The Proj Construction


 Marcia Copeland
 3 years ago
 Views:
Transcription
1 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 Projective Space 12 1 Basic Properties Definition 1. Let be a ring. graded algebra is an algebra R which is also a graded ring in such a way that if r R d then ar R d for all a. That is, R d is an submodule of R for all d 0. Equivalently a graded algebra is a morphism of graded rings R where we grade by setting 0 =, n = 0 for n > 0. morphism of graded algebras is a morphism of algebras which preserves grade. Equivalently, this is a morphism of graded rings which is also a morphism of modules, or a morphism of graded rings R S making the following diagram commute R S Let S be a graded algebra, and let X = P rojs. Then there is a canonical morphism of rings θ : Γ(X, P rojs) defined by θ(a)(p) = a/1 S (p) Where as usual, a denotes the element a 1 of S. By Ex 2.4 this induces a morphism of schemes f : P rojs Spec. If γ : S gives the algebra structure on S, then an element a is mapped to the maximal ideal of O P rojs,p via Γ(X, P rojs) O P rojs,p if and only if a γ 1 p. We denote this prime ideal by p. Let κ p : f(p) S (p) be defined by κ p (a/s) = a/s. Then f : P rojs Spec f(p) = p f # U (t)(p) = κ p(t(f(p))) 1
2 So the projective space over any graded algebra is a scheme over. In particular there is a canonical morphism of schemes P n Spec. Notice that if e S + is homogenous then the composite of the open immersion Spec(S (e) ) = D + (e) P rojs with f is the morphism of schemes corresponding to the canonical map S (e) defined by a a/1. Proposition 1. If S is a graded algebra then the morphism f : P rojs Spec is separated. In particular P rojs is separated for any graded ring S. Proof. The open sets D + (a) for homogenous a S + are an open cover of P rojs. Therefore the open sets D + (a) D + (b) are an open cover of P rojs P rojs, and the inverse image of this open set under the diagonal P rojs P rojs P rojs is clearly D + (a) D + (b) = D + (ab). Since the property of being a closed immersion is local on the base, we reduce to showing that the morphism D + (ab) D + (a) D + (b) is a closed immersion. Since we know that this gives the canonical morphisms D + (ab) D + (a), D + (ab) D + (b) when composed with the projections, the corresponding morphism of algebras is the morphism α : S (a) S (b) S (ab) induced by the canonical morphisms S (a) S (ab), S (b) S (ab). Since α is clearly surjective, the proof is complete. Proposition 2. If S is a noetherian graded ring then P rojs is a noetherian scheme. Proof. For any homogenous f S + the ring S (f) is noetherian (GRM,Corollary 15). Since S is noetherian the ideal S + admits a finite set of homogenous generators f 1,..., f p and the schemes D + (f i ) = SpecS (fi) cover P rojs. Therefore P rojs is noetherian. Proposition 3. If S is a finitely generated graded algebra, then the scheme P rojs is of finite type over Spec. Proof. The hypothesis imply that S 0 is a finitely generated algebra, and that S is a finitely generated S 0 algebra. Therefore S + is a finitely generated ideal (GRM,Corollary 8). So using the argument of Proposition 2 we reduce to showing that for homogenous f S +, S (f) is a finitely generated algebra. By (GRM,Proposition 14) it suffices to show that S (d) is a finitely generated algebra, which follows from (GRM,Lemma 11). Proposition 4. If S is a graded domain with S + 0 then P rojs is an integral scheme. Proof. By (3.1) it is enough to show that P rojs is reduced and irreducible. By assumption the zero ideal 0 is a homogenous prime ideal, so P rojs has a generic point and is therefore irreducible. The scheme P rojs is covered by open subsets isomorphic to schemes Spec(S (f) ) for nonzero homogenous f S +, and it is clear that S (f) is an integral domain. This shows that P rojs is reduced and therefore integral. Corollary 5. Let be a ring and consider the scheme P n for n 1. Then (i) The morphism P n Spec is separated and of finite type. (ii) If is noetherian then so is P n. (iii) If is an integral domain, then P n is an integral scheme. 2 Functorial Properties Let ϕ : S T be a morphism of graded rings, G(ϕ) the open subset {p P rojt p ϕ(s + )} and induce the following morphism of schemes using Ex 2.14: Φ : G(ϕ) P rojs Φ(p) = ϕ 1 p Φ # V (s)(p) = ϕ (p)(s(ϕ 1 p)) 2
