Foundations of Algebraic Geometry Class 13
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Boundary and Shape of Cohen-Macaulay Cone
BOUNDARY AND SHAPE OF COHEN-MACAULAY CONE HAILONG DAO AND KAZUHIKO KURANO Abstract. Let R be a Cohen-Macaulay local domain. In this paper we study the cone of Cohen-Macaulay modules inside the Grothendieck group of finitely generated R-modules modulo numerical equivalences, introduced in [3]. We prove a result about the boundary of this cone for Cohen-Macaulay domain admitting de Jong’s alterations, and use it to derive some corollaries on finite- ness of isomorphism classes of maximal Cohen-Macaulay ideals. Finally, we explicitly compute the Cohen-Macaulay cone for certain isolated hypersurface singularities defined by ξη − f(x1; : : : ; xn). 1. Introduction Let R be a Noetherian local ring and G0(R) the Grothendieck group of finitely generated R-modules. Using Euler characteristic of perfect complexes with finite length homologies (a generalized version of Serre’s intersection multiplicity pair- ings), one could define the notion of numerical equivalence on G0(R) as in [17]. See Section 2 for precise definitions. When R is the local ring at the vertex of an affine cone over a smooth projective variety X, this notion can be deeply related to that of numerical equivalences on the Chow group of X as in [17] and [20]. Let G0(R) be the Grothendieck group of R modulo numerical equivalences. A simple result in homological algebra tells us that if M is maximal Cohen- Macaulay (MCM), the Euler characteristic function will always be positive. Thus, maximal Cohen-Macaulay modules all survive in G0(R), and it makes sense to talk about the cone of Cohen-Macaulay modules inside G (R) : 0 R X C (R) = [M] ⊂ G (R) : CM R≥0 0 R M:MCM The definition of this cone and some of its basic properties was given in [3]. -
Math 632: Algebraic Geometry Ii Cohomology on Algebraic Varieties
MATH 632: ALGEBRAIC GEOMETRY II COHOMOLOGY ON ALGEBRAIC VARIETIES LECTURES BY PROF. MIRCEA MUSTA¸TA;˘ NOTES BY ALEKSANDER HORAWA These are notes from Math 632: Algebraic geometry II taught by Professor Mircea Musta¸t˘a in Winter 2018, LATEX'ed by Aleksander Horawa (who is the only person responsible for any mistakes that may be found in them). This version is from May 24, 2018. Check for the latest version of these notes at http://www-personal.umich.edu/~ahorawa/index.html If you find any typos or mistakes, please let me know at [email protected]. The problem sets, homeworks, and official notes can be found on the course website: http://www-personal.umich.edu/~mmustata/632-2018.html This course is a continuation of Math 631: Algebraic Geometry I. We will assume the material of that course and use the results without specific references. For notes from the classes (similar to these), see: http://www-personal.umich.edu/~ahorawa/math_631.pdf and for the official lecture notes, see: http://www-personal.umich.edu/~mmustata/ag-1213-2017.pdf The focus of the previous part of the course was on algebraic varieties and it will continue this course. Algebraic varieties are closer to geometric intuition than schemes and understanding them well should make learning schemes later easy. The focus will be placed on sheaves, technical tools such as cohomology, and their applications. Date: May 24, 2018. 1 2 MIRCEA MUSTA¸TA˘ Contents 1. Sheaves3 1.1. Quasicoherent and coherent sheaves on algebraic varieties3 1.2. Locally free sheaves8 1.3. -
Relative 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 associates a scheme to any sheaf of algebras. The contents of this note are roughly EGA II §1.2, §1.3. Contents 1 Affine Morphisms 1 2 The Spec Construction 4 3 The Sheaf Associated to a Module 8 1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets −1 Vα such that for every α, f Vα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the affine property is stable under composition with isomorphisms on either end. Example 1. Any closed immersion X −→ Y is an affine morphism by our solution to (Ex 4.3). Remark 1. A scheme X affine over S is not necessarily affine (for example X = S) and if an affine scheme X is an S-scheme, it is not necessarily affine over S. However, if S is separated then an S-scheme X which is affine is affine over S. Lemma 1. An affine morphism is quasi-compact and separated. Any finite morphism is affine. Proof. Let f : X −→ Y be affine. Then f is separated since any morphism of affine schemes is separated, and the separatedness condition is local. -
Arxiv:1807.03665V3 [Math.AG]
DEMAILLY’S NOTION OF ALGEBRAIC HYPERBOLICITY: GEOMETRICITY, BOUNDEDNESS, MODULI OF MAPS ARIYAN JAVANPEYKAR AND LJUDMILA KAMENOVA Abstract. Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang con- jecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a pro- jective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties. 1. Introduction The aim of this paper is to provide evidence for Demailly’s conjecture which says that a projective algebraically hyperbolic variety over C is Kobayashi hyperbolic. We first define the notion of an algebraically hyperbolic projective scheme over an alge- braically closed field k of characteristic zero which is not assumed to be C, and could be Q, for example. Then we provide indirect evidence for Demailly’s conjecture by showing that algebraically hyperbolic schemes share many common features with Kobayashi hyperbolic complex manifolds. -
