also affine. Proof. The condition is clearly sufficient, so suppose that f is affine. Using Lemma 2 we can reduce to the case where Y = SpecA is affine and we only have to show that X is affine. Cover Y with open affines Y α = SpecBα with f 1 Y α affine. If x Y α then there is g A with x D(g) Y α, and since D(g) corresponds to D(h) for some h B α the open set f 1 D(g) is affine. Therefore we can assume Y α = D(g α ) and further that there is only a finite number D(g 1 ),..., D(g n ) with the g i generating the unit ideal of A. Let t i be the image of g i under the canonical ring morphism A O X (X). Then the t i generate the unit ideal of O X (X) and the open sets X ti = f 1 D(g i ) cover X, so by Ex the scheme X is affine. Proposition 5. Affine morphisms are stable under pullback. That is, suppose we have a pullback diagram g X Y If f is an affine morphism then so is g. p G f H Proof. By Lemma 3 it suffices to find an open neighborhood V of each point y Y such that g 1 V V is affine. Given y Y let S be an affine open neighborhood of q(y) and set U = f 1 S, V = q 1 S. By assumption U is affine and g 1 V = p 1 U so we have a pullback diagram g 1 V g V V q U S If W V is affine then g 1 V W = U S W is a pullback of affine schemes and therefore affine. Therefore g V is an affine morphism, as required. Lemma 6. Let X be a scheme and L an invertible sheaf with global section f Γ(X, L ). Then the inclusion X f X is affine. Proof. The open set X f is defined in (MOS,Lemma 29), and it follows from Remark (MOS,Remark 2) that we can find an affine open cover of X whose intersections with X f are all affine. This shows that the inclusion X f X is an affine morphism of schemes. If X is a scheme over S with structural morphism f : X S then we denote by A (X) the canonical O S -algebra f O X, when there is no chance of confusion. If X is an S-scheme affine over S then it follows from Lemma 1 and (5.18) that A (X) is a quasi-coherent sheaf of O S -algebras. If X, Y are two schemes over S with structural morphisms f, g then a morphism h : X Y of schemes over S consists of a continuous map ψ : X Y and a morphism of sheaves of rings θ : O Y h O X. The composite g O Y g h O X = f O X is a morphism of O S -algebras A (h) : A (Y ) A (X). This defines a contravariant functor A ( ) : Sch/S Alg(S) from the category of schemes over S to the category of sheaves of commutative O S -algebras. 2

3 X h = (ψ, θ) Y g f S O S g OYg θ f O X We showed in (Ex 2.4) that for a ring B, a B-algebra A and B-scheme X there is a bijection Hom B (X, SpecA) = Hom B (A, O X (X)) between morphisms of schemes over SpecB and morphisms of B-algebras. We now present a relative version of this result: Proposition 7. Let X, Y be S-schemes with Y affine over S. Then the map h A (h) gives a bijection Hom S (X, Y ) = Hom Alg(S) (A (Y ), A (X)). Proof. We begin with the case where Y = SpecA and S = SpecB are affine for a B-algebra A. We know from (Ex 2.4) that there is a bijection Hom S (X, Y ) = Hom B (A, O X (X)). Let f : Y S be the structural morphism, so that A (Y ) = f O Y. Then by (5.2) we have A (Y ) = Ã as sheaves of algebras (see our Modules over a Ringed Space notes for the properties of the functor : BAlg Alg(S)). Therefore the adjunction Γ (these are functors between BAlg and Alg(S)) gives us a bijection Hom Alg(S) (A (Y ), A (X)) = Hom Alg(S) (Ã, A (X)) = Hom B (A, O X (X)) It is not difficult to check that this has the desired form, so we have the desired bijection in this special case. One extends easily to the case where Y = SpecA and S = SpecB. In the general case, cover S with open affines S α with g 1 S α an affine open subset of Y (by Proposition 4 we may as well assume {S α } α Λ is the set of all affine subsets of S). Let f α : f 1 S α S α, g α : g 1 S α S α be induced by f, g respectively. Then by the special case already treated, the following maps are bijections Hom Sα (f 1 S α, g 1 S α ) Hom Alg(Sα)(A (Y ) Sα, A (X) Sα ) (1) Suppose that h, h : X Y are morphisms of schemes over S with A (h) = A (h ). Then A (h) Sα = A (h ) Sα and therefore h α = h α. Since the g 1 S α cover Y it follows that h = h. This shows that the map Hom S (X, Y ) Hom Alg(S) (A (Y ), A (X)) is injective. To see that it is surjective, let φ : A (Y ) A (X) be given, and let h α : f 1 S α g 1 S α correspond to φ Sα via (1). Since every affine open subset of S occurs among the S α, we can use injectivity of (1) to see that the h α can be glued to give h : X Y with A (h) = φ, which completes the proof. Let Sch/ A S denote the full subcategory of Sch/S consisting of all schemes affine over S. Then there is an induced functor A ( ) : Sch/ A S QcoAlg(S) into the full subcategory of all quasi-coherent sheaves of commutative O S -algebras. 3