3 Here ϕ (p) : S (ϕ 1 p) T (p) is defined by a/s ϕ(a)/ϕ(s). If ϕ is an isomorphism of graded rings with inverse ψ then clearly G(ϕ) = P rojt, G(ψ) = P rojs and one checks that if Ψ : P rojs P rojt is induced by ψ then ΦΨ = 1 and ΨΦ = 1 so that P rojs = P rojt. More generally, let ϕ : S T and ψ : T Z be morphisms of graded rings. Then G(ψϕ) G(ψ). Let Φ : G(ϕ) P rojs and Ψ : G(ψ) P rojt be induced as above, and let Ω be induced by the composite ψϕ. Then ΨG(ψϕ) G(ϕ) and it is readily checked that the following diagram of schemes commutes: Ω P rojs G(ψϕ) Φ Ψ G(ψϕ) G(ϕ) If ϕ : S T is a morphism of graded algebras for some ring, then Φ : G(ϕ) P rojs is a morphism of schemes over, where G(ϕ), P rojs are schemes in the canonical way. lso if ϕ : S S is the identity, clearly G(ϕ) = P rojs and Φ is the identity. For any graded ring S and homogenous f S + there is an isomorphism of schemes Spec(S (f) ) = D + (f). We claim that this isomorphism is natural with respect to morphisms of graded rings, in the following sense. Lemma 6. Let ϕ : S T be a morphism of graded rings and Φ : G(ϕ) P rojs the induced morphism of schemes. If f S + is homogenous then Φ 1 D + (f) = D + (ϕ(f)) and the following diagram commutes: Spec(T (ϕ(f)) ) Spec(S (f) ) (1) D + (ϕ(f)) D + (f) where the top morphism corresponds to ϕ (f) : S (f) T (ϕ(f)) defined by s/f n ϕ(s)/ϕ(f) n, and the bottom morphism is the restriction of Φ. Proof. By (H, Ex.2.4) it suffices to check that the diagram commutes on global sections of Spec(S (f) ), which follows immediately from the definition of Φ and the explicit form of the vertical isomorphisms given in our Section 2.2 notes. Lemma 7. Let S be a graded ring and f, g S + homogenous elements. diagram commutes Spec(S (f) ) D + (f) Then the following Spec(S (fg) ) D + (fg) where the vertical morphism on the left corresponds to the ring morphism S (f) S (fg) defined by s/f n sg n /(fg) n. Proof. By (H, Ex.2.4) it suffices to check that the diagram commutes on global sections of Spec(S (f) ). But this is easy to check, using the explicit form of the isomorphisms given in our Section 2.2 notes. Let H be a graded ring and suppose f H 0. Then ϕ f (h) = d 0 f d h d defines a morphism of graded rings ϕ f : H H. If f is a unit with inverse g, then ϕ f is an isomorphism with inverse ϕ g. If f is a unit, then it is easy to see that for a homogenous prime ideal p, ϕ 1 f p = p and the induced morphism (ϕ f ) (p) : H (p) H (p) is the identity. Therefore ϕ f induces the identity morphism P rojh P rojh. This simple observation has the following important consequence 3
4 Lemma 8. Let ϕ : S T be a morphism of graded rings and suppose f T 0 is a unit. Let ϕ f : S T be the following morphism of graded rings ϕ f (s) = d 0 f d ϕ(s d ) Then G(ϕ) = G(ϕ f ) and the induced morphisms G(ϕ) P rojs are the same. Proposition 9. Let S be a graded ring and e > 0. The morphism of graded rings ϕ : S [e] S induces an isomorphism of schemes Φ : P rojs P rojs [e] natural in S. Proof. We defined the graded ring S [e] in (GRM,Definition 6). The inclusion ϕ : S [e] S is a morphism of graded rings. It is clear that G(ϕ) = P rojs, so we get a morphism of schemes Φ : P rojs P rojs [e]. If a is a homogenous ideal of S then a S [e] is homogenous ideal of S [e], and it is not hard to see that for a homogenous prime ideal p we have p a if and only if p S [e] a S [e]. In particular this shows that Φ is injective. To see that it is surjective, let q be a homogenous prime ideal of S [e]. Then the ideal q generated by q in S is homogenous and it is not difficult to check that q S [e] = q. We claim that the homogenous ideal p = q is prime. If a S m and b S n are such that ab p then (ab) ke q for some k > 0 and therefore either a ke q or b ke q, which shows that a p or b p. Clearly p S [e] = q and if q S [e] + then p S +, which shows that Φ is surjective. Our arguments in the previous paragraph also imply that Φ(V (a)) = V (a S [e] ) for any homogenous ideal a, so Φ is in fact a homeomorphism. For homogenous f S + we have D + (f) = D + (f e ) and it is clear that for f S [e] + we have Φ(D + (f)) = D + (f). The induced morphism of rings S [e] (f) S (f) is the morphism ϕ (f) which is clearly an isomorphism. This proves that Φ : P rojs P rojs [e] is an isomorphism. This isomorphism is natural in S in the following sense. Suppose ψ : S T is a morphism of graded rings, ψ [e] : S [e] T [e] the induced morphism of graded rings. It is not hard to check that the isomorphism P rojt = P rojt [e] identifies the open subsets G(ψ) and G(ψ [e] ) and that the following diagram commutes G(ψ) Ψ P rojs G(ψ [e] ) Ψ [e] P rojs [e] which completes the proof. Definition 2. Let S be a graded ring and e > 0. Let S e denote the graded ring with the same ring structure as S, but with the inflated grading S e = S S 1 0 so (S e) 0 = S 0, (S e) e = S 1, (S e) 2e = S 2 and so on. It is not hard to see that an ideal a S is homogenous in S iff. it is homogenous in S e, so as topological spaces we have P rojs = P roj(s e). Moreover for a homogenous prime ideal p we have an equality of rings S (p) = (S e) (p), and therefore the sheaves of rings are also the same. Hence we have an equality of schemes P rojs = P roj(s e). Example 1. For a graded ring S and e > 0 it is clear that S (e) e = S [e]. Therefore as schemes there is an equality P rojs (e) = P rojs [e]. Corollary 10. Let S be a graded ring and e > 0. There is a canonical isomorphism of schemes Ψ : P rojs P rojs (e) natural in S. 4