3. Ample and Semiample We Recall Some Very Classical Algebraic Geometry
3. Ample and Semiample We recall some very classical algebraic geometry. Let D be an in- 0 tegral Weil divisor. Provided h (X; OX (D)) > 0, D defines a rational map: φ = φD : X 99K Y: The simplest way to define this map is as follows. Pick a basis σ1; σ2; : : : ; σm 0 of the vector space H (X; OX (D)). Define a map m−1 φ: X −! P by the rule x −! [σ1(x): σ2(x): ··· : σm(x)]: Note that to make sense of this notation one has to be a little careful. Really the sections don't take values in C, they take values in the fibre Lx of the line bundle L associated to OX (D), which is a 1-dimensional vector space (let us assume for simplicity that D is Carier so that OX (D) is locally free). One can however make local sense of this mor- phism by taking a local trivialisation of the line bundle LjU ' U × C. Now on a different trivialisation one would get different values. But the two trivialisations differ by a scalar multiple and hence give the same point in Pm−1. However a much better way to proceed is as follows. m−1 0 ∗ P ' P(H (X; OX (D)) ): Given a point x 2 X, let 0 Hx = f σ 2 H (X; OX (D)) j σ(x) = 0 g: 0 Then Hx is a hyperplane in H (X; OX (D)), whence a point of 0 ∗ φ(x) = [Hx] 2 P(H (X; OX (D)) ): Note that φ is not defined everywhere. -
Scheme X of Finite Type Is Noetherian
MATH 8253 ALGEBRAIC GEOMETRY WEEK 12 CIHAN_ BAHRAN 3.2.1. 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 : X ! Y for the structure morphism which is assumed to be of finite type. To show that X is Noetherian, it suffices to show that it is quasi-compact and locally Noetherian. Since Y is quasi-compact, it can be covered by finitely many affine open subsets, say −1 −1 V1;:::;Vn. Then f (Vj)'s cover X. Now f is quasi-compact, so each f (Vj) is quasi- compact and hence their finite union X is quasi-compact. To show that every point in X has a Noetherian neighborhood, let x 2 X. Since Y is locally Noetherian, f(x) is contained in an affine open subset V = Spec B of Y such that B is Noetherian. Let U = Spec A be an affine neighborhood of x such that U ⊆ f −1(V ). Then since f is of finite-type, A is a finitely generated B-algebra, hence A is Noetherian. Moreover, in the above notation, if Y has finite dimension, then dim B < 1 and so by Corollary 2.5.17 the finitely generated B-algebra A has finite dimension. So every x 2 X has a neighborhood with finite dimension. Since X is quasi-compact, X itself has finite dimension. 3.2.2. Show that any open immersion into a localy Noetherian scheme is a mor- phism of finite type. -
Foundations of Algebraic Geometry Class 20
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 20 RAVI VAKIL CONTENTS 1. Dimension and transcendence degree 1 2. Images of morphisms, and Chevalley's theorem 5 3. Fun in codimension one: Krull's Principal Ideal Theorem, Algebraic Hartogs' Lemma, and more 6 4. Proof of Krull's Principal Ideal Theorem 3.3 ?? 10 This is the last class of the quarter! We will finish with dimension theory today. 1. DIMENSION AND TRANSCENDENCE DEGREE We now prove an alternative interpretation for dimension for irreducible varieties. 1.1. Theorem (dimension = transcendence degree). — Suppose A is a finitely-generated domain over a field k. Then dim Spec A is the transcendence degree of the fraction field FF(A) over k. By “finitely generated domain over k”, we mean “a finitely generated k-algebra that is an integral domain”. In case you haven't seen the notion of transcendence degree, here is a quick summary of the relevant facts. Suppose K=k is a finitely generated field extension. Then any two maximal sets of algebraically independent elements of K over k (i.e. any set with no al- gebraic relation) have the same size (a non-negative integer or ). If this size is finite, 1 say n, and x1, . , xn is such a set, then K=k(x1; : : : ; xn) is necessarily a finitely generated algebraic extension, i.e. a finite extension. (Such a set x1, . , xn is called a transcendence basis, and n is called the transcendence degree.) An In particular, we see that dim k = n. However, our proof of Theorem 1.1 will go through this fact, so it isn't really a Corollary. -
Lecture (8).Pdf
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 (K.S. Kedlaya, MIT, Spring 2009) More properties of morphisms (updated 5 Mar 09) Note that finite presentation is not discussed in EGA 1; see EGA 4.1 instead. 1 More about separated morphisms Lemma. The composition of closed immersions is a closed immersion. Proof. Let f : X ! Y and g : Y ! Z be closed immersions. Since the property of being a closed immersion is local on the base, we may assume Z = Spec(A) is affine. Then Y = Spec(B) for B a quotient of A, so X = Spec(C) for C a quotient of B. Hence C is a quotient of A, proving the claim. (A similar argument shows that a composition of finite morphisms is finite.) Lemma. (a) Any closed immersion is separated. (b) A composition of separated morphisms is separated. (c) Separatedness is stable under base change. (d) A product of separated morphisms is separated. (e) If f : X ! Y and g : Y ! Z are morphisms, g � f is separated, and g is separated, then f is separated. (f) If f : X ! Y is separated, then fred : Xred ! Yred is separated. Proof. We know (a) because closed immersions are affine and affine morphisms are separated. We know (c) from the previous handout. Parts (d)-(f) follow once we also have (b); see exercises. It remains to prove (b). Let f : X ! Y and g : Y ! Z be separated morphisms. -
