Derived Functors of /-Adic Completion and Local Homology
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Derived Functors for Hom and Tensor Product: the Wrong Way to Do It
Derived Functors for Hom and Tensor Product: The Wrong Way to do It UROP+ Final Paper, Summer 2018 Kevin Beuchot Mentor: Gurbir Dhillon Problem Proposed by: Gurbir Dhillon August 31, 2018 Abstract In this paper we study the properties of the wrong derived functors LHom and R ⊗. We will prove identities that relate these functors to the classical Ext and Tor. R With these results we will also prove that the functors LHom and ⊗ form an adjoint pair. Finally we will give some explicit examples of these functors using spectral sequences that relate them to Ext and Tor, and also show some vanishing theorems over some rings. 1 1 Introduction In this paper we will discuss derived functors. Derived functors have been used in homo- logical algebra as a tool to understand the lack of exactness of some important functors; two important examples are the derived functors of the functors Hom and Tensor Prod- uct (⊗). Their well known derived functors, whose cohomology groups are Ext and Tor, are their right and left derived functors respectively. In this paper we will work in the category R-mod of a commutative ring R (although most results are also true for non-commutative rings). In this category there are differ- ent ways to think of these derived functors. We will mainly focus in two interpretations. First, there is a way to concretely construct the groups that make a derived functor as a (co)homology. To do this we need to work in a category that has enough injectives or projectives, R-mod has both. -
The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction 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/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all >> homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
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. -
Poisson Geometry, Deformation Quantisation and Group Representations
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin, Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford OX1 3LB, United Kingdom The title below are available from booksellers, or from Cambridge University Press at www.cambridge.org 161 Lectures on block theory, BURKHARD KULSHAMMER¨ 163 Topics in varieties of group representations, S.M. VOVSI 164 Quasi-symmetric designs, M.S. SHRlKANDE & S.S. SANE 166 Surveys in combinatorics, 1991, A.D. KEEDWELL (ed) 168 Representations of algebras, H. TACHIKAWA & S. BRENNER (eds) 169 Boolean function complexity, M.S. PATERSON (ed) 170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK 171 Squares, A.R. RAJWADE 172 Algebraic varieties, GEORGE R. KEMPF 173 Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds) 174 Lectures on mechanics, J.E. MARSDEN 175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) 176 Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) 177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) 178 Lower K- and L-theory, A. RANlCKl 179 Complex projective geometry, G. ELLlNGSRUD et al 180 Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT 181 Geometric group theory I, G.A. NlBLO & M.A. ROLLER (eds) 182 Geometric group theory II, G.A. NlBLO & M.A. ROLLER (eds) 183 Shintani Zeta Functions, A. YUKlE 184 Arithmetical functions, W. SCHWARZ & J. SPlLKER 185 Representations of solvable groups. O. MANZ & T.R. WOLF 186 Complexity: knots, colourings and counting, D.J.A. -
Agnieszka Bodzenta
June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY AGNIESZKA BODZENTA Contents 1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA 5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4. -
Intersection Homology Duality and Pairings: Singular, PL, and Sheaf