5 Proof. The isomorphism is the equality P rojs (e) = P rojs [e] followed by the isomorphism of Proposition 9. Naturality in S means that for a morphism of graded rings θ : S T with induced morphism of graded rings θ (e) : S (e) T (e) the isomorphism P rojs = P rojs (e) identifies G(θ) and G(θ (e) ) and the following diagram commutes G(θ) Θ P rojs G(θ (e) ) Θ (e) P rojs (e) This follows easily from the naturality of Proposition 9. Proposition 11. Let S be a graded algebra generated by S 1 as an algebra. Let S be the graded algebra defined by S 0 =, S d = S d for d > 0. Then the canonical morphism of graded algebras S S induces an isomorphism of schemes P rojs P rojs natural in S. Proof. We make the direct sum S = d 1 S d into a graded algebra in the canonical way (see our note on Constructing Graded lgebras). Let ϕ : S S be induced by S and all the inclusions S d. This is a morphism of graded algebras. It is clear that G(ϕ) = P rojs, so let Φ : P rojs P rojs be the induced morphism of schemes. It follows from (GRM,Lemma 17) that Φ is injective and from (GRM,Proposition 18), (GRM,Lemma 17) that Φ is surjective. For a homogenous ideal a of S we have Φ(V (a)) = V (ϕ 1 a) (GRM,Lemma 16) so Φ is a homeomorphism. To show that Φ is an isomorphism of schemes it suffices by Lemma 1 to show that for d > 0 and f S d the following ring morphism is bijective ϕ (f) : S (f) S (f) a/f n a/f n This map is clearly surjective. To see that it is injective, suppose a/f n maps to zero in S (f). If n > 0 then a S nd and f k a = 0 for some k 0 implies that a/f n = 0 in S (f). If n = 0 and a then we can write a/1 = af/f and reduce to the case where n > 0. This shows that ϕ (f) is an isomorphism and completes the proof that Φ is an isomorphism of schemes. To prove naturality, let ψ : S T be a morphism of graded algebras, where T is also generated by T 1 as an algebra. There is an induced morphism of graded algebras ψ = 1 d 1 ψ d : S T making the following diagram commute S ψ T S ψ T We claim that the isomorphisms Φ S : P rojs P rojs and Φ T : P rojt P rojt are natural in the sense that Φ T identifies the open subsets G(ψ) and G(ψ ) and the following diagram commutes Ψ G(ψ) P rojs The details are easily checked. G(ψ ) Φ S Ψ P rojs Definition 3. Let ϕ : S T be a morphism of graded rings. We say that ϕ is a quasimonomorphism, quasiepimorphism or quasiisomorphism if it has this property as a morphism of graded Smodules (GRM,Definition 10). 5
6 Corollary 12. Let ϕ : S T be a morphism of graded algebras where S is generated by S 1 as an algebra and T is generated by T 1 as an algebra. If Φ : G(ϕ) P rojs is the induced morphism of schemes then (i) If ϕ is a quasiepimorphism then G(ϕ) = P rojt and Φ is a closed immersion. (ii) If further ϕ is a quasiisomorphism then Φ is an isomorphism. Proof. Using Proposition 11 we can assume that S 0 = and T 0 =. If ϕ is a quasiepimorphism (resp. quasiisomorphism) then for some e > 0 the morphism ϕ (e) : S (e) T (e) is surjective (resp. bijective) so by Corollary 10 we can reduce to proving that if ϕ is surjective then Φ is a closed immersion, and further that if ϕ is an isomorphism then so is Φ. But we proved the former claim in our solution to (H,Ex.3.12) and the latter claim is trivial. Remark 1. Note that if S is a graded algebra and S denotes the graded algebra S 0 =, S d = S d for d > 0 defined in Proposition 11 then the canonical morphism of graded algebras ϕ : S S is trivially a quasiisomorphism. Proposition 13. Let ϕ : S T be a quasiisomorphism of graded algebras with S 0 = and T 0 =. Then there is an integer E > 0 such that for all e E, ϕ (e) : S (e) T (e) is an isomorphism of graded algebras. Proof. Let E > 0 be large enough that ϕ n : S n T n is bijective for all n E. Then it is not hard to see that ϕ (e) : S (e) T (e) is an isomorphism for e E. 3 Products Lemma 14. Let, B be rings, S a graded algebra, and T a graded Balgebra. Let ψ : B be a morphism of rings and ϕ : S T a morphism of graded rings making the following diagram commute: ψ B (2) S ϕ T Then the following diagram commutes, where Ψ = Spec(ψ) and the vertical morphisms are canonical: Φ G(ϕ) P rojs (3) SpecB Ψ Spec Proof. The open sets D + (f) P rojs for homogenous f S + are an open cover, so the open sets D + (ϕ(f)) = Φ 1 D + (f) for homogenous f S + must be an open cover of G(ϕ). So it suffices to show that the two legs of the diagram agree when composed with Spec(T (ϕ(f)) ) G(ϕ) for every homogenous f S +. Using the commutativity of (1), this reduces to checking commutativity of the following diagram of rings, where the vertical morphisms are canonical: T (ϕ(f)) ϕ (f) S (f) (4) B ϕ But this square is trivially commutative, completing the proof. lternatively, use the fact that morphisms into Spec are in bijection with ring morphisms out of and just check the diagram on global sections. 6
