Math 216A. A gluing construction of Proj(S)
|
|
- Cecilia Jenkins
- 6 years ago
- Views:
Transcription
1 Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does not include 0). Morphisms between N-graded rings are understood to respect the grading. The irrelevant ideal is S + = n>0 S n ; keep in mind that we allow S 0 to have a nontrivial ideal theory (that is, it need not be a ield). An element S is homogeneous i S d or some d, and then d is unique i 0; we call d the degree o (when 0), and we consider 0 as having arbitrary degree. Note that the equation deg(g) = deg() + deg g is valid even i one o, g, or g vanishes, using the convention that 0 may be considered to have arbitrary degree. For example, S + is exactly the set o elements (including 0) with positive degree. For a general element S, the homogeneous parts o are the projections d o into each S d (so d = 0 or all but initely many d). An ideal I in S is homogeneous i an element = n 0 n o S lies in I i and only i each homogeneous part n lies in I. It is a simple exercise (inducting on degrees) to check that an ideal generated by homogeneous elements is a homogeneous ideal, and that homogeneous ideal I in S is prime i and only i it is a proper ideal and g I I or g I or homogeneous, g S. It is also clear that the kernel o a morphism o N-graded rings is a homogeneous ideal, and that or any homogeneous ideal I o S there is a natural N-grading on S/I. Deinition 1.1. Let S be an N-graded ring. The topological space Proj(S) has underlying set Proj(S) = {p a homogeneous prime such that S + p}, and the closed sets are the loci V (I) = {p Proj(S) I p} or homogeneous ideals I o S (context will prevent conusion with the analogous V (I) notation or aine schemes). It is easy to check that the V (I) s do satisy the axioms to deine the closed sets or a topology on Proj(S) (the empty set is V (S) and Proj(S) = V (0)). A homogeneous prime p ails to contain I i and only i there exists a homogeneous element I that does not lie in p (here we use crucially that I is homogeneous). Thus, a base o open sets or the topology on Proj(S) is given by loci D + () = {p Proj(S) p} = Proj(S) V (S) or homogeneous S. A crucial act is that it is even enough to take with positive degree: Lemma 1.2. A base o open sets or the topology on Proj(S) is given by loci D + () or homogeneous S +. Proo. Consider a homogeneous S and a point p D + (). We need to ind a homogeneous element g S + such that p D + (g) D + (). Since p does not contain S + (by the deinition o Proj(S)), there exists h S + not in p. Thus, the condition p implies h p, and h is homogenous with positive degree (possibly even h = 0) since both and h are homogeneous and deg h > 0. We conclude that p D + (h), and clearly D + (h) D + (). Beware that Proj(S) is generally not quasi-compact! For example, Proj(k[x 1, x 2,... ]) with ininitely many indeterminates o degree 1 is not quasi-compact, as it is covered by opens D + (x i ) and there is evidently no inite subcover. This issue is best understood as ollows: 1
2 2 Theorem 1.3. For an N-graded ring S, Proj(S) is empty i and only i all elements o S + are nilpotent. More generally, or positive-degree homogeneous elements and { i } i I in S, D + () D + ( i ) i and only i some power o lies in the homogeneous ideal generated by the i s. In particular, a collection o D + ( i ) s with all deg i > 0 covers Proj(S) i and only i every element o S + has some power lying in the homogeneous ideal generated by the i s. The contrast with Spec is o course that Spec(A i ) s cover Spec A i and only i the i s generate the unit ideal. The intererence o S + in the analogous covering criterion or Proj, coupled with the possibility that S + might not be initely generated, is the reason why Proj(S) can ail to be quasi-compact. On the other hand, in most interesting situations the ideal S + is initely generated and hence Proj(S) is quasi-compact. However, we note that some undamental constructions o Mumord in the study o moduli o abelian varieties rest crucially on the use o non-quasi-compact Proj s. Proo. Let I be the homogeneous ideal generated by the i s, so the complement o D + ( i ) is the set o p Proj(S) that contain I. Hence, we need to determine when D + () is disjoint rom the set o such p s, or equivalently when