Intersection homology duality and pairings: singular, PL, and sheaf-theoretic Greg Friedman and James E. McClure July 22, 2020 Contents 1 Introduction 2 2 Conventions and some background 11 2.1 Pseudomanifolds and intersection homology . .. 11 3 Hypercohomology in the derived category 13 4 Some particular sheaf complexes 16 4.1 The Verdier dualizing complex and Verdier duality . 16 4.2 TheDelignesheaf................................. 17 4.3 The sheaf complex of intersection chains . .. 19 4.4 Thesheafofintersectioncochains . .. 20 ∗ 4.4.1 Ip¯C is quasi-isomorphic to PDp¯ ..................... 22 5 Sheafification of the cup product 24 5.1 Applications: A multiplicative de Rham theorem for perverse differential forms and compatibility with the blown-up cup product . 26 arXiv:1812.10585v3 [math.GT] 22 Jul 2020 5.1.1 Perversedifferentialforms . 26 5.1.2 Blown-up intersection cohomology . 31 6 Classical duality via sheaf maps 32 6.1 Orientationsandcanonicalproducts. ... 33 6.2 ProofofTheorem6.1............................... 35 7 Compatibility of cup, intersection, and sheaf products 50 8 Classical duality and Verdier duality 55 A Co-cohomology 59 1 B A meditation on signs 60 2000 Mathematics Subject Classification: Primary: 55N33, 55N45 Secondary: 55N30, 57Q99 Keywords: intersection homology, intersection cohomology, pseudomanifold, cup product, cap product, intersection product, Poincar´e duality, Verdier dual- ity, sheaf theory Abstract We compare the sheaf-theoretic and singular chain versions of Poincar´eduality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the sin- gular intersection cohomology cup product, the hypercohomology product induced by the Goresky-MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky- MacPherson PL intersection product. -
Arxiv:1910.05574V3 [Math.AT] 21 Jun 2021 2.7
REPRESENTATION STABILITY, SECONDARY STABILITY, AND POLYNOMIAL FUNCTORS JEREMY MILLER, PETER PATZT, AND DAN PETERSEN Abstract. We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two generalizations of classical homological stability theorems with twisted coefficients. We apply our results to prove homological stability for hyperelliptic mapping class groups with twisted coefficients, prove new representation stability results for congruence subgroups, establish secondary homological stability for groups of diffeomorphisms of surfaces viewed as discrete groups, and improve the known stable range for homological stability for general linear groups of the sphere spectrum. Contents 1. Introduction 2 1.1. Homological stability with polynomial coefficients2 1.2. Representation stability with polynomial coefficients3 1.3. Secondary homological stability with polynomial coefficients4 1.4. Stability for polynomial coefficients5 1.5. Applications 6 1.6. Outline of the paper9 1.7. Acknowledgments9 2. Categorical and algebraic preliminaries9 2.1. Tensor products as coends9 2.2. Stability categories 10 2.3. Central stability homology and degree-wise coherence 12 2.4. Stability short exact sequences 15 2.5. Rings, modules, and Tor groups 15 2.6. Splitting complexes and the Koszul complex 18 arXiv:1910.05574v3 [math.AT] 21 Jun 2021 2.7. Polynomial coefficient systems 21 3. Polynomial modules and derived representation stability 23 3.1. Via central stability complexes 23 3.2. Via the Koszul resolution 28 Date: June 22, 2021. Jeremy Miller was supported in part by NSF grant DMS-1709726 and a Simons Foundation Collaboration Grant. -
Right Exact Functors
Journal ofPure and Applied Algebra 4 (I 974) 1 —8. © North-Holland Publishing Company RIGHT EXACT FUNCTORS Michael BARR Department ofMathematics, McGill University, Montreal, Canada Communicated by R]. Hilton Received 20 January 1973 Revised 30 May 1973 0. Introduction Let X be a category with finite limits. We say that X is exact if: (1) whenever X1_’g X2 lfl X3_" X4 is a pullback diagram and f is a regular epimorphism (the coequalizer of some pair of maps), so is g; (2) wheneverE C X X X is an equivalence relation (see Section 1 below), then the two projection mapsE 3 X have a coequalizer; moreover, E is its kernel pair. When X lacks all finite limits, a somewhat finer definition can be given. We refer to [3, I(1.3)] for details. In this note, we explore some of the properties of right exact functors — those which preserve the coequalizers of equivalence relations. For functors which pre- serve kernel pairs, this is equivalent (provided that the domain is an exact category) to preserving regular epimorphisms. I had previously tried — in vain — to show that functors which preserve regular epirnorphisms have some of the nice properties which right exact functors do have. For example, what do you need to assume about a triple on an exact category to insure that the category of algebras is exact? The example of the torsion-free-quotient triple on abelian groups shows that pre- serving regular epimorphisms is not enough. On the other hand, as I showed in [3, 1.5.11], the algebras for a finitary theory in any exact category do form an exact category. -