7 Proposition 15. Let, B be rings, S a graded algebra, ψ : B a morphism of rings. Then T = S B is a graded Balgebra and P rojt = P rojs SpecB. Proof. Consider, B as graded rings in the canonical way. Then S and B become graded  modules, so the tensor product S B becomes a graded Balgebra (GRM,Section 6) with grading { } (S B) n = s i b i s i S n, b i B i The map ϕ : S T defined by ϕ(s) = s 1 is a morphism of graded rings, which makes the diagram (2) commute. Since T + is generated as a Bmodule by ϕ(s + ) it follows that G(ϕ) = P rojt and we get a morphism of schemes Φ : P rojt P rojs fitting into a commutative diagram P rojt Φ P rojs (5) SpecB Ψ Spec We have to show that this diagram is a product of schemes over Spec. The open sets D + (f) for homogenous f S + cover P rojs and by our earlier notes on the local nature of products, it suffices to show that D + (ϕ(f)) = Φ 1 D + (f) is a product for D + (f) and SpecB over Spec for all homogenous f S +. Using commutativity of (1) we reduce this to showing that the following diagram is a pullback: Spec(T (ϕ(f)) ) Spec(S (f) ) Spec(B) Spec() Which is equivalent to showing that (4) is a pushout of rings. The map S f B T ϕ(f) defined by (s/f n, b) (s b)/ϕ(f) n is welldefined and bilinear, so induces S f B T ϕ(f), which is easily checked to be a morphism of rings. The canonical map S S f is a morphism of  modules, so there is a welldefined morphism T = S B S f B defined by s b s/1 b. This is a morphism of rings mapping multiples of ϕ(f) = f 1 to units, so it induces a morphism of rings T ϕ(f) S f B defined by (s b)/ϕ(f) n s/f n b. Since we have already constructed the inverse, this is an isomorphism of rings T ϕ(f) = Sf B. The ring S f is a Zgraded ring, and thus can be considered as a graded module. The ring S f B is therefore also Zgraded. Putting the canonical Zgrading on T ϕ(f) it is easily checked that T ϕ(f) = Sf B is an isomorphism of Zgraded rings, which restricts to give an isomorphism of the degree zero subrings T (ϕ(f)) = (Sf B) 0. We have to show that this latter ring is isomorphic as a ring to S (f) B. Let α : S f Z B S f B be canonical (GRM,Section 6). The kernel of α is the abelian group P generated by elements (a x) b x (a b) where x is homogenous. The morphism of groups S (f) Z B S f Z B is injective since S (f) is a direct summand of S f and tensor products preserve colimits. Therefore the group S (f) Z B is isomorphic to its image in S f Z B, which is mapped by α onto (S f B) 0. So there is an isomorphism of abelian groups (S (f) Z B)/P = (S f B) 0 where P = P (S (f) Z B). But since S (f) Z B can be identified with the degree zero subring of S f Z B, P is generated as an abelian group by elements (a x) b x (a b) where x is homogenous of degree zero, that is, x S (f). Hence there is an isomorphism of abelian groups S (f) B = (S (f) Z B)/P = (Sf B) 0 s/f n b s/f n b So finally we have an isomorphism of rings T (ϕ(f)) = S(f) B defined by (s b)/ϕ(f) n s/f n b. Using this isomorphism one checks easily that (4) is a pushout, completing the proof. 7
8 Let ψ : B be a morphism of rings (n 0) and ϕ : [x 0,..., x n ] B[x 0,..., x n ] the morphism of graded rings induced by ψ. Then it is not difficult to check that the following diagram is a pushout of rings: ψ B [x 0,..., x n ] ϕ B[x 0,..., x n ] By uniqueness of the pushout, there is an isomorphism of rings α B[x 0,..., x n ] B [x 0,..., x n ] f(x 0,..., x n ) f(1 x 0,..., 1 x n ) f(α)x α0 0 xαn n α f(α) x α0 0 xαn n This is the morphism of Balgebras induced by x i 1 x i. Similarly x i x i 1 induces an isomorphism of rings θ : B[x 0,..., x n ] [x 0,..., x n ] B. If we give the tensor product the canonical graded structure, these are isomorphisms of graded rings. Note that θϕ is just the canonical map of [x 0,..., x n ] into the tensor product. Now let n 1 and T = [x 0,..., x n ] B. Then G(ϕ) = P n B and G(θ) = G(θϕ) = P rojt. Let Ω : P rojt Pn be induced by θϕ. Consider the following diagram: P n B (6) Φ Θ Ω P rojt P n SpecB Ψ Spec ll unmarked morphisms are the canonical ones, and by the previous Proposition the square is a pullback. The top triangle commutes by the results of the previous section. To check commutativity of the left triangle, we need only check the two legs give the same morphism on global sections (morphisms into SpecB are in bijection with ring morphisms out of B). n element b B determines the global section q b/1 of P n B and p (1 b)/1 of P rojt. These clearly correspond under the localisations of θ, so (6) commutes. Hence the outside square is a pullback, and we have proved the following result. Proposition 16. Let ψ : B be a morphism of rings and for n 1 let ϕ : [x 0,..., x n ] B[x 0,..., x n ] be the corresponding morphism of graded rings. Then the following diagram is a pullback: Φ P n B P n SpecB Ψ Spec In other words, P n B = SpecB Spec P n. In particular, for any ring we have Pn = Spec Pn Z. P n B One could probably run all the proofs with n = 0, but since in that case P n Spec and SpecB are isomorphisms, the diagram in the proposition is trivially a pullback. 8
