DOCSLIB.ORG

Reflexivity Revisited

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Reflexivity Revisited REFLEXIVITY REVISITED MOHSEN ASGHARZADEH ABSTRACT. We study some aspects of reflexive modules. For example, we search conditions for which reflexive modules are free or close to free modules. 1. INTRODUCTION In this note (R, m, k) is a commutative noetherian local ring and M is a finitely generated R- module, otherwise specializes. The notation stands for a general module. For simplicity, M the notation ∗ stands for HomR( , R). Then is called reflexive if the natural map ϕ : M M M M is bijection. Finitely generated projective modules are reflexive. In his seminal paper M→M∗∗ Kaplansky proved that projective modules (over local rings) are free. The local assumption is really important: there are a lot of interesting research papers (even books) on the freeness of projective modules over polynomial rings with coefficients from a field. In general, the class of reflexive modules is extremely big compared to the projective modules. As a generalization of Seshadri’s result, Serre observed over 2-dimensional regular local rings that finitely generated reflexive modules are free in 1958. This result has some applications: For instance, in the arithmetical property of Iwasawa algebras (see [45]). It seems freeness of reflexive modules is subtle even over very special rings. For example, in = k[X,Y] [29, Page 518] Lam says that the only obvious examples of reflexive modules over R : (X,Y)2 are the free modules Rn. Ramras posed the following: Problem 1.1. (See [19, Page 380]) When are finitely generated reflexive modules free? Over quasi-reduced rings, problem 1.1 was completely answered (see Proposition 4.22). Ram- ras proved any finitely generated reflexive module over BNSI (betti numbers strictly increasing) rings is free. We present some applications of this result. Also, we introduce the class of eventually BNSI rings and we study freeness of reflexive modules over them. We know any nonzero free module decomposes into a direct sum of rank one submodules. Treger conjectured (see [50, Page arXiv:1812.00830v4 [math.AC] 23 Dec 2019 462]): Conjecture 1.2. Let (R, m, k) be a complete local (singular and containing a field) normal domain of dimension 2 where k = k and char(k) = 2. Then R is a cone such as k[[x,y,z]] if and only if 6 (x2+y2+z2) every nonzero reflexive module M decomposes into a direct sum of rank one submodules. 2 connects reflexivity to the finiteness conditions. As an application, we study the following: § When are quasi-reflexive modules flat? 3 collects different notions of reflexivity. Also, Propo- § sition 3.8 supports a question of Iyengar. 4 deals with freeness of reflexive modules over some § 2010 Mathematics Subject Classification. Primary 13D02; 13C10 . Key words and phrases. Betti numbers; free modules; reflexive modules; splitting. 1 2 classes of rings. In 5 we investigate the reflexivity of dual modules. In 6 we settle Conjecture § § 1.2. In 7 we deal with a question of Braun: Let I ✁ R be a reflexive ideal of a normal domain with § id (I) < ∞. Is I ω ? R ≃ R In 8 we descent freeness (resp. reflexivity) from the endomorphism ring to the module. This § is inspired by the paper of Auslander-Goldman. Similarly, we descent some data from the higher tensor products to the module. In particular, we slightly extend some results of Vasconcelos, Huneke-Wiegand and the recent work of Cesnaviˇcius.ˇ Grothendieck solved a conjecture of Samuel, see Theorem 9.1. In 9 we try to understand § this miracles by looking at the mentioned result of Auslander-Goldman. In particular, there is a connection between Problem 1.1 and UFD property of regular (complete-intersection) rings, see Corollary 9.3 as a sample. Samuel remarked that there is no symmetric analogue of Auslander’s theorem on torsion part of tensor powers. We present a tiny remark (also see Corollary 9.9): Observation. Let (R, m) be a regular local ring and M be of rank one. If Symn(M) is reflexive for some n max 2, dim R , then M is free. ≥ { } 2. REFLEXIVITY AND FINITENESS All rings are noetherian. Following Bass, is called torsion-less if ϕ is injective (this some M M times is called semi-reflexive). A torsion-less module is noetherian if and only if is noe- M M∗ therian. Submodules of a torsion-less module are torsion-less. We say is weakly reflexive if ϕ M M is surjective. In general, neither submodule nor quotient of a weakly reflexive module is weakly reflexive. Observation 2.1. Let (R, m) be a zero-dimensional Gorenstein local ring. Then