Abstract Homotopy Theory, Generalized Sheaf Cohomology, Homotopical Algebra, Sheaf of Spectra, Homotopy Category, Derived Functor
TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 186, December 1973 ABSTRACTHOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the "homotopical algebra" of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in gen- eralized cohomology theory) and the "local to global" spectral sequence (in sheaf cohomology theory). Introduction. In this paper we will study the homotopy theory of sheaves of simplicial sets and sheaves of spectra. This homotopy theory will be used to give a derived functor definition of generalized sheaf cohomology groups H^iX, E), where X is a topological space and E is a sheaf of spectra on X, subject to certain finiteness conditions. These groups include as special cases the usual generalized cohomology of X defined by a spectrum [22] and the cohomology of X with coefficients in a complex of abelian sheaves. The cohomology groups have all the properties one would expect, the most important one being a spectral sequence Ep2q= HpiX, n_ E) =■»Hp+*iX, E), which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized coho- mology) and the "local to global" spectral sequence (in sheaf cohomology), and Received by the editors June 8, 1972 and, in revised form, October 15, 1972. -
Homological Algebra
MATH 613: HOMOLOGICAL ALGEBRA LECTURES BY PROF. HARM DERKSEN; NOTES BY ALEKSANDER HORAWA These are notes from the course Math 613: Homological Algebra taught by Prof. Harm Derksen in Winter 2017 at the University of Michigan. They were LATEX'd by Aleksander Horawa. This version is from July 1, 2017. Please check if a new version is available at my website https://sites.google.com/site/aleksanderhorawa/. If you find a mistake, please let me know at [email protected]. The textbook for this course was [Wei94], and the notes largely follow this book without specific reference. Citations are made where other resources were used. Contents 1. Review of category theory2 2. Algebraic topology 16 3. Homological algebra 22 4. Homological δ-functors 31 5. Projectives and left derived functors 33 6. Injectives and right derived functors 42 7. Limits 45 8. Sheaves and sheaf cohomology 47 9. Adjoint functors 50 10. Tor and Ext 55 11. Universal coefficients theorem 66 12. Quivers 69 13. Homological dimension 76 14. Local cohomology 89 15. Spectral sequences 90 16. Triangulated categories 106 17. Derived categories 115 References 120 1 2 HARM DERKSEN 1. Review of category theory We begin with a short review of the necessary category theory. Definition 1.1. A category C is (1) a class of objects, Obj C, and (2) for all A; B 2 Obj C, a set HomC(A; B) of morphisms from A to B, (3) for any A; B; C 2 Obj C, a composition map HomC(A; B) × HomC(B; C) ! HomC(A; C); (f; g) 7! g ◦ f = gf; (4) for any A 2 Obj C, a morphism idA 2 HomC(A; A), such that (a) for any A; B 2 Obj C and all f 2 HomC(A; B) idBf = f = fidA; (b) for any A; B; C; D 2 Obj C and any f : A ! B, g : B ! C, h: C ! D, the composi- tion is associative: (hg)f = h(gf): Note that Obj C may not be a set: for example, the category of sets cannot have the set of all objects (Russel paradox). -
Eilenberg-Watts Calculus for Finite Categories and a Bimodule Radford
ZMP-HH/16-26 Hamburger Beitr¨age zur Mathematik Nr. 630 Eilenberg-Watts calculus for finite categories and a bimodule Radford S4 theorem J¨urgen Fuchs a,c, Gregor Schaumann b, Christoph Schweigert c a Teoretisk fysik, Karlstads Universitet Universitetsgatan 21, S–65188 Karlstad b Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A–1090 Wien c Fachbereich Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D–20146 Hamburg Abstract We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne prod- ucts of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories over finite tensor categories. This leads to a generalization of Radford’s S4-theorem to bimodule categories. We also explain the relation of our construction to relative arXiv:1612.04561v3 [math.RT] 14 Nov 2019 Serre functors on module categories that are constructed via inner Hom functors. 1 Introduction A classical result in algebra, the Eilenberg-Watts theorem [Ei, Wa], states that, given two unital rings R and S, any right exact functor F : R-mod → S-mod that preserves small coproducts is naturally isomorphic to tensoring with an S-R-bimodule.