9 4 Linear Morphisms Let be a ring n, m 0 and f 0 (x 0,..., x n ),..., f m (x 0,..., x n ) [x 0,..., x n ] homogenous polynomials of degree 1. Then we define φ : [y 0,..., y m ] [x 0,..., x n ] φ(y i ) = f i (x 0,..., x n ) This is a morphism of graded algebras, hence induces a morphism of schemes Φ : G(φ) P m. Here G(φ) = {p P n p φ(s + )} = {p P n p {f 0,..., f m }} m = D + (f i ) i=0 For example, suppose m 1 and 0 i m and let φ i : [x 0,..., x m ] [x 0,..., x m 1 ] be the morphism of graded algebras determined by the following assignments: x 0 x 0,..., x i 1 x i 1, x i+1 x i,..., x m x m 1 and x i 0. Then clearly G(φ i ) = P m 1 and so we have a morphism of schemes Φ i : P m 1 P m The morphism φ i is surjective, so by Ex 3.12 the morphism Φ i is a closed immersion whose image is the closed set V (Kerφ) = V ((x i )) = P m D +(x i ). So for m 1 there are m + 1 closed immersions of P m 1 in P m. Notice that the following diagram also commutes for any 0 i m: P m 1 Φ i Spec In particular there are two closed immersions Spec = P 0 P1. lso note that for n 1 and 0 i n the open set D + (x i ) of P n is isomorphic to Spec[x 0,..., x n ] (xi) and thus to Spec[x 1,..., x n ]. In fact these are isomorphisms of schemes over Spec. nother example are automorphisms arising from invertible matrices. Fix n 1 and let = (a ij ) be an invertible (n + 1) (n + 1) matrix over a field k. For convenience we use the indices 0 i, j n. Then the map x i a ij x j determines a kalgebra automorphism P m ϕ : k[x 0,..., x n ] k[x 0,..., x n ] which gives rise to an automorphism of schemes over k, Φ = P rojϕ : P n k Pn k. 5 Projective Morphisms Definition 4. Let be a scheme and n 0. projective nspace over is a pullback SpecZ P n Z. This consists of two morphisms Z, Z P n Z making the following diagram a pullback: Z P n Z SpecZ Two projective nspaces over are canonically isomorphic as schemes over and P n Z, and we denote any projective nspace over by P n. morphism X is projective if it factors as a 9
10 closed immersion X P n followed by the projection Pn for some n 1 and projective nspace P n. This definition is independent of the projective nspace chosen, in the sense that if it factors through one projection via a closed immersion, then it factors through all of them by a closed immersion. We refer to this situation by saying that X factors via a closed immersion through projective nspace. For a ring and n 0 the scheme P roj[x 0,..., x n ] is a projective nspace over Spec with the canonical morphisms, so the notation P n is unambiguous. Since the morphism P0 Z SpecZ is an isomorphism, projective 0spaces over correspond to isomorphisms Z, in the sense that if Z, Z P 0 Z is a projective 0space then Z is an isomorphism, and any isomorphism Z can be paired with Z SpecZ P 0 Z to make a projective 0space. For any ring and n 1 there are n + 1 closed immersions P n 1 P n of schemes over Spec. We can extend this to projective space over any scheme: Lemma 17. For any scheme there are n + 1 canonical closed immersions P n 1 P n of schemes over for any n 1. Proof. Let P n 1 and P n be any projective n 1 (resp. n)spaces over. We induce a morphism P n 1 P n into the pullback so that the following diagram commutes: P n Z P n 1 Z SpecZ P n P n 1 Where for 0 i n we let P n 1 Z P n Z be the ith closed immersion. Using the usual pullback argument, we see that P n 1 P n is the pullback of Pn 1 Z P n Z, and since closed immersions are stable under pullback the proof is complete. Lemma 18. If for some n 0 a morphism X factors via a closed immersion through projective nspace, then it factors via a closed immersion through projective mspace for any m n. The lemma implies that we could just as well define a projective morphism X to be a morphism factoring via a closed immersion through a projection P n for some n 0, since if X factors via a closed immersion through projective 0space, then it factors through projective 1space by a closed immersion, and is thus projective. So our final definition is: Definition 5. morphsim X is projective if it factors via a closed immersion through projective nspace over for some n 1 (equivalently, some n 0). morphism X is quasiprojective if it factors via an immersion through projective nspace for some n 1 (equivalently, some n 0). Notice that a morphism X factors through projective 0space over iff. it is a closed immersion. In particular any closed immersion is projective. Lemma 19. Let be a ring and S a graded algebra finitely generated as an algebra by S 1. Then the structural morphism P rojs Spec is projective. Proof. By hypothesis we can find a surjective morphism of graded algebras [x 0,..., x n ] S with n 1, so the induced morphism P rojs P n is a closed immersion of schemes over, as required. 10