the properties of weakly reflexive and finitely generated are the same. In particular, (weakly) reflexivity is not closed under taking direct limits. Proof. Here we use the concept of Gorenstein-projective. For its definition, see [14, 4.2.1]. Since R is zero-dimensional and Gorenstein, any module is Gorenstein-projective (see [14, 4.4.8]). It follows by definition that any module is a submodule of a projective module. Over local rings, and by the celebrated theorem of Kaplansky, any projective module is free. Combining these, any module is a submodule of a free module. It follows by definition that any module is torsion-less. Now, let be weakly reflexive. Thus, is reflexive. It is shown in [44, Corollary 2.4.(4)] that M M over any commutative artinian ring, every reflexive module is finitely generated. By this, is M finitely generated. Conversely, assume that M is finitely generated. Since R is zero-dimensional and Gorenstein, M is reflexive. To see the particular case, we remark that any module can be written as a directed limit of finitely generated modules. We use this along with the first part to get the claim. Discussion 2.2. The zero dimensional assumption is important. Indeed, by [29, Ex. 2.8’(2)], LN Z is reflexive. This is one of two extra-credit exercises in the book [29]. There are flat modules that are not reflexive, e.g. the vector space LN Q and the abelian group Q. In order to handle this drawback, let R be a normal domain of dimension bigger than zero with 3 fraction field Q(R). Following Samuel, is called quasi-reflexive if = Tp Spec(R),ht(p)=1 p M M ∈ M where we compute the intersection in := Q(R). Yuan proved that any flat module is M0 M ⊗R quasi-reflexive (see [57, Lemma 2]). Lemma 2.3. Let (R, m) be a normal domain of dimension bigger than zero and be quasi-reflexive. Then M there is a family of finitely generated and reflexive modules Mi i I such that = limMi. { } ∈ M −→ Proof. There is a directed family Ni i I of finitely generated submodules of such that = 1 { } ∈ M M limN . Let Spec (R) := p Spec(R), ht(p) = 1 . For each i, we set M := 1 (N )p i { ∈ } i Tp Spec (R) i −→ 1 ∈ where we compute the intersection in . Let p Spec (R). We note that M (N )p p M0 ∈ i ⊂ i ⊂ M and so Mi Tp Spec1(R) p = . Suppose Ni Nj. It follows that (Ni)p (Nj)p p. ⊂ ∈ M M ⊂ ⊂ ⊂ M Consequently, \ (N )p \ (N )p \ p = . i ⊂ j ⊂ M M p Spec1(R) p Spec1(R) p Spec1(R) ∈ ∈ ∈ This says that M M . Let x . There is a j I such that x N . Since N M , we i ⊂ j ⊂ M ∈ M ∈ ∈ j j ⊆ j have x M limM , i.e., limM . The reverse inclusion holds by definition. In sum, = ∈ j ⊂ j M ⊆ i M −→ −→ limMi. Clearly, Ni∗∗ is finitely generated. In view of Fact 5.1 (see below) Ni∗∗ is reflexive. Again, 1 let−→p Spec (R). Note that Rp is a discrete valuation ring. Over such a ring any torsion-free is ∈ free, and so reflexive. Since Ni is finitely generated, HomR(Ni, R) commutes with the localization. Thus, (Ni)p (Ni)p∗∗ (Ni∗∗)p. By e.g. [57, Proposition 1], Ni∗∗ Tp Spec1(R)(Ni∗∗)p. We put ≃ ≃ ≃ ∈ these together, N∗∗ \ (N∗∗)p \ (N )p. i ≃ i ≃ i p Spec1(R) p Spec1(R) ∈ ∈ From this, M N . In particular, M is finitely generated and reflexive. This completes the i ≃ i∗∗ i proof. Proposition 2.4. Let (R, m) be a regular local ring of dimension at most two. There is no difference between flat modules and quasi-reflexive modules. In particular, any direct limit of quasi-reflexive modules is quasi-reflexive. Proof. Recall that any flat module is quasi-reflexive. Conversely, let be quasi-reflexive. By M the above lemma, there is a family of finitely generated reflexive modules M such that = { i} M limMi. By the mentioned result of Serre, each Mi is free. Clearly, direct limit of free modules is flat.−→ We apply this to observe that is flat. To see the particular case, we mention that direct M limit of flat modules is again flat. Corollary 2.5. Let (R, m, k) be a normal domain of dimension bigger than zero. The following are equiva- lent: i) there is no difference between flat modules and quasi-reflexive modules, ii) R is a regular local ring of dimension at most two. Proof. i) ii): The second syzygy of k is reflexive and so quasi-reflexive. By the assumption it ⇒ is flat. Finitely generated flat modules over local rings are free. Thus, second syzygy of k is free. This in turn is equivalent with p. dim(k) 2. In view of local-global-principle, R is regular and ≤ of dimension at most two. 4 ii) i): This is in the previous Proposition.
Load more

Recommended publications
  • Modules and Vector Spaces
    Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module.