4 Corollary 8. The functor A ( ) : Sch/ A S QcoAlg(S) is fully faithful. In particular, if X, Y are S-schemes affine over S then an S-morphism h : X Y is an isomorphism if and only if A (h) : A (Y ) A (X) is an isomorphism. 2 The Spec Construction In this section X is a scheme and B is a quasi-coherent sheaf of commutative O X -algebras. Then for an affine open subset U X the scheme SpecB(U) is canonically a scheme over U, via the morphism π U : SpecB(U) SpecO X (U) = U. It is clear that π U is affine, and given another affine open set V we denote π 1 U (U V ) by X U,V. If U V are affine open subsets, then let ρ U,V : SpecB(U) SpecB(V ) be induced by the restriction B(V ) B(U). Our aim in this section is to glue the schemes SpecB(U) to define a canonical scheme affine over X associated to B. We begin with an important technical Lemma (which will also be used in our construction of Proj). Lemma 9. Let X be a scheme and U V affine open subsets. If F is a quasi-coherent sheaf of modules on X then the following morphism of O X (U)-modules is an isomorphism F (V ) OX (V ) O X (U) F (U) a b b a U Proof. We reduce immediately to the case where X = V, and then to the case where X = SpecA is actually an affine scheme. So there is a ring B and an open immersion i : Y = SpecB X whose open image in X is U. Factor U X as an isomorphism j : U = Y followed by i. Let F be a quasi-coherent sheaf of modules on X and induce a morphism of O X (U)-modules F (X) OX (X) O X (U) F (U) with a b b a U. We have to show this is an isomorphism. Using various isomorphisms defined in our Section 2.5 notes we have an isomorphism of abelian groups F (X) OX (X) O X (U) = F (X) A B = (F (X) A B) (Y ) = i F (X)(Y ) = i F (Y ) = (j F U )(Y ) = F (U) Using the explicit calculations of of these isomorphisms, one can check that it sends a b to b a U, which proves our claim. Proposition 10. Let X be a scheme and U V affine open subsets. If B is a quasi-coherent sheaf of commutative O X -algebras then the following diagram is a pullback SpecB(U) ρ U,V SpecB(V ) In particular ρ U,V π U U V is an open immersion inducing an isomorphism of SpecB(U) with π 1 V U. Proof. It follows from Lemma 9 that the following diagram is a pushout of rings π V O X (V ) O X (U) B(V ) B(U) That is, the morphism B(V ) OX (V ) O X (U) B(U) given by a b b a U is an isomorphism of O X (V )-algebras (and O X (U)-algebras). Applying Spec gives the desired pullback. Since open 4

