RESOLUTION of SURFACES 0ADW Contents 1. Introduction 1 2. a Trace
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A NOTE on COMPLETE RESOLUTIONS 1. Introduction Let
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 11, November 2010, Pages 3815–3820 S 0002-9939(2010)10422-7 Article electronically published on May 20, 2010 A NOTE ON COMPLETE RESOLUTIONS FOTINI DEMBEGIOTI AND OLYMPIA TALELLI (Communicated by Birge Huisgen-Zimmermann) Abstract. It is shown that the Eckmann-Shapiro Lemma holds for complete cohomology if and only if complete cohomology can be calculated using com- plete resolutions. It is also shown that for an LHF-group G the kernels in a complete resolution of a ZG-module coincide with Benson’s class of cofibrant modules. 1. Introduction Let G be a group and ZG its integral group ring. A ZG-module M is said to admit a complete resolution (F, P,n) of coincidence index n if there is an acyclic complex F = {(Fi,ϑi)| i ∈ Z} of projective modules and a projective resolution P = {(Pi,di)| i ∈ Z,i≥ 0} of M such that F and P coincide in dimensions greater than n;thatis, ϑn F : ···→Fn+1 → Fn −→ Fn−1 → ··· →F0 → F−1 → F−2 →··· dn P : ···→Pn+1 → Pn −→ Pn−1 → ··· →P0 → M → 0 A ZG-module M is said to admit a complete resolution in the strong sense if there is a complete resolution (F, P,n)withHomZG(F,Q) acyclic for every ZG-projective module Q. It was shown by Cornick and Kropholler in [7] that if M admits a complete resolution (F, P,n) in the strong sense, then ∗ ∗ F ExtZG(M,B) H (HomZG( ,B)) ∗ ∗ where ExtZG(M, ) is the P-completion of ExtZG(M, ), defined by Mislin for any group G [13] as k k−r r ExtZG(M,B) = lim S ExtZG(M,B) r>k where S−mT is the m-th left satellite of a functor T . -
The Geometry of Syzygies
The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points . -
How to Make Free Resolutions with Macaulay2
How to make free resolutions with Macaulay2 Chris Peterson and Hirotachi Abo 1. What are syzygies? Let k be a field, let R = k[x0, . , xn] be the homogeneous coordinate ring of Pn and let X be a projective variety in Pn. Consider the ideal I(X) of X. Assume that {f0, . , ft} is a generating set of I(X) and that each polynomial fi has degree di. We can express this by saying that we have a surjective homogenous map of graded S-modules: t M R(−di) → I(X), i=0 where R(−di) is a graded R-module with grading shifted by −di, that is, R(−di)k = Rk−di . In other words, we have an exact sequence of graded R-modules: Lt F i=0 R(−di) / R / Γ(X) / 0, M |> MM || MMM || MMM || M& || I(X) 7 D ppp DD pp DD ppp DD ppp DD 0 pp " 0 where F = (f0, . , ft). Example 1. Let R = k[x0, x1, x2]. Consider the union P of three points [0 : 0 : 1], [1 : 0 : 0] and [0 : 1 : 0]. The corresponding ideals are (x0, x1), (x1, x2) and (x2, x0). The intersection of these ideals are (x1x2, x0x2, x0x1). Let I(P ) be the ideal of P . In Macaulay2, the generating set {x1x2, x0x2, x0x1} for I(P ) is described as a 1 × 3 matrix with gens: i1 : KK=QQ o1 = QQ o1 : Ring 1 -- the class of all rational numbers i2 : ringP2=KK[x_0,x_1,x_2] o2 = ringP2 o2 : PolynomialRing i3 : P1=ideal(x_0,x_1); P2=ideal(x_1,x_2); P3=ideal(x_2,x_0); o3 : Ideal of ringP2 o4 : Ideal of ringP2 o5 : Ideal of ringP2 i6 : P=intersect(P1,P2,P3) o6 = ideal (x x , x x , x x ) 1 2 0 2 0 1 o6 : Ideal of ringP2 i7 : gens P o7 = | x_1x_2 x_0x_2 x_0x_1 | 1 3 o7 : Matrix ringP2 <--- ringP2 Definition. -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
Depth, Dimension and Resolutions in Commutative Algebra
Depth, Dimension and Resolutions in Commutative Algebra Claire Tête PhD student in Poitiers MAP, May 2014 Claire Tête Commutative Algebra This morning: the Koszul complex, regular sequence, depth Tomorrow: the Buchsbaum & Eisenbud criterion and the equality of Aulsander & Buchsbaum through examples. Wednesday: some elementary results about the homology of a bicomplex Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. Claire Tête Commutative Algebra I will begin with a little example. Let us consider the ideal a = hX1, X2, X3i of A = k[X1, X2, X3]. What is "the" resolution of A/a as A-module? (the question is deliberatly not very precise) We would like to find something like this dm dm−1 d1 · · · Fm Fm−1 · · · F1 F0 A/a with A-modules Fi as simple as possible and s.t. Im di = Ker di−1. We say that F· is a resolution of the A-module A/a Claire Tête Commutative Algebra I will begin with a little example. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q. -
Computations in Algebraic Geometry with Macaulay 2
Computations in algebraic geometry with Macaulay 2 Editors: D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels Preface Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re- cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith- mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv- ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi- mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. -
Introduction to Algebraic Geometry