every p Proj(S) that contains I also contains ; we want to show that this condition is exactly the condition that a power o lies in I. Passing to the N-graded S/I, we are reduced to proving that a homogeneous S + lies in p or all p Proj(S) i and only i is nilpotent; keep in mind that has positive degree. One direction is obvious, and conversely we must prove that i S + is homogeneous o degree d > 0 and is not nilpotent, then there exists a homogeneous prime p such that p (and so S + p too, so p Proj(S)). We will make use o an auxiliary construction that will play an important role later. Let S (d) = n 0 S dn (so S (d) = S i d = 1). This is naturally an N-graded ring with vanishing graded pieces in degrees not divisible by d. Consider the localized ring (S (d) ) ; since (S (d) ) = S (d) [T ]/(1 T ), by assigning T degree d we see that (S (d) ) naturally has a Z-grading (with vanishing terms away rom degrees divisible by d). For example, s/ n is assigned degree deg(s) nd or homogeneous elements s S (d). Let (S (d) ) () (S (d) ) denote the direct summand o degree-0 elements in the Z-graded (S (d) ). This is a ring, and i is not nilpotent in S then it is not nilpotent in S (d), so then (S (d) ) 0 and hence the subring (S (d) ) () is nonzero. It then ollows that there exists a prime ideal q in (S (d) ) (). We will use this to construct a homogeneous prime p in S (d) that does not contain (and so in particular does not contain S (d) + since deg > 0); the ideal generated by the homogeneous a S such that a d S (d) lies in p is then readily checked to be a homogeneous prime ideal o S that does not contain (this rests crucially on the act that membership in the homogeneous p may be checked on component-parts). Let p be the contraction o q(s (d) ) under S (d) (S (d) ). The ideal p o S (d) does not contain, since otherwise q(s (d) ) would contain the degree-0 element 1, which is absurd since (q(s (d) ) ) (S (d) ) () = q is a proper ideal. To check that p is homogeneous prime, irst observe that (by contstruction) q(s (d) ) is a homogeneous ideal o the Z-graded (S (d) ), so p is a homogeneous ideal o the N-graded S (d). Hence, to veriy primality it is suicient to work with homogeneous elements. That is, we consider homogeneous a, a S (d) with respective degrees dn and dn and we assume aa p. Our goal is to prove a p or a p. Since aa p, the homogenous image o aa in (S (d) ) is contained in q(s (d) ), so aa = (x/ e ) r with r Z, x S kd, and x/ e q (S (d) ) (). Thus, by comparing degrees we get dn + dn = dr, so n + n = r. Hence, aa / r = (a/ n )(a / n ) (S (d) ) is a product o terms with degree 0. However, a a n = x n e (S(d) ) () (q(s (d) ) ) = q, so by primality o q in (S (d) ) () we conclude that at least o the degree-0 elements a/ n or a / n lies in q! Hence, either a or a in S (d) map into q(s (d) ) upon inverting, so by deinition either a or a lie in p.
3 3 2. First steps towards a scheme structure For homogeneous S +, we get an open set D + () Proj(S) consisting o those p Proj(S) that do not contain. These are a base o open sets, and we claim that D + () is naturally homeomorphic to Spec S (), where S () S is the degree-0 part o the Z-graded localization o S at the homogeneous. To deine a homeomorphism ϕ : D + () Spec S (), to each p D + () we associate the prime ideal p () = (ps ) S () Spec S () ; this is prime because it is the contraction o the prime ps = p o S under the ring map S () S (note that p is prime since p is a prime o S not containing ). Theorem 2.1. The map ϕ : D + () Spec(S () ) is a homeomorphism. Proo. For any homogeneous ideal a o S, we generalize the above operation on homogeneous prime ideals by deining ϕ(a) = (as ) S (). For any p D + (), we claim (1) ϕ(a) ϕ(p) a p. Once this is proved, it will ollow that ϕ is at least injective. The ( ) implication is obvious, and or the converse it suices to prove that i a a is a homogeneous element then a p. Let n = deg a 0 and let d = deg > 0. It ollows that a d n as S () = ϕ(a) ϕ(p) = ps S (), so there exists a homogeneous x p such that a d / n = x/ m in S with md = deg(x). Thus, or some e 0 we have e ( m a d n x) = 0 in S, and since p we must have m a d n x p. However, x p, so m a d p. Since p is prime, p, and d is positive, we conclude a p as desired. This completes