5 immersions are stable under pullback, we see that ρ U,V is an open immersion. Since π 1 V U is another candidate for the pullback U V SpecB(V ) it follows that the open image of ρ U,V is π 1 V U. For open affine U X we denote by B(U) the sheaf of algebras on U obtained by taking the direct image along SpecO X (U) = U of the sheaf of algebras on SpecO X (U) corresponding to the O X (U)-algebra B(U). There is an isomorphism of sheaves of algebras on U δ U : A (SpecB(U)) = B(U) defined as the direct image of the isomorphism of sheaves on SpecO X (U) obtained from the algebra analogue of (5.2d) (see our Modules over a Ringed Space notes). The sheaves of algebras are compatible with restriction in the following sense Lemma 11. Let X be a scheme, W U affine open subsets and B a quasi-coherent sheaf of commutative O X -algebras. There is a canonical isomorphism of sheaves of algebras on W ε W,U : B(U) W B(W ) a/s a W /s W Proof. We have the following commutative diagram SpecO X (W ) k W j i SpecO X (U) U l Since j is an open immersion it induces an isomorphism SpecO X (W ) = Imj. Let g : Imj SpecO X (W ) be the inverse of this isomorphism, and let s : Imj W be the morphism induced by l. Let ε W,U be the following isomorphism of sheaves of algebras (using some results from our Modules over Ringed space notes) ε W,U : B(U) W = s ( B(U) Imj ) = k g ( B(U) Imj ) = k j ( B(U)) = k (B(U) OX (U) O X (W )) = k B(W ) = B(W ) Suppose we are given an open set T W, a B(U) and s O X (U) with l 1 T D(s). Then using the explicit forms of the above isomorphisms, one checks that the section a/s B(U)(T ) maps to the section a W /s W of B(W )(T ). Since B is quasi-coherent, there is an isomorphism µ U : B(U) = B U of sheaves of algebras on U. Together with δ U this gives an isomorphism of sheaves of algebras on U β U = µ U δ U : A (SpecB(U)) = B(U) = B U In particular for affine W U and b B(U) we have the following action of β U : (β U ) W : O SpecB(U) (X U,W ) B(W ) b/1 b W (2) The schemes X U,V and X V,U are both affine over U V by Lemma 2. And on U V we have an isomorphism of sheaves of algebras A (X U,V ) = A (SpecB(U)) U V = B U V = A (SpecB(V )) U V = A (X V,U ) 5

6 Therefore by Corollary 8 there is an isomorphism θ U,V : X U,V X V,U of schemes over U V with A (θ U,V ) = (β U ) 1 U V (β V ) U V. If U V then X U,V = SpecB(U) and we claim that the following diagram commutes θ U,V SpecB(U) X V,U ρ U,V SpecB(V ) The two legs agree on global sections of SpecB(V ) by (2), and they are therefore equal. It is clear that θ U,V = θ 1 V,U and for affine opens U, V, W X we have θ U,V (X U,V X U,W ) = X V,U X V,W since θ U,V is a morphism of schemes over U V. So to glue the SpecB(U) it only remains to check that θ U,W = θ V,W θ U,V on X U,V X U,W. But these are all morphisms of schemes affine over U V W, so by using the injectivity of A ( ) the verification is straightforward. Thus our family of schemes and patches satisfies the conditions of the Glueing Lemma (Ex. 2.12), and we have a scheme Spec(B) together with open immersions ψ U : SpecB(U) Spec(B) for each affine open subset U X. These morphisms have the following properties: (a) The open sets Imψ U cover Spec(B). (b) For affine open subsets U, V X we have ψ U (X U,V ) = Imψ U Imψ V and ψ V XV,U θ U,V = ψ U XU,V. In particular for affine open subsets U V we have a commutative diagram SpecB(V ) ψ V Spec(B) ρ U,V ψ U SpecB(U) (3) The open sets Imψ U are a nonempty open cover of Spec(B), and it is a consequence of (b) above and the fact that θ U,V is a morphism of schemes over U V that the morphisms Imψ U = SpecB(U) U X can be glued (that is, for open affines U, V the corresponding morphisms agree on Imψ U Imψ V ). Therefore there is a unique morphism of schemes π : Spec(B) X with the property that for every affine open subset U X the following diagram commutes SpecB(U) ψ U Spec(B) (4) π U π U X In fact it is easy to see that π 1 U = Imψ U, the above diagram is also a pullback. Moreover π is affine by Lemma 3 since all the morphisms π U are affine. Our next task is to show that A (Spec(B)) = B. For open affine U X, let j U : Imψ U U be the morphism induced by π and ψ U : SpecB(U) = Imψ U the isomorphism induced by ψ U, so that j U ψ U = π U. Then there is an isomorphism of sheaves of algebras on U ω U : A (Spec(B)) U = (j U ) (O Spec(B) ImψU ) = (j U ) ((ψ U ) O SpecB(U) ) = (π U ) O SpecB(U) = A (SpecB(U)) 6