MODEL ANSWERS to HWK #1 4.1. Suppose That F : X −→ Y Is a Finite
MODEL ANSWERS TO HWK #1 4.1. Suppose that f : X −! Y is a finite morphism of schemes. Since properness is local on the base, we may assume that Y = Spec B is affine. By (3.4) it follows that X = Spec A is affine and A is a finitely generated B-module. It follows that A is integral over B. There are two ways to proceed. Here is the first. f is separated as X and Y are affine. As A is a finitely generated B-module it is certainly a finitely generated B-algebra and so f is of finite type. Since the property of being finite is stable under base extension, to show that f is universally closed it suffices to prove that f is closed. Let I E A be an ideal and let J E B be the inverse image of I. I claim that f(V (I)) = V (J). One direction is clear, the LHS is contained in the RHS. Otherwise suppose q 2 V (J), that is, J ⊂ q. We want to produce I ⊂ p whose image is q. Equivalently we want to lift prime ideals of B=J to prime ideals of A=I. But A=I is integral over B=J and what we want is the content of the Going up Theorem in commutative algebra. Here is the second. Pick a1; a2; : : : ; an 2 A which generate A as a B- module. Let C = B[a1] and let Z = Spec C. Then there are finite morphisms X −! Z and Z −! Y . -
Flat Modules in Algebraic Geometry Compositio Mathematica, Tome 24, No 1 (1972), P
COMPOSITIO MATHEMATICA M. RAYNAUD Flat modules in algebraic geometry Compositio Mathematica, tome 24, no 1 (1972), p. 11-31 <http://www.numdam.org/item?id=CM_1972__24_1_11_0> © Foundation Compositio Mathematica, 1972, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 24, Fasc. 1, 1972, pag. 11-31 Wolters-Noordhoff Publishing Printed in the Netherlands FLAT MODULES IN ALGEBRAIC GEOMETRY by M. Raynaud 5th Nordic Summerschool in Mathematics Oslo, August 5-25, 1970 Consider the following data: a noetherian scheme S, a morphism of finite type f: X ~ S, a coherent sheaf of Ox-modules vit. If x is a point of X and s = f(x), recall that M is flat over S at the point x, if the stalk Mx is a flat OS, s-module; M is flat over S, or is S-flat, if vit is flat over S at every point of X. Grothendieck has investigated, in great details, the properties of the morphism f when -4Y is S-flat (EGA IV, 11 12 ... ), and some of its results are now classical. For instance we have: a) the set of points x of X where JI is flat over S is open (EGA IV 11.1.1). -
The Cohomology of Coherent Sheaves
CHAPTER VII The cohomology of coherent sheaves 1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space = U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii) Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S We will write an i-cochain s = s(α0,...,αi), i.e., s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map (U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2: Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define: 211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES s(σβ0, σβ1) defined here U σβ0 Uσβ1 Vβ1 Vβ0 ref s(β0, β1) defined here Figure VII.1 (iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles, Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to . -
Foundations of Algebraic Geometry Class 21
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 21 RAVI VAKIL CONTENTS 1. Integral extensions, the going-up theorem, Noether normalization, and a proof of the big dimension theorem (that transcendence degree = Krull dimension) 1 2. Images of morphisms 5 3. Important example: Morphisms to projective (and quasiprojective) schemes, and invertible sheaves 7 Today: integral extensions, Going-up theorem, Noether Normalization, proof that transcendence degree = Krull dimension, proof of Chevalley's theorem. Invertible sheaves and morphisms to (quasi)projective schemes Welcome back everyone! This is the second quarter in a three-quarter experimental sequence on algebraic geometry. We know what schemes are, their properties, quasicoherent sheaves on them, and mor- phisms between them. This quarter, we're going to talk about fancier concepts: fibered products; normalization; separatedness and the definition of a variety; rational maps; classification of curves; cohomology; differentials; and Riemann-Roch. I'd like to start with some notions that I now think I should have done in the middle of last quarter. They are some notions that I think are easier than are usually presented. 1. INTEGRAL EXTENSIONS, THE GOING-UP THEOREM, NOETHER NORMALIZATION, AND A PROOF OF THE BIG DIMENSION THEOREM (THAT TRANSCENDENCE DEGREE = KRULL DIMENSION) Recall the maps of sets corresponding to a map of rings. If we have φ : B A, we get a map Spec A Spec B as sets (and indeed as topological spaces, and schemes),! which sends p ⊂ A to!φ-1p ⊂ B. The notion behaves well under quotients and localization of both the source and target affine scheme. A ring homomorphism φ : B A is integral if every element of A is integral over φ(B).