the proo o injectivity or ϕ. Once we prove ϕ is surjective, and hence is bijective, (1) implies ϕ(v (a) D + ()) = V (ϕ(a)). Hence, or any ideal b o S (), the preimage a o bs in S is a homogeneous ideal satisying ϕ(a) = b. We may thereore conclude that every closed set V (b) in Spec S () corresponds (under the bijection ϕ) to a closed set V (a) D + () in D + (). However, all closed sets in D + () (with the subspace topology rom Proj(S)) have such a orm or some a, so we thereby get that ϕ is a homeomorphism. It remains to check that ϕ is surjective. A key observation is that the natural map (S (d) ) () S () is an isomorphism. The basic idea is that a degree-0 element in S () must have the orm x/ n with homogeneous x o degree deg(x) = nd S nd, so x is in S (d) ; the straightorward details are let to the reader (hint: equality o subrings o S ). Via this identiication, any prime ideal o S () may be considered as a prime ideal in (S (d) ) (). However, in the proo o Theorem 1.3 it was proved (check!) that every prime ideal q o (S (d) ) () has the orm ϕ(p) or some homogeneous prime p o S not containing (that is, or some p D + ()). Let us now write ϕ : D + () Spec(S () ) to denote the homeomorphism constructed above, with S + any positive-degree homogeneous element (so ϕ (p) = ps S () ). We shall use this homeomorphism to endow D + () with a structure o aine scheme, using the structure shea on Spec(S () ). In view o the act that the D + () s orm a base o opens in Proj(S), the key issue is to identiy S () as the ring o sections
4 4 on the open subset D + () Proj(S), and to this end it is useul to note that S () may be described entirely in terms o the subset D + () Proj(S) and the ring S without mentioning : Theorem 2.2. For homogeneous S +, let T be the multiplicative set o homogeneous elements g S such that g p or all p D + () Proj(S) (despite the notation, T only depends on D + () and not on ). The natural map S () (T 1 S) 0 to the degree-0 part o the Z-graded T 1 S induced by S T 1 S is an isomorphism. Proo. Let d = deg > 0. For injectivity, suppose x S is homogeneous o degree nd and the degree-0 element x/ n S maps to 0 in T 1 S. Hence, there exists g T such that gx = 0 in S. Replacing g with g d i necessary, we can assume deg g = md. Thus, (g/ m )(x/ n ) = 0 in S, and hence this equality holds in S (). By the deinition o T, or all p D + () we have g p, so g/ m is not contained in the prime ideal ϕ (p) = (ps ) S () o S () (as, g p). But ϕ is bijective onto Spec S (), so g/ m S () is not contained in any primes. It ollows that g/ m S () is a unit, so the vanishing o (g/ m )(x/ n ) in S () orces x/ n = 0 in S (). This gives exactly the desired injectivity. Now choose g T and x S with deg(x) = deg(g), so x/g (T 1 S) 0 makes sense. We seek a homogeneous a S o some degree nd (or some n 0) such that a/ n S () maps to x/g in T 1 S. We may replace x with g d 1 x and g with g d to get to the case deg g = md or some m 0. Thus, using the deinition o T and the bijectivity o ϕ we see that g/ m S () is not contained in any prime ideals, so it is a unit. In the degree-0 part o the Z-graded T 1 S we have x g = m g g m x g = m g so (g/ m ) 1 (x/ m ) S () maps to x/g (T 1 S) 0. This proves the desired surjectivity. x m, 3. A scheme structure on Proj(S) By Theorem 2.2, whenever, h S + are homogeneous elements such that D + (h) D + () inside o Proj(S) we have (by the deinitions!) T T h inside S, and so we get a canonical map (2) S () = (T 1 S) 0 (T 1 S) 0 = S (h) on degree-0 parts induced by the map T 1 S T 1 h S o Z-graded localizations. We may thereore consider the diagram o topological spaces D + () ϕ h Spec((T 1 S) 0 ) D + (h) ϕ h Spec((T 1 h S) 0) where the let column is the inclusion within Proj(S). One readily checks (upon reviewing the deinitions o the various maps) that this diagram commutes with the right side an open embedding, ultimately because the canonical equality (S () ) h deg / deg h = S (h) = (S (h) ) deg h /h deg inside o S h (check!) and the act that deg h /h deg S (h) (since T T h ) implies that (2) induces an isomorphism (S () ) h deg / S (h). deg h Clearly D + () D + (g) = D + (g), and by taking h = g above we see that this open subset o D + () is carried by ϕ onto the open subset Spec((S () ) g deg / deg g) Spec(S ()).