7 For open affines W U let π U XU,W : X U,W W be induced from π U. Then there is an isomorphism of sheaves of algebras on W ζ W,U : A (SpecB(U)) W = (π U XU,W ) (O SpecB(U) XU,W ) = (π U XU,W ) ((θ W,U ) O SpecB(W ) ) = A (SpecB(W )) Despite the complicated notation, if one draws a picture it is straightforward to check that the following diagram commutes ω U W A (Spec(B)) W A (SpecB(U)) W ω W ζ W,U A (SpecB(W )) We claim that the isomorphism ζ W,U is compatible with the isomorphism ε W,U defined earlier. Lemma 12. Let X be a scheme, W U affine open subsets and B a quasi-coherent sheaf of commutative O X -algebras. Then the following diagram of sheaves of algebras on W commutes (5) A (SpecB(U)) W ζ W,U A (SpecB(W )) δ U W B(U) W µ U W ε W,U δ W B(W ) µ W B W Proof. First we check that the square on the left commutes by beginning at B(U) W and showing at the two morphisms to A (SpecB(W )) agree. For this, we need only check they agree on sections of the form a/s, and this is straightforward. We can use the same trick to check commutativity of the triangle on the right. For an affine open subset U X let φ U : A (Spec(B)) U B U be the isomorphism of sheaves of algebras on U given by the composite φ U = β U ω U. Lemma 12 and (5) show that φ U W = φ W for open affines W U, so together the φ U give an isomorphism of sheaves of algebras φ : A (Spec(B)) B with φ U = φ U. In summary: Definition 2. Let X be a scheme and B a commutative quasi-coherent sheaf of O X -algebras. Then we can canonically associate to B a scheme π : Spec(B) X affine over X. For every open affine subset U X there is an open immersion ψ U : SpecB(U) Spec(B) with the property that the diagram (4) is a pullback and the diagram (3) commutes for any open affines U V. There is also a canonical isomorphism of sheaves of algebras A (Spec(B)) = B. Spec(B) SpecB(U) ψ U = 7

8 Corollary 13. Let S be a scheme. The functor A ( ) : Sch/ A S QcoAlg(S) is an equivalence. In particular schemes X, Y affine over S are S-isomorphic if and only if A (X) = A (Y ). Proof. We know from Corollary 8 that the functor is fully faithful, and the above construction together with the fact that A (Spec(B)) = B shows that it is representative. Therefore it is an equivalence. If A is a commutative ring, then the composite Γ( )A ( ) gives an equivalence Sch/ A SpecA AAlg. Any quasi-coherent sheaf of commutative algebras on SpecA is isomorphic to B for some commutative A-algebra B. The morphism SpecB SpecA is affine and A (SpecB) = B, so it follows that Spec( B) = SpecB as schemes over A. In particular any scheme X affine over SpecA is A-isomorphic to SpecB for some commutative A-algebra B. Therefore Lemma 14. Let S be an affine scheme. Then an S-scheme X is affine over S if and only if X is an affine scheme. 3 The Sheaf Associated to a Module Let X be a scheme and B a commutative quasi-coherent sheaf of O X -algebras. Let Mod(B) denote the category of all sheaves of B-modules and Qco(B) the full subcategory of quasi-coherent sheaves of B-modules (SOA,Definition 1). Note that these are precisely the sheaves of B-modules that are quasi-coherent as sheaves of O X -modules, so in this section there is no harm in simply calling these sheaves quasi-coherent (SOA,Proposition 19). In this section we define for every finite morphism f : X Y a functor : Qco(A (X)) Mod(X) which is the relative version of the functor AMod Mod(SpecA) for a ring A. Qco(A (X)) is an abelian category (SOA,Corollary 20). Note that Lemma 15. Let X be a scheme and U V affine open subsets. If B is a commutative quasicoherent sheaf of O X -algebras, and M a quasi-coherent sheaf of B-modules, then the following morphism of B(U)-modules is an isomorphism M (V ) B(V ) B(U) M (U) (6) a b b a U (7) Proof. Such a morphism of B(U)-modules certainly exists. We know from Lemma 9 that there are isomorphisms of O X (U)-modules B(V ) OX (V ) O X (U) = B(U) M (V ) OX (V ) O X (U) = M (U) So at least we have an isomorphism of abelian groups M (V ) B(V ) B(U) = M (V ) B(V ) (B(V ) OX (V ) O X (U)) = (M (V ) B(V ) B(V )) OX (V ) O X (U) = M (V ) OX (V ) O X (U) = M (U) It is easily checked that this map agrees with (6), which is therefore an isomorphism. Throughout the remainder of this section, f : X Y is a finite morphism and A (X) = f O X the corresponding commutative quasi-coherent sheaf of O Y -algebras. 8