11 Let f : X be a morphism of schemes, fix an integer n 1 and pullbacks P n X, Pn. Then the morphisms P n X Pn Z and Pn X X induce a morphism Pn f : Pn X Pn which is the unique morphism of schemes over P n Z fitting into the following commutative diagram P n X P n f P n X f In fact using standard properties of pullbacks it is not hard to see that this is a pullback diagram. Clearly P n 1 = 1 and if we have a morphism g : Z and fix a pullback P n Z then Pn g P n f = Pn gf. If X = Spec, = SpecB and f is induced by a ring morphism ψ : B, and if we choose P n, Pn B as our pullbacks, then Pn f is the morphism induced by the canonical ring morphism [x 0,..., x n ] B[x 0,..., x n ] as in Proposition Dimensions of Schemes We want to translate some results about varieties into results about schemes. To be specific, recall the following (proved in I, 3.4): Proposition 20. Let P n be a projective variety and 0 i n such that i = U i, where U i = n is the open set x i 0. Then i corresponds to an affine variety in n and S( ) (xi) = ( i ), where S( ) = k[x 0,..., x n ]/I( ) and ( i ) = k[y 1,..., y n ]/I( i ). Proof. In outline: let S = k[x 0,..., x n ]. Then k[y 1,..., y n ] = S (xi) and this isomorphism identifies I( i ) with the prime ideal I( )S xi S (xi) (which is denoted I( )S (xi) in the proof of I, 3.4). It follows that ( i ) = k[y 1,..., y n ]/I( i ) = S (xi)/i( )S (xi)) = (S/I( )) (xi) = S( ) (xi) Let k be a field, and for n 1 set S = k[x 0,..., x n ] and consider the the projective space P n k = P rojs. There are n + 1 open affine subsets D +(x i ) = SpecS (xi) = Speck[x 0 /x i,..., x n /x i ] which cover P n k. We know from our divisor notes that the closed irreducible subsets of Pn k are in bijection with the homogenous primes of S other than S +. Let P n k be a closed irreducible subset, and suppose D + (x i ). Then I( ) D + (x i ) and D + (x i ) is the closure of I( ). The prime I( ) corresponds to the prime I( )S (xi) = I( )S xi S (xi) of SpecS (xi). Let i Speck[x 0 /x i,..., x n /x i ] be the closed irreducible subset corresponding to D + (x i ). Then I( i ) is the prime ideal of k[x 0 /x i,..., x n /x i ] corresponding to I( )S (xi). D + (x i ) = = SpecS (xi) Speck[x 0 /x i,..., x n /x i ] 11
12 Let ( i ) denote k[x 0 /x i,..., x n /x i ]/I( i ) and S( ) denote S/I( ). Then immediately we have an isomorphism of kalgebras ( i ) = S (xi)/i( )S (xi). It follows from the next Lemma that S (xi)/i( )S (xi) = (S/I( )) (xi). So finally we have an isomorphism of kalgebras ( i ) = S( ) (xi). Lemma 21. Let be a ring and S a graded algebra. If I is a homogenous ideal of S and x S homogenous, then there is a canonical isomorphism of algebras α : (S/I) (x+i) S (x) /(IS x S (x) ) (f + I)/(x n + I) f/x n + IS x S (x) Proof. It is easy to check the given map is an isomorphism of algebras. We proved in our Chapter 1, Section 3 notes that for a graded domain S and nonzero x S 1 there is an isomorphism of S 0 algebras S x = S(x) [z, z 1 ] (the ring of Laurent polynomials). s in the proof of Ex I, 2.6, this implies that for the case S = k[x 0,..., x n ] we have an isomorphism of kalgebras S( ) xi = S( )(xi)[z, z 1 ] = ( i )[z] z It follows that the quotient field of S( ) is kisomorphic to the quotient field of ( i )[z]. Since both S( ) and ( i )[z] are finitely generated domains over the field k, this implies that dims( ) = dim( i )[z] = dim( i ) + 1. Proposition 22. Let S = k[x 0,..., x n ] for a field k and n 1. If P n k irreducible closed subset then dims( ) = 1 + dim where I( ) is the homogenous prime associated with and S( ) = S/I( ). = P rojs is an Proof. The previous discussion shows that is covered by open subsets homeomorphic to schemes of the form Spec( i ) where dim( i ) = dims( ) 1. The claim now follows from I, Ex1.10 and the fact that for a ring, dim = dim(spec). Since dims( ) = coht.i( ), another way to state the result is dim = coht.i( ) 1. This is in contrast to the affine case, where for an irreducible closed we would have dim = coht.i( ). 7 Points of Projective Space Let S = k[x 0,..., x n ] where k is a field and n 1. If k is algebraically closed, then we know from our Varieties as Schemes notes that the closed points of P n k = P rojs are homeomorphic to the variety P n defined in Chapter 1. Corresponding to a point P = (a 0,..., a n ) with a i 0 is the homogenous prime ideal I(P ) = (a i x 0 a 0 x i,..., a i x n a n x i ). In fact, we can still use this idea of points when k is not algebraically closed. Throughout this section k is an arbitrary field and S = k[x 0,..., x n ] for fixed n 1. Lemma 23. Given a nonzero tuple P = (a 0,..., a n ) k n+1 with say a i 0, I(P ) = (a i x 0 a 0 x i,..., a i x n a n x i ) is a homogenous prime ideal of S of height n 1. Proof. First we show that the ideal I(P ) does not depend on i. That is, if also a j 0 then we claim I(P ) = (a j x 0 a 0 x j,..., a j x n a n x j ). This follows easily from the following formula for any 0 k n a j x k a k x j = a j /a i (a i x k a k x i ) a k /a i (a i x j a j x i ) Since I(P ) is generated by linear polynomials, it is a homogenous prime ideal of height n 1 by our Linear Variety notes. It is clear that if P = λq for some nonzero λ k then I(P ) = I(Q), as one would expect for points of projective space. Since ht.s + = n it is clear that I(P ) P rojs, so we have associated points of P n k with tuples of kn+1. 12