    [Show full text]
  • Pub . Mat . UAB Vol . 27 N- 1 on the ENDOMORPHISM RING of A
    Pub . Mat . UAB Vol . 27 n- 1 ON THE ENDOMORPHISM RING OF A FREE MODULE Pere Menal Throughout, let R be an (associative) ring (with 1) . Let F be the free right R-module, over an infinite set C, with endomorphism ring H . In this note we first study those rings R such that H is left coh- erent .By comparison with Lenzing's characterization of those rings R such that H is right coherent [8, Satz 41, we obtain a large class of rings H which are right but not left coherent . Also we are concerned with the rings R such that H is either right (left) IF-ring or else right (left) self-FP-injective . In particular we pro- ve that H is right self-FP-injective if and only if R is quasi-Frobenius (QF) (this is an slight generalization of results of Faith and .Walker [31 which assure that R must be QF whenever H is right self-i .njective) moreover, this occurs if and only if H is a left IF-ring . On the other hand we shall see that if' R is .pseudo-Frobenius (PF), that is R is an . injective cogenera- tortin Mod-R, then H is left self-FP-injective . Hence any PF-ring, R, that is not QF is such that H is left but not right self FP-injective . A left R-module M is said to be FP-injective if every R-homomorphism N -. M, where N is a -finitély generated submodule of a free module F, may be extended to F .
    [Show full text]
  • 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 .
    [Show full text]
  • On Fusion Categories
    Annals of Mathematics, 162 (2005), 581–642 On fusion categories By Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik Abstract Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not nec- essarily hermitian) modular category is unitary. We also show that the cate- gory of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion cat- egory. At the end of the paper we generalize some of these results to positive characteristic. 1. Introduction Throughout this paper (except for §9), k denotes an algebraically closed field of characteristic zero. By a fusion category C over k we mean a k-linear semisimple rigid tensor (=monoidal) category with finitely many simple ob- jects and finite dimensional spaces of morphisms, such that the endomorphism algebra of the neutral object is k (see [BaKi]). Fusion categories arise in several areas of mathematics and physics – conformal field theory, operator algebras, representation theory of quantum groups, and others. This paper is devoted to the study of general properties of fusion cate- gories. This has been an area of intensive research for a number of years, and many remarkable results have been obtained.
    [Show full text]
  • A NOTE on SHEAVES WITHOUT SELF-EXTENSIONS on the PROJECTIVE N-SPACE
    A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-SPACE. DIETER HAPPEL AND DAN ZACHARIA Abstract. Let Pn be the projective n−space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on Pn has a trivial endomorphism algebra. This generalizes a result of Drezet for n = 2: Let Pn be the projective n-space over the complex numbers. Recall that an exceptional sheaf is a coherent sheaf E such that Exti(E; E) = 0 for all i > 0 and End E =∼ C. Dr´ezetproved in [4] that if E is an indecomposable sheaf over P2 such that Ext1(E; E) = 0 then its endomorphism ring is trivial, and also that Ext2(E; E) = 0. Moreover, the sheaf is locally free. Partly motivated by this result, we prove in this short note that if E is an indecomposable coherent sheaf over the projective n-space such that Ext1(E; E) = 0, then we automatically get that End E =∼ C. The proof involves reducing the problem to indecomposable linear modules without self extensions over the polynomial algebra. In the second part of this article, we look at the Auslander-Reiten quiver of the derived category of coherent sheaves over the projective n-space. It is known ([10]) that if n > 1, then all the components are of type ZA1. Then, using the Bernstein-Gelfand-Gelfand correspondence ([3]) we prove that each connected component contains at most one sheaf. We also show that in this case the sheaf lies on the boundary of the component.
    [Show full text]
  • Vector Bundles on Projective Space
    Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X.
    [Show full text]
  • October 2013
    LONDONLONDON MATHEMATICALMATHEMATICAL SOCIETYSOCIETY NEWSLETTER No. 429 October 2013 Society MeetingsSociety 2013 ELECTIONS voting the deadline for receipt of Meetings TO COUNCIL AND votes is 7 November 2013. and Events Members may like to note that and Events NOMINATING the LMS Election blog, moderated 2013 by the Scrutineers, can be found at: COMMITTEE http://discussions.lms.ac.uk/ Thursday 31 October The LMS 2013 elections will open on elections2013/. Good Practice Scheme 10th October 2013. LMS members Workshop, London will be contacted directly by the Future elections page 15 Electoral Reform Society (ERS), who Members are invited to make sug- Friday 15 November will send out the election material. gestions for nominees for future LMS Graduate Student In advance of this an email will be elections to Council. These should Meeting, London sent by the Society to all members be addressed to Dr Penny Davies 1 page 4 who are registered for electronic who is the Chair of the Nominat- communication informing them ing Committee (nominations@lms. Friday 15 November that they can expect to shortly re- ac.uk). Members may also make LMS AGM, London ceive some election correspondence direct nominations: details will be page 5 from the ERS. published in the April 2014 News- Monday 16 December Those not registered to receive letter or are available from Duncan SW & South Wales email correspondence will receive Turton at the LMS (duncan.turton@ Regional Meeting, all communications in paper for- lms.ac.uk). Swansea mat, both from the Society and 18-21 December from the ERS. Members should ANNUAL GENERAL LMS Prospects in check their post/email regularly in MEETING Mathematics, Durham October for communications re- page 11 garding the elections.