9 Proposition 16. Let M a quasi-coherent sheaf of A (X)-modules. There is a canonical quasicoherent sheaf of modules M on X with the property that for every affine open subset V Y there is an isomorphism µ V : M f 1 V (ψ V ) M (V ) where ψ V : SpecO X (f 1 V ) f 1 V is the canonical isomorphism. Proof. Let U be the set of all open affine subsets of Y, so that U = {f 1 V } V U is an indexed affine open cover of X. For each V U we have the Γ(V, A (X)) = O X (f 1 V )-module M (V ) and therefore a sheaf of modules M (V ) on X V = SpecO X (f 1 V ). Taking the direct image along ψ V we have a sheaf of modules M V = (ψ V ) M (V ) on f 1 V. We want to glue the sheaves M V. Let W V be affine open susbets of Y and ρ W,V : X W X V the canonical open immersion. Using Lemma 15 we have an isomorphism of sheaves of modules on X W α W,V : ρ W,V (M (V ) ) = (M (V ) OX (f 1 V ) O X (f 1 W )) = M (W ) [V, m/s] b/t (b m W ) /ts W Let X V,W be the affine open subset (ψ V ) 1 (f 1 W ) of X V, denote by ρ W,V : X W X V,W the isomorphism induced by ρ W,V, and let ψ V,W : X V,W f 1 W be the isomorphism induced by ψ V. Notice that ψ V,W = ψ W (ρ W,V ) 1. We have an isomorphism of sheaves of modules on f 1 W M W = (ψ W ) (M (W ) ) = (ψ W ) ρ W,V (M (V ) ) = (ψ W ) (ρ W,V ) 1 (M (V ) XV,W ) = (ψ W (ρ W,V ) 1 ) (M (V ) XV,W ) = (ψ V,W ) (M (V ) XV,W ) = (ψ V ) (M (V ) ) f 1 W = M V f 1 W using (MRS,Proposition 107) and (MRS,Proposition 111). W V we have an isomorphism So for every affine open inclusion ϕ V,W : M V f 1 W M W m/s m W /s W Clearly ϕ U,U = 1 and if Q W V are open affine subsets then ϕ V,Q = ϕ W,Q ϕ V,W f 1 Q. This means that for open affine U, V Y the isomorphisms ϕ 1 U,W ϕ V,W for open affine W U V glue together to give an isomorphism of sheaves of modules ϕ V,U : M V f 1 U f 1 V M U f 1 U f 1 V ϕ U,W ϕ V,U f 1 W = ϕ V,W for affine open W U V The notation is unambiguous, since this definition agrees with the earlier one if U V. By construction these isomorphisms can be glued (GS,Proposition 1) to give a canonical sheaf of modules M on X and a canonical isomorphism of sheaves of modules µv : M f 1 V M V for every open affine V Y. These isomorphisms are compatible in the following sense: we have µ V = ϕ U,V µ U on f 1 U f 1 V for any open affine U, V Y. It is clear that M is quasi-coherent since the modules M (U) are. Proposition 17. If β : M N is a morphism of quasi-coherent sheaves of A (X)-modules then there is a canonical morphism β : M N of sheaves of modules on X and this defines an additive functor : Qco(A (X)) Mod(X). Proof. For every affine open V Y, β V : M (V ) N (V ) is a morphism of Γ(V, A (X)) = O X (f 1 V )-modules, and therefore gives a morphism (β V ) : M (V ) N (V ) of sheaves of 9