13 Lemma 24. Given a nonzero tuple P = (a 0,..., a n ) k n+1 there is an automorphism Φ : P n k P n k of schemes over k mapping P to (1, 0,..., 0). Proof. Suppose that a i k[x 0,..., x n ] by 0 and define an automorphism of kalgebras ψ : k[x 0,..., x n ] x 0 x i x 1 a i x 0 a 0 x i. x n a i x n a n x i It is not hard to see that this induces a kautomorphism of P n k identifying P and (1, 0,..., 0) (that is, identifying the homogenous prime ideals I(P ) and (x 1,..., x n )). 13
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 Aalgebras then the tensor product S A T becomes a graded Aalgebra
More informationSynopsis 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 quasicoherent O Y module.
More informationSection 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
More informationSection 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
More informationThe 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
More informationSynopsis 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 quasicoherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf
More informationSection 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
More informationAlgebraic 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
More informationModules over a Scheme
Modules over a Scheme Daniel Murfet October 5, 2006 In these notes we collect various facts about quasicoherent sheaves on a scheme. Nearly all of the material is trivial or can be found in [Gro60]. These
More informationMATH 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
More informationReid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.
Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y
More informationSynopsis of material from EGA Chapter II, 5
Synopsis of material from EGA Chapter II, 5 5. Quasiaffine, quasiprojective, proper and projective morphisms 5.1. Quasiaffine morphisms. Definition (5.1.1). A scheme is quasiaffine if it is isomorphic
More informationSchemes via Noncommutative Localisation
Schemes via Noncommutative Localisation Daniel Murfet September 18, 2005 In this note we give an exposition of the wellknown results of Gabriel, which show how to define affine schemes in terms of the
More informationRelative Affine Schemes
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
More informationConcentrated Schemes
Concentrated Schemes Daniel Murfet July 9, 2006 The original reference for quasicompact and quasiseparated morphisms is EGA IV.1. Definition 1. A morphism of schemes f : X Y is quasicompact if there
More informationSCHEMES. David Harari. Tsinghua, FebruaryMarch 2005
SCHEMES David Harari Tsinghua, FebruaryMarch 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............
More informationn P say, then (X A Y ) P
COMMUTATIVE ALGEBRA 35 7.2. The Picard group of a ring. Definition. A line bundle over a ring A is a finitely generated projective Amodule such that the rank function Spec A N is constant with value 1.
More informationSection 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
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine nspace the collection A n k of points P = a 1, a,..., a
More informationFormal power series rings, inverse limits, and Iadic completions of rings
Formal power series rings, inverse limits, and Iadic 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
More information1. 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 quasiprojective varieties over a field k Affine Varieties 1.
More informationCHAPTER 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
More informationMATH 221 NOTES BRENT HO. Date: January 3, 2009.
MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................
More informationMatsumura: 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
More informationFORMAL 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
More informationExercises 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 Email address: alex.massarenti@sissa.it These notes collect a series of
More informationHARTSHORNE 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
More informationArithmetic Algebraic Geometry
Arithmetic Algebraic Geometry 2 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016 2 Contents 1 Preliminaries 1 1.1 Affine Varieties.......................... 1 1.2 Projective Varieties........................
More informationAzumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras
Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More information0.1 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.
More informationAlgebraic 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
More information214A 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 }.
