From Sheaf Cohomology to the Algebraic De Rham Theorem
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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. -
The Calabi Complex and Killing Sheaf Cohomology
The Calabi complex and Killing sheaf cohomology Igor Khavkine Department of Mathematics, University of Trento, and TIFPA-INFN, Trento, I{38123 Povo (TN) Italy [email protected] September 26, 2014 Abstract It has recently been noticed that the degeneracies of the Poisson bra- cket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of dif- ferential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of lo- cally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of general- ized Poincar´eduality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds. Contents 1 Introduction2 2 The Calabi complex4 2.1 Tensor bundles and Young symmetrizers..............5 2.2 Differential operators.........................7 2.3 Formal adjoint complex....................... 11 2.4 Equations of finite type, twisted de Rham complex........ 14 3 Cohomology of locally constant sheaves 16 3.1 Locally constant sheaves....................... 16 3.2 Acyclic resolution by a differential complex............ 18 3.3 Generalized Poincar´eduality................... -
Sheaf Theory
Sheaf Theory Anne Vaugon December 20, 2013 The goals of this talk are • to define a generalization denoted by RΓ(F) of de Rham cohomology; • to explain the notation RΓ(F) (here F is a sheaf and RΓ is a derived functor). 1 Presheaves and sheaves 1.1 Definitions and examples Let X be a topological space. Definition 1.1. A presheaf of k-modules F on X is defined by the following data: • a k-module F(U) for each open set U of X; • a map rUV : F(U) → F(V ) for each pair V ⊂ U of open subsets such that – rWV ◦ rVU = rWU for all open subsets W ⊂ V ⊂ U; – rUU = Id for all open subsets U. Therefore, a presheaf is a functor from the opposite category of open sets to the category of k-modules. If F is a presheaf, F(U) is called the set of sections of U and rVU the restriction from U to V . Definition 1.2. A presheaf F is a sheaf if • for any family (Ui)i∈I of open subsets of X • for any family of elements si ∈ F(Ui) such that rUi∩Uj ,Ui (si) = rUi∩Uj ,Uj (sj) for all i, j ∈ I there exists a unique s ∈ F(U) where U = ∪i∈I Ui such that rUi,U (s) = si for all i ∈ I. This means that we can extend a locally defined section. Definition 1.3. A morphism of presheaves f : F → G is a natural trans- formation between the functors F and G: for each open set U, there exists a morphism f(U): F(U) → G(U) such that the following diagram is commutative for V ⊂ U. -
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. -
SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter. -
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. -
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:1710.09830V1 [Math.AC] 26 Oct 2017
COMPUTATIONS OVER LOCAL RINGS IN MACAULAY2 MAHRUD SAYRAFI Thesis Advisor: David Eisenbud Abstract. Local rings are ubiquitous in algebraic geometry. Not only are they naturally meaningful in a geometric sense, but also they are extremely useful as many problems can be attacked by first reducing to the local case and taking advantage of their nice properties. Any localization of a ring R, for instance, is flat over R. Similarly, when studying finitely generated modules over local rings, projectivity, flatness, and freeness are all equivalent. We introduce the packages PruneComplex, Localization and LocalRings for Macaulay2. The first package consists of methods for pruning chain complexes over polynomial rings and their localization at prime ideals. The second package contains the implementation of such local rings. Lastly, the third package implements various computations for local rings, including syzygies, minimal free resolutions, length, minimal generators and presentation, and the Hilbert–Samuel function. The main tools and procedures in this paper involve homological methods. In particular, many results depend on computing the minimal free resolution of modules over local rings. Contents I. Introduction 2 I.1. Definitions 2 I.2. Preliminaries 4 I.3. Artinian Local Rings 6 II. Elementary Computations 7 II.1. Is the Smooth Rational Quartic a Cohen-Macaulay Curve? 12 III. Other Computations 13 III.1. Computing Syzygy Modules 13 arXiv:1710.09830v1 [math.AC] 26 Oct 2017 III.2. Computing Minimal Generators and Minimal Presentation 16 III.3. Computing Length and the Hilbert-Samuel Function 18 IV. Examples and Applications in Intersection Theory 20 V. Other Examples from Literature 22 VI. -
2 Sheaves and Cohomology
2 Sheaves and Cohomology 2.1 Sheaves and Presheaves We fix a topological space X. Later we will include assumptions that are satisfied by smooth manifolds. 2.1.1 Definitions and Examples Definition 2.1. A presheaf of abelian groups F on X assigns to each open U ⊆ X an abelian group F (U) = Γ(U; F ) and for every inclusion of open sets V ⊆ U a homomorphism of abelian groups F ρUV : F (U) ! F (V ), often called the restriction map, satisfying F 1 [P1] ρUU = F(U) F F F [P2] for W ⊆ V ⊆ U, we have ρVW ◦ ρUV = ρUW . If F and G are two presheaves (of abelian groups) on X, then a morphism ' : F ! G consists of the data of a morphism 'U : F (U) ! G (U) for each open set U ⊆ X such that if V ⊆ U is an inclusion, then we have commutative diagrams 'U F (U) / G (U) F G ρUV ρUV (V ) / (V ): F 'V G Remark 2.2. We may form a category TopX whose objects are open sets in X and whose mor- phisms are simply inclusions of open sets. Then the above definition says that a presheaf is a ◦ contravariant functor TopX ! Ab, and that a morphism of presheaves is a natural transforma- tion of the associated functors. Definition 2.3. A sheaf F of abelian groups on X is a presheaf which, for any open set U ⊆ X and any open covering fUigi2I of U, satisfies the two additional properties: [S1] if s 2 F (U) is such that sjUi = 0 for all i 2 I, then s = 0; [S2] if si 2 F (Ui) such that sijUi\Uj = sjjUi\Uj for all i; j 2 I, then there exists s 2 F (U) such that sjUi = si for each i. -
On Sheaf Theory
Lectures on Sheaf Theory by C.H. Dowker Tata Institute of Fundamental Research Bombay 1957 Lectures on Sheaf Theory by C.H. Dowker Notes by S.V. Adavi and N. Ramabhadran Tata Institute of Fundamental Research Bombay 1956 Contents 1 Lecture 1 1 2 Lecture 2 5 3 Lecture 3 9 4 Lecture 4 15 5 Lecture 5 21 6 Lecture 6 27 7 Lecture 7 31 8 Lecture 8 35 9 Lecture 9 41 10 Lecture 10 47 11 Lecture 11 55 12 Lecture 12 59 13 Lecture 13 65 14 Lecture 14 73 iii iv Contents 15 Lecture 15 81 16 Lecture 16 87 17 Lecture 17 93 18 Lecture 18 101 19 Lecture 19 107 20 Lecture 20 113 21 Lecture 21 123 22 Lecture 22 129 23 Lecture 23 135 24 Lecture 24 139 25 Lecture 25 143 26 Lecture 26 147 27 Lecture 27 155 28 Lecture 28 161 29 Lecture 29 167 30 Lecture 30 171 31 Lecture 31 177 32 Lecture 32 183 33 Lecture 33 189 Lecture 1 Sheaves. 1 onto Definition. A sheaf S = (S, τ, X) of abelian groups is a map π : S −−−→ X, where S and X are topological spaces, such that 1. π is a local homeomorphism, 2. for each x ∈ X, π−1(x) is an abelian group, 3. addition is continuous. That π is a local homeomorphism means that for each point p ∈ S , there is an open set G with p ∈ G such that π|G maps G homeomorphi- cally onto some open set π(G).