10 modules on X V. Let b V : M V N V be the morphism (ψ V ) (β V ). One checks easily that for open affine U, V Y and T = f 1 (U V ) the following diagram commutes b V T M V T N V T M U T bu T N U T Therefore there is a unique morphism of sheaves of modules β with the property that for every affine open V X the following diagram commutes (GS,Proposition 6) M f 1 V eβ f 1 V N f 1 V µ V M V N V b V Using this unique property it is easy to check that defines an additive functor. Proposition 18. The additive functor : Qco(A (X)) Mod(X) is exact. Proof. Since Qco(A (X)) is an abelian subcategory of Mod(A (X)) (SOA,Corollary 20), a sequence is exact in the former category if and only if it is exact in the latter, which is if and only if it is exact as a sequence of sheaves of abelian groups. So suppose we have an exact sequence of quasi-coherent A (X)-modules M ϕ M ψ M This is exact in Mod(X), so using (MOS,Lemma 5) we have for every open affine V Y an exact sequence of O X (f 1 V )-modules µ V M (V ) ϕ V M (V ) ψ V M (V ) Since the functor : O X (f 1 V )Mod Mod(X V ) is exact, we have an exact sequence of sheaves of modules on X V M (V ) fϕ V M (V ) fψ V M (V ) Applying (ψ V ) and using the natural isomorphism (ψ V ) M (V ) = M f 1 V following sequence of sheaves of modules on f 1 V is exact we see that the M f 1 V eϕ f 1 U M f 1 V eψ f 1 V M f 1 V It now follows from (MRS,Lemma 38) that the functor is exact. Definition 3. Let f : X Y be a finite morphism of noetherian schemes and F a quasicoherent sheaf of modules on Y. Then A (X) coherent (H, II Ex.5.5) and therefore the sheaf of O Y -modules H om OY (A (X), F ) is quasi-coherent (MOS,Corollary 44). This sheaf becomes a quasi-coherent sheaf of A (X)-modules with the action (a φ) W (t) = φ W (a f 1 W t). Therefore we have a quasi-coherent sheaf of modules on X f!(f ) = H om OY (A (X), F ) This defines an additive functor f!( ) : Qco(Y ) Qco(X). Remark 2. For any closed immersion f : X Y of schemes there is a right adjoint f! : Mod(Y ) Mod(X) to the direct image functor f (MRS,Proposition 97). In the special case where X, Y are noetherian we have just defined another functor f! : Qco(Y ) Qco(X) and as the notation suggests, these two functors are naturally equivalent. 10

The Relative Proj Construction

The Relative Proj Construction Daniel Murfet October 5, 2006 Earlier we defined the Proj of a graded ring. In these notes we introduce a relative version of this construction, which is the Proj of a sheaf

Modules over a Scheme

Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasi-coherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These

Concentrated Schemes

Concentrated Schemes Daniel Murfet July 9, 2006 The original reference for quasi-compact and quasi-separated morphisms is EGA IV.1. Definition 1. A morphism of schemes f : X Y is quasi-compact if there

Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

Synopsis of material from EGA Chapter II, 3

Synopsis of material from EGA Chapter II, 3 3. Homogeneous spectrum of a sheaf of graded algebras 3.1. Homogeneous spectrum of a graded quasi-coherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf

Section Blowing Up

Section 2.7.1 - Blowing Up Daniel Murfet October 5, 2006 Now we come to the generalised notion of blowing up. In (I, 4) we defined the blowing up of a variety with respect to a point. Now we will define

Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

Modules over a Ringed Space

Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted

Schemes via Noncommutative Localisation

Schemes via Noncommutative Localisation Daniel Murfet September 18, 2005 In this note we give an exposition of the well-known results of Gabriel, which show how to define affine schemes in terms of the

Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

Direct Limits. Mathematics 683, Fall 2013

Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry

Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

FORMAL GLUEING OF MODULE CATEGORIES

FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on

Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

The Proj Construction

The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of

Scheme theoretic vector bundles

Scheme theoretic vector bundles The best reference for this material is the first chapter of [Gro61]. What can be found below is a less complete treatment of the same material. 1. Introduction Let s start

Section Projective Morphisms

Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism

Derived Categories Of Quasi-coherent Sheaves

Derived Categories Of Quasi-coherent Sheaves Daniel Murfet October 5, 2006 In this note we give a careful exposition of the basic properties of derived categories of quasicoherent sheaves on a scheme.