More informationAlgebraic 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
More informationAlgebraic 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
More informationALGEBRAIC 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
More informationLINE BUNDLES ON PROJECTIVE SPACE
LINE BUNDLES ON PROJECTIVE SPACE DANIEL LITT We wish to show that any line bundle over P n k is isomorphic to O(m) for some m; we give two proofs below, one following Hartshorne, and the other assuming
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY. Throughout these notes all rings will be commutative with identity. k will be an algebraically
INTRODUCTION TO ALGEBRAIC GEOMETRY STEVEN DALE CUTKOSKY Throughout these notes all rings will be commutative with identity. k will be an algebraically closed field. 1. Preliminaries on Ring Homomorphisms
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationA 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
More informationHomework 2  Math 603 Fall 05 Solutions
Homework 2  Math 603 Fall 05 Solutions 1. (a): In the notation of AtiyahMacdonald, Prop. 5.17, we have B n j=1 Av j. Since A is Noetherian, this implies that B is f.g. as an Amodule. (b): By Noether
More informationHilbert function, Betti numbers. Daniel Gromada
Hilbert function, Betti numbers 1 Daniel Gromada References 2 David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry 19, 110 David Eisenbud: The Geometry of Syzygies 1A, 1B My own notes
More informationThis is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:
Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationNONSINGULAR CURVES BRIAN OSSERMAN
NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that
More informationLECTURE IV: PERFECT PRISMS AND PERFECTOID RINGS
LECTURE IV: PERFECT PRISMS AND PERFECTOID RINGS In this lecture, we study the commutative algebra properties of perfect prisms. These turn out to be equivalent to perfectoid rings, and most of the lecture
More informationALGEBRAIC GEOMETRY CAUCHER BIRKAR
ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasiprojective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction
More informationSUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More informationAlgebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...
Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationCommutative Algebra. Contents. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...
Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 4 1.1 Rings & homomorphisms.............................. 4 1.2 Modules........................................ 6 1.3 Prime & maximal ideals...............................
More information12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism 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 Amorphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an Sgroup. A representation of G is a morphism of Sgroups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More information1. Valuative Criteria Specialization vs being closed
1. Valuative Criteria 1.1. Specialization vs being closed Proposition 1.1 (Specialization vs Closed). Let f : X Y be a quasicompact Smorphisms, and let Z X be closed nonempty. 1) For every z Z there
More informationSystems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,
Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.
More informationMATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1
MATH 8254 ALGEBRAIC GEOMETRY HOMEWORK 1 CİHAN BAHRAN I discussed several of the problems here with Cheuk Yu Mak and Chen Wan. 4.1.12. Let X be a normal and proper algebraic variety over a field k. Show
More informationSUMMER 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
More informationSmooth morphisms. Peter Bruin 21 February 2007
Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,
More informationNOTES ON ALGEBRAIC GEOMETRY MATH 202A. Contents Introduction Affine varieties 22
NOTES ON ALGEBRAIC GEOMETRY MATH 202A KIYOSHI IGUSA BRANDEIS UNIVERSITY Contents Introduction 1 1. Affine varieties 2 1.1. Weak Nullstellensatz 2 1.2. Noether s normalization theorem 2 1.3. Nullstellensatz
More informationCommutative Algebra. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...
Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 2 1.1 Rings & homomorphisms................... 2 1.2 Modules............................. 4 1.3 Prime & maximal ideals....................
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationUNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of socalled formulas. A formula produces simultaneously, for
More informationCOHOMOLOGY 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
More informationCHEVALLEY S THEOREM AND COMPLETE VARIETIES
CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized
More information7 Orders in Dedekind domains, primes in Galois extensions
18.785 Number theory I Lecture #7 Fall 2015 10/01/2015 7 Orders in Dedekind domains, primes in Galois extensions 7.1 Orders in Dedekind domains Let S/R be an extension of rings. The conductor c of R (in
More informationLecture 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 pathconnected topological space, and
More information1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An Amodule M is flat if the (rightexact) functor A M is exact. It is faithfully flat if a complex of Amodules P N Q
More informationA NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS
A NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS RAVI JAGADEESAN AND AARON LANDESMAN Abstract. We give a new proof of Serre s result that a Noetherian local ring is regular if
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an Ngraded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local
More informationWe can choose generators of this kalgebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a prespecified embedding in a projective space. However, an ample line bundle
More informationAN 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,
More informationSummer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More informationModules 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
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationMath 203A  Solution Set 3
Math 03A  Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More informationDIVISORS ON NONSINGULAR CURVES
DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationMATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties
MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,
More informationChern 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
More informationCohomology 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 halfexact if whenever 0 M M M 0 is an exact sequence of Amodules, the sequence T (M ) T (M)
More informationwhich 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
More informationAPPENDIX 3: AN OVERVIEW OF CHOW GROUPS
APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationIntroduction to Algebraic Geometry. Jilong Tong
Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets.................................. 11 1.1.1 Some definitions................................
More informationThe most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
More informationMatrix factorisations
28.08.2013 Outline Main Theorem (Eisenbud) Let R = S/(f ) be a hypersurface. Then MCM(R) MF S (f ) Survey of the talk: 1 Define hypersurfaces. Explain, why they fit in our setting. 2 Define. Prove, that
More information14 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
More informationCOURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA
COURSE SUMMARY FOR MATH 504, FALL QUARTER 20178: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties
More information(dim Z j dim Z j 1 ) 1 j i
Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated kalgebra. In class we have seen that the dimension theory of A is linked to the
More informationHomological Dimension
Homological Dimension David E V Rose April 17, 29 1 Introduction In this note, we explore the notion of homological dimension After introducing the basic concepts, our two main goals are to give a proof
More informationLecture 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,
More informationYuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99
Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationCOMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski
More information