    [Show full text]
  • FLAT MODULES OVER VALUATION RINGS 3 S1,...,Sn ∈ R(R) and Y1,...,Yn ∈ F Such That Φ(X) = S1y1 + ··· + Snyn
    FLAT MODULES OVER VALUATION RINGS FRANC¸OIS COUCHOT Abstract. Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent. A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect. In this paper, we consider the following properties of modules: P-flatness, flat- ness, content flatness, local projectivity, finite projectivity and single projectivity. We investigate the relations between these properties when R is a PP-ring or a valu- ation ring. Garfinkel ([11]), Zimmermann-Huisgen ([22]), and Gruson and Raynaud ([13]) introduced the concepts of locally projective modules and strongly coherent rings and developed important theories on these. The notions of finitely projec- tive modules and π-coherent rings are due to Jones ([15]).
    [Show full text]
  • Math 615: Lecture of April 16, 2007 We Next Note the Following Fact
    Math 615: Lecture of April 16, 2007 We next note the following fact: n Proposition. Let R be any ring and F = R a free module. If f1, . , fn ∈ F generate F , then f1, . , fn is a free basis for F . n n Proof. We have a surjection R F that maps ei ∈ R to fi. Call the kernel N. Since F is free, the map splits, and we have Rn =∼ F ⊕ N. Then N is a homomorphic image of Rn, and so is finitely generated. If N 6= 0, we may preserve this while localizing at a suitable maximal ideal m of R. We may therefore assume that (R, m, K) is quasilocal. Now apply n ∼ n K ⊗R . We find that K = K ⊕ N/mN. Thus, N = mN, and so N = 0. The final step in our variant proof of the Hilbert syzygy theorem is the following: Lemma. Let R = K[x1, . , xn] be a polynomial ring over a field K, let F be a free R- module with ordered free basis e1, . , es, and fix any monomial order on F . Let M ⊆ F be such that in(M) is generated by a subset of e1, . , es, i.e., such that M has a Gr¨obner basis whose initial terms are a subset of e1, . , es. Then M and F/M are R-free. Proof. Let S be the subset of e1, . , es generating in(M), and suppose that S has r ∼ s−r elements. Let T = {e1, . , es} − S, which has s − r elements. Let G = R be the free submodule of F spanned by T .
    [Show full text]
  • Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings
    City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2005 Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings Mahdi Majidi-Zolbanin Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/1765 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of NewYork. 2005 UMI Number: 3187456 Copyright 2005 by Majidi-Zolbanin, Mahdi All rights reserved. UMI Microform 3187456 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ii c 2005 Mahdi Majidi-Zolbanin All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy. Lucien Szpiro Date Chair of Examining Committee Jozek Dodziuk Date Executive Officer Lucien Szpiro Raymond Hoobler Alphonse Vasquez Ian Morrison Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iv Abstract Splitting of Vector Bundles on Punctured Spectrum of Regular Local Rings by Mahdi Majidi-Zolbanin Advisor: Professor Lucien Szpiro In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings.
    [Show full text]
  • 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.
    [Show full text]
  • The Depth Theory of Hopf Algebras and Smash Products
    The Depth Theory of Hopf Algebras and Smash Products Christopher J. Young Supervised by Dr. Lars Kadison November 25, 2016 To all my loved ones. Abstract The work done during this doctoral thesis involved advancing the theory of algebraic depth. Subfactor depth is a concept which had already existed for decades, but research papers discovered a purely algebraic analogue to the concept. The main uses of algebraic depth, which applies to a ring and subring pair, have been in Hopf-Galois theory, Hopf algebra actions and to some extent group theory. In this thesis we consider the application of algebraic depth to finite dimensional Hopf algebras, and smash products. This eventually leads to a striking discovery, a concept of module depth. Before explaining this work the thesis will go through many know results of depth up to this point historically. The most important result of the thesis: we discover a strong connection between the algebraic depth of a smash product A#H and the module depth of an H-module algebra A. In separate work L. Kadison discovered a connection between algebraic depth R ⊆ H for Hopf algebras and the module depth of V ∗, another important H-module algebra. The three concepts are related. Another important achievement of the work herein, we are able to calculate for the first time depth values of polynomial algebras, Taft algebras with certain subgroups and specific smash products of the Taft algebras. We also give a bound for the depth of R=I ⊆ H=I, which is extremely useful in general. Contents Background iii Contribution iii 1 The Depth Theory of a Ring Extension 1 1.1 Tensor Products .
    [Show full text]