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

The Segre Embedding. Daniel Murfet May 16, 2006

The Segre Embedding Daniel Murfet May 16, 2006 Throughout this note all rings are commutative, and A is a fixed ring. If S, T are graded A-algebras then the tensor product S A T becomes a graded A-algebra

which is a group homomorphism, such that if W V U, then

4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

Sel. math., New ser. Online First c 2006 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s00029-006-0022-4 Selecta Mathematica New Series Dualizing complexes and perverse sheaves on noncommutative ringed

AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

SOME OPERATIONS ON SHEAVES

SOME OPERATIONS ON SHEAVES R. VIRK Contents 1. Pushforward 1 2. Pullback 3 3. The adjunction (f 1, f ) 4 4. Support of a sheaf 5 5. Extension by zero 5 6. The adjunction (j!, j ) 6 7. Sections with support

Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

What is an ind-coherent sheaf?

What is an ind-coherent sheaf? Harrison Chen March 8, 2018 0.1 Introduction All algebras in this note will be considered over a field k of characteristic zero (an assumption made in [Ga:IC]), so that we

DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

Derived Algebraic Geometry IX: Closed Immersions

Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of T-Structures 7 3 The Proof of Proposition 1.0.10 14 4 Closed

0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

Matsumura: Commutative Algebra Part 2

Matsumura: Commutative Algebra Part 2 Daniel Murfet October 5, 2006 This note closely follows Matsumura s book [Mat80] on commutative algebra. Proofs are the ones given there, sometimes with slightly more

Cohomology and Base Change

Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

PART II.1. IND-COHERENT SHEAVES ON SCHEMES

PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image

A course in. Algebraic Geometry. Taught by C. Birkar. Michaelmas 2012

A course in Algebraic Geometry Taught by C. Birkar Michaelmas 2012 Last updated: May 30, 2013 1 Disclaimer These are my notes from Caucher Birkar s Part III course on algebraic geometry, given at Cambridge

Derived Categories Of Sheaves

Derived Categories Of Sheaves Daniel Murfet October 5, 2006 We give a standard exposition of the elementary properties of derived categories of sheaves on a ringed space. This includes the derived direct

DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites

DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

Descent on the étale site Wouter Zomervrucht, October 14, 2014

Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 47 AND 48 RAVI VAKIL CONTENTS 1. The local criterion for flatness 1 2. Base-point-free, ample, very ample 2 3. Every ample on a proper has a tensor power that

ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF

ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

SMA. Grothendieck topologies and schemes

SMA Grothendieck topologies and schemes Rafael GUGLIELMETTI Semester project Supervised by Prof. Eva BAYER FLUCKIGER Assistant: Valéry MAHÉ April 27, 2012 2 CONTENTS 3 Contents 1 Prerequisites 5 1.1 Fibred

Basic Modern Algebraic Geometry

Version of December 13, 1999. Final version for M 321. Audun Holme Basic Modern Algebraic Geometry Introduction to Grothendieck s Theory of Schemes Basic Modern Algebraic Geometry Introduction to Grothendieck

Section Divisors

Section 2.6 - Divisors Daniel Murfet October 5, 2006 Contents 1 Weil Divisors 1 2 Divisors on Curves 9 3 Cartier Divisors 13 4 Invertible Sheaves 17 5 Examples 23 1 Weil Divisors Definition 1. We say a

THE SMOOTH BASE CHANGE THEOREM

THE SMOOTH BASE CHANGE THEOREM AARON LANDESMAN CONTENTS 1. Introduction 2 1.1. Statement of the smooth base change theorem 2 1.2. Topological smooth base change 4 1.3. A useful case of smooth base change

D-manifolds and derived differential geometry

D-manifolds and derived differential geometry Dominic Joyce, Oxford University September 2014 Based on survey paper: arxiv:1206.4207, 44 pages and preliminary version of book which may be downloaded from

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

Factorization of birational maps for qe schemes in characteristic 0

Factorization of birational maps for qe schemes in characteristic 0 AMS special session on Algebraic Geometry joint work with M. Temkin (Hebrew University) Dan Abramovich Brown University October 24, 2014

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

Sheaves of Modules. Chi-yu Cheng. This is a supplement for the study of Hartshorne s Algebraic Geometry

Sheaves of Modules Chi-yu Cheng This is a supplement for the study of Hartshorne s Algebraic Geometry Currently a graduate student at the University of Washington 1 1 Preliminary Observations Observation

SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY

SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes

PART II.2. THE!-PULLBACK AND BASE CHANGE

PART II.2. THE!-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.