5 5 Likewise, as an open subset o D + (g) it is carried by ϕ g onto the open subset Spec((S (g) ) deg g /g deg ) Spec(S (g)). We now have put three scheme structures on D + (g), namely Spec S (g) and the two as basic opens in Spec S () and in Spec S (g). These three structures are identiied by means o the ring isomorphisms (3) (S () ) g deg / deg g S (g) (S (g) ) deg g /g deg that are really equalities as subrings o S g. Consequently, the cocycle condition or gluing is satisied (it comes down to transitivity or equality among three subrings o S (gh) or any three homogeneous, g, h S + ), so we may glue the structure sheaves O Spec(S() ) over the D + () s via (3). That is, we are gluing the Spec S () s (as ringed spaces) along the Spec S (g) s, where the underlying topological space Proj(S) o the gluing was made at the start. The glued structure shea over P = Proj(S) will be denoted O P, and so the ringed space (P, O P ) is covered by open subspaces (D + (), O P D+()) Spec(S () ) or homogeneous S +. Hence, (P, O P ) is a scheme. Deinition 3.1. Let S be an N-graded ring. The scheme Proj(S) is the topological space denoted Proj(S) above, equipped with the unique shea o rings O P whose restriction to D + () is O Spec(S() ) (using ϕ ) or all homogeneous S +, with the overlap-gluing isomorphism O Spec((S() ) g deg / deg g ) = O Spec(S() ) D+() D +(g) O Spec(S(g) ) D+(g) D +() = O Spec((S(g) ) deg g /g deg ) deined by the isomorphism Spec((S () ) g deg / g) Spec((S (g)) deg deg g /gdeg ) arising rom the canonical ring isomorphism in (3) or homogeneous, g S +. By Theorem 1.3, we obtain a useul alternative description: Corollary 3.2. Let { i } be a collection o homogeneous elements in S + such that every element o S + has some power contained in the ideal generated by the i s. The scheme Proj(S) is obtained by gluing the aine schemes Spec(S (i)) along the open aine overlaps Spec(S (i j)) Spec(S (i)) deined by the isomorphisms S (i j) (S (i)) deg i j / deg j. i Remark 3.3. We emphasize that there is content in this construction, namely that the above ring isomorphisms satisy triple overlap compatibility; this is most painlessly seen in terms o a triple equality o subrings o S (i j k ). Example 3.4. Let S = A[X 0,..., X n ] be an N-graded ring by putting A in degree 0 and declaring each X i to be homogeneous o degree 1. It ollows that Proj(S) is covered by the opens D + (X i ) = Spec S (Xi) = Spec A[X 0 /X i,..., X n /X i ] or 0 i n, and the gluing isomorphism is determined by the isomorphism (S (Xi)) Xj/X i (S (Xj)) Xi/X j deined by X k /X i (X k /X j ) (X i /X j ) 1 or k i. These are exactly the standard ormulas that express projective n-space as the gluing o n + 1 copies o aine n-space along certain open overlaps deined by non-vanishing o various coordinate unctions. Inspired by the above example, or any ring A we deine projective n-space over A to be P n A = Proj(A[X 0,..., X n ]) with the usual grading on A[X 0,..., X n ]. This is naturally a scheme over Spec A since each basic open aine D + () is naturally an A-scheme (as A has degree 0 in the N-grading being used) and the open-aine gluing data is one o A-algebras (more generally, Proj(S) is always naturally a scheme over Spec S 0 ). As a particularly degenerate example, we have P 0 A = Spec A[X 0] (X0) = Spec A.
6 6 Remark 3.5. I we assign A[X 0,..., X n ] an N-graded structure by putting A in degree 0 and assigning X i some positive degree d i, the resulting N-graded rings are generally not isomorphic as N-graded rings or dierent n-tuples d = (d 0,..., d n ), and their A-scheme Proj s (so-called weighted projective n-spaces over A with weights d) are generally not isomorphic to each other.
SEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationDescent 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
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationThe 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
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
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 informationwhere 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
More information3 Lecture 3: Spectral spaces and constructible sets
3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible
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 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 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 path-connected topological space, and
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 informationThe 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
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 quasi-coherent O Y -module.
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 information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More informationIn the index (pages ), reduce all page numbers by 2.
Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by
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, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a
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 informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
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 informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More informationReview of Linear Algebra
Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space
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 informationk k would be reducible. But the zero locus of f in A n+1
Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along
More informationMath 203A, Solution Set 6.
Math 203A, Solution Set 6. Problem 1. (Finite maps.) Let f 0,..., f m be homogeneous polynomials of degree d > 0 without common zeros on X P n. Show that gives a finite morphism onto its image. f : X P
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 informationAdic Spaces. Torsten Wedhorn. June 19, 2012
Adic Spaces Torsten Wedhorn June 19, 2012 This script is highly preliminary and unfinished. It is online only to give the audience of our lecture easy access to it. Therefore usage is at your own risk.
More informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationHomogeneous Coordinate Ring
Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in
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 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 quasi-coherent O Y algebra. (3.1.1). Let Y be a prescheme. A sheaf
More informationVariations on a Casselman-Osborne theme
Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne
More informationSynopsis 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
More informationMath 396. Quotient spaces
Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition
More information2 Coherent D-Modules. 2.1 Good filtrations
2 Coherent D-Modules As described in the introduction, any system o linear partial dierential equations can be considered as a coherent D-module. In this chapter we ocus our attention on coherent D-modules
More informationCLASS NOTES MATH 527 (SPRING 2011) WEEK 6
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6 BERTRAND GUILLOU 1. Mon, Feb. 21 Note that since we have C() = X A C (A) and the inclusion A C (A) at time 0 is a coibration, it ollows that the pushout map i
More information3. 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
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationRepresentation Theory of Hopf Algebroids. Atsushi Yamaguchi
Representation Theory o H Algebroids Atsushi Yamaguchi Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions
More informationMath 145. Codimension
Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in
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 informationHow to glue perverse sheaves
How to glue perverse sheaves A.A. Beilinson The aim o this note [0] is to give a short, sel-contained account o the vanishing cycle constructions o perverse sheaves; e.g., or the needs o [1]. It diers
More informationLIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset
5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used,
More information12. 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
More information7 Lecture 7: Rational domains, Tate rings and analytic points
7 Lecture 7: Rational domains, Tate rings and analytic points 7.1 Introduction The aim of this lecture is to topologize localizations of Huber rings, and prove some of their properties. We will discuss
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 information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationNotas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018
Notas de Aula Grupos Profinitos Martino Garonzi Universidade de Brasília Primeiro semestre 2018 1 Le risposte uccidono le domande. 2 Contents 1 Topology 4 2 Profinite spaces 6 3 Topological groups 10 4
More informationMath 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm
Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel
More informationALGEBRAIC GEOMETRY CAUCHER BIRKAR
ALGEBRAIC GEOMETRY CAUCHER BIRKAR Contents 1. Introduction 1 2. Affine varieties 3 Exercises 10 3. Quasi-projective varieties. 12 Exercises 20 4. Dimension 21 5. Exercises 24 References 25 1. Introduction
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 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 informationSCHEMES. 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.............
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 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 k-algebra. In class we have seen that the dimension theory of A is linked to the
More informationMath 396. Bijectivity vs. isomorphism
Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1
More informationWe simply compute: for v = x i e i, bilinearity of B implies that Q B (v) = B(v, v) is given by xi x j B(e i, e j ) =
Math 395. Quadratic spaces over R 1. Algebraic preliminaries Let V be a vector space over a field F. Recall that a quadratic form on V is a map Q : V F such that Q(cv) = c 2 Q(v) for all v V and c F, and
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 210B. Profinite group cohomology
Math 210B. Profinite group cohomology 1. Motivation Let {Γ i } be an inverse system of finite groups with surjective transition maps, and define Γ = Γ i equipped with its inverse it topology (i.e., the
More informationFOUNDATIONS 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.
More informationHSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS
HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence
More informationThe homology of real subspace arrangements
Journal o Topology Advance Access published October 14, 2010 Journal o Topology (2010) Page 1 o 33 C 2010 London Mathematical Society doi:10.1112/jtopol/jtq027 The homology o real subspace arrangements
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 quasi-compact S-morphisms, and let Z X be closed non-empty. 1) For every z Z there
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationZORN S LEMMA AND SOME APPLICATIONS
ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More informationGENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
More informationFormal 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
More informationA NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3185 3189 S 0002-9939(97)04112-9 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by
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 E-mail address: alex.massarenti@sissa.it These notes collect a series of
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 informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
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 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 quasi-projective varieties over a field k Affine Varieties 1.
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
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 informationFinite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract)
Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 www.elsevier.com/locate/entcs Finite Dimensional Hilbert Spaces are Complete or Dagger Compact Closed Categories (Extended bstract)
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)
More informationNAP Module 4. Homework 2 Friday, July 14, 2017.
NAP 2017. Module 4. Homework 2 Friday, July 14, 2017. These excercises are due July 21, 2017, at 10 pm. Nepal time. Please, send them to nap@rnta.eu, to laurageatti@gmail.com and schoo.rene@gmail.com.
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 informationMath 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).
More informationMath 6510 Homework 10
2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained
More informationRemarks on the existence of Cartier divisors
arxiv:math/0001104v1 [math.ag] 19 Jan 2000 Remarks on the existence of Cartier divisors Stefan Schröer October 22, 2018 Abstract We characterize those invertible sheaves on a noetherian scheme which are
More informationGLUING 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
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationne varieties (continued)
Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we
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 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 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 informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.
ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationAnalytic continuation in several complex variables
Analytic continuation in several complex variables An M.S. Thesis Submitted, in partial ulillment o the requirements or the award o the degree o Master o Science in the Faculty o Science, by Chandan Biswas
More information