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 . 11 1.1.2 Hilbert's Nullstellensatz . 12 1.1.3 Zariski topology on an affine algebraic set . 14 1.1.4 Coordinate ring of an affine algebraic set . 16 1.2 Projective algebraic sets . 19 1.2.1 Definitions . 19 1.2.2 Homogeneous Nullstellensatz . 21 1.2.3 Homogeneous coordinate ring . 22 1.2.4 Exercise: plane curves . 22 1.3 Morphisms of algebraic sets . 24 1.3.1 Affine case . 24 1.3.2 Quasi-projective case . 26 2 The Language of schemes 29 2.1 Sheaves and locally ringed spaces . 29 2.1.1 Sheaves on a topological spaces . 29 2.1.2 Ringed space . 34 2.2 Schemes . 36 2.2.1 Definition of schemes . 36 2.2.2 Morphisms of schemes . 40 2.2.3 Projective schemes . 43 2.3 First properties of schemes and morphisms of schemes . 49 2.3.1 Topological properties . 49 2.3.2 Noetherian schemes . 50 2.3.3 Reduced and integral schemes . 51 2.3.4 Finiteness conditions . 53 2.4 Dimension . 54 2.4.1 Dimension of a topological space . 54 2.4.2 Dimension of schemes and rings . 55 2.4.3 The noetherian case . 57 2.4.4 Dimension of schemes over a field . 61 2.5 Fiber products and base change . 62 2.5.1 Sum of schemes . 62 2.5.2 Fiber products of schemes . -
THAN YOU NEED to KNOW ABOUT EXT GROUPS If a and G Are
MORE THAN YOU NEED TO KNOW ABOUT EXT GROUPS If A and G are abelian groups, then Ext(A; G) is an abelian group. Like Hom(A; G), it is a covariant functor of G and a contravariant functor of A. If we generalize our point of view a little so that now A and G are R-modules for some ring R, then n 0 we get not two functors but a whole sequence, ExtR(A; G), with ExtR(A; G) = HomR(A; G). In 1 n the special case R = Z, module means abelian group and ExtR is called Ext and ExtR is trivial for n > 1. I will explain the definition and some key properties in the case of general R. To be definite, let's suppose that R is an associative ring with 1, and that module means left module. If A and G are two modules, then the set HomR(A; G) of homomorphisms (or R-linear maps) A ! G has an abelian group structure. (If R is commutative then HomR(A; G) can itself be viewed as an R-module, but that's not the main point.) We fix G and note that A 7! HomR(A; G) is a contravariant functor of A, a functor from R-modules ∗ to abelian groups. As long as G is fixed, we sometimes denote HomR(A; G) by A . The functor also takes sums of morphisms to sums of morphisms, and (therefore) takes the zero map to the zero map and (therefore) takes trivial object to trivial object. -
Arxiv:1708.05877V4 [Math.AC]
NORMAL HYPERPLANE SECTIONS OF NORMAL SCHEMES IN MIXED CHARACTERISTIC JUN HORIUCHI AND KAZUMA SHIMOMOTO Abstract. The aim of this article is to prove that, under certain conditions, an affine flat normal scheme that is of finite type over a local Dedekind scheme in mixed characteristic admits infinitely many normal effective Cartier divisors. For the proof of this result, we prove the Bertini theorem for normal schemes of some type. We apply the main result to prove a result on the restriction map of divisor class groups of Grothendieck-Lefschetz type in mixed characteristic. Dedicated to Prof. Gennady Lyubeznik on the occasion of his 60th birthday. 1. Introduction Let X be a connected Noetherian normal scheme. Then does X have sufficiently many normal Cartier divisors? The existence of such a divisor when X is a normal projective variety over an algebraically closed field is already known. This fact follows from the classical Bertini theorem due to Seidenberg. In birational geometry, it is often necessary to compare the singularities of X with the singularities of a divisor D X, which is ⊂ known as adjunction (see [12] for this topic). In this article, we prove some results related to this problem in the mixed characteristic case. The main result is formulated as follows (see Corollary 5.4 and Remark 5.5). Theorem 1.1. Let X be a normal connected affine scheme such that there is a surjective flat morphism of finite type X Spec A, where A is an unramified discrete valuation ring → of mixed characteristic p > 0. Assume that dim X 2, the generic fiber of X Spec A ≥ → is geometrically connected and the residue field of A is infinite. -
Moving Codimension-One Subvarieties Over Finite Fields
Moving codimension-one subvarieties over finite fields Burt Totaro In topology, the normal bundle of a submanifold determines a neighborhood of the submanifold up to isomorphism. In particular, the normal bundle of a codimension-one submanifold is trivial if and only if the submanifold can be moved in a family of disjoint submanifolds. In algebraic geometry, however, there are higher-order obstructions to moving a given subvariety. In this paper, we develop an obstruction theory, in the spirit of homotopy theory, which gives some control over when a codimension-one subvariety moves in a family of disjoint subvarieties. Even if a subvariety does not move in a family, some positive multiple of it may. We find a pattern linking the infinitely many obstructions to moving higher and higher multiples of a given subvariety. As an application, we find the first examples of line bundles L on smooth projective varieties over finite fields which are nef (L has nonnegative degree on every curve) but not semi-ample (no positive power of L is spanned by its global sections). This answers questions by Keel and Mumford. Determining which line bundles are spanned by their global sections, or more generally are semi-ample, is a fundamental issue in algebraic geometry. If a line bundle L is semi-ample, then the powers of L determine a morphism from the given variety onto some projective variety. One of the main problems of the minimal model program, the abundance conjecture, predicts that a variety with nef canonical bundle should have semi-ample canonical bundle [15, Conjecture 3.12]. -
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.