Lectures on algebraic stacks

Rend. Mat. Appl. (7). Volume 38, (2017), 1 169 RENDICONTI DI MATEMATICA E DELLE SUE APPLICAZIONI Lectures on algebraic stacks Alberto Canonaco Abstract. These lectures give a detailed and almost self-contained

1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

SCHEMES. David Harari. Tsinghua, February-March 2005

SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

Deformation theory of representable morphisms of algebraic stacks

Deformation theory of representable morphisms of algebraic stacks Martin C. Olsson School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, molsson@math.ias.edu Received:

Introduction to Chiral Algebras

Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS

GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS KARL SCHWEDE Abstract. We first construct and give basic properties of the fibered coproduct in the category of ringed spaces. We then look at some special

COHOMOLOGY AND DIFFERENTIAL SCHEMES. 1. Schemes

COHOMOLOG AND DIFFERENTIAL SCHEMES RAMOND HOOBLER Dedicated to the memory of Jerrold Kovacic Abstract. Replace this text with your own abstract. 1. Schemes This section assembles basic results on schemes

Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

Basic results on Grothendieck Duality

Basic results on Grothendieck Duality Joseph Lipman 1 Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 2007 1 Supported in part by NSA Grant

IND-COHERENT SHEAVES AND SERRE DUALITY II. 1. Introduction

IND-COHERENT SHEAVES AND SERRE DUALITY II 1. Introduction Let X be a smooth projective variety over a field k of dimension n. Let V be a vector bundle on X. In this case, we have an isomorphism H i (X,

D-MODULES: AN INTRODUCTION

D-MODULES: AN INTRODUCTION ANNA ROMANOVA 1. overview D-modules are a useful tool in both representation theory and algebraic geometry. In this talk, I will motivate the study of D-modules by describing

214A HOMEWORK KIM, SUNGJIN

214A HOMEWORK KIM, SUNGJIN 1.1 Let A = k[[t ]] be the ring of formal power series with coefficients in a field k. Determine SpecA. Proof. We begin with a claim that A = { a i T i A : a i k, and a 0 k }.

THE GROTHENDIECK GROUP OF A QUANTUM PROJECTIVE SPACE BUNDLE

THE GROTHENDIECK GROUP OF A QUANTUM PROJECTIVE SPACE BUNDLE IZURU MORI AND S. PAUL SMITH Abstract. We compute the Grothendieck group of non-commutative analogues of projective space bundles. Our results

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar

RAPHAËL ROUQUIER. k( )

GLUING p-permutation MODULES 1. Introduction We give a local construction of the stable category of p-permutation modules : a p- permutation kg-module gives rise, via the Brauer functor, to a family of

Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

Lectures on Galois Theory. Some steps of generalizations

= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame

14 Lecture 14: Basic generallities on adic spaces

14 Lecture 14: Basic generallities on adic spaces 14.1 Introduction The aim of this lecture and the next two is to address general adic spaces and their connection to rigid geometry. 14.2 Two open questions

LECTURE NOTES IN EQUIVARIANT ALGEBRAIC GEOMETRY. Spec k = (G G) G G (G G) G G G G i 1 G e

LECTURE NOTES IN EQUIVARIANT ALEBRAIC EOMETRY 8/4/5 Let k be field, not necessaril algebraicall closed. Definition: An algebraic group is a k-scheme together with morphisms (µ, i, e), k µ, i, Spec k, which

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)

Representable presheaves

Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom

1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

arxiv:math/ v3 [math.ag] 10 Nov 2006

Construction of G-Hilbert schemes Mark Blume arxiv:math/0607577v3 [math.ag] 10 Nov 2006 Abstract One objective of this paper is to provide a reference for certain fundamental constructions in the theory

Zariski s Main Theorem and some applications

Zariski s Main Theorem and some applications Akhil Mathew January 18, 2011 Abstract We give an exposition of the various forms of Zariski s Main Theorem, following EGA. Most of the basic machinery (e.g.

PERVERSE SHEAVES. Contents

PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a