Local Duality for Structured Ring Spectra 2
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Local Cohomology and Matlis Duality – Table of Contents 0Introduction
Local Cohomology and Matlis duality { Table of contents 0 Introduction . 2 1 Motivation and General Results . 6 1.1 Motivation . 6 1.2 Conjecture (*) on the structure of Ass (D( h (R))) . 10 R H(x1;:::;xh)R h 1.3 Regular sequences on D(HI (R)) are well-behaved in some sense . 13 1.4 Comparison of two Matlis Duals . 15 2 Associated primes { a constructive approach . 19 3 Associated primes { the characteristic-free approach . 26 3.1 Characteristic-free versions of some results . 26 dim(R)¡1 3.2 On the set AssR(D(HI (R))) . 28 4 The regular case and how to reduce to it . 35 4.1 Reductions to the regular case . 35 4.2 Results in the general case, i. e. h is arbitrary . 36 n¡2 4.3 The case h = dim(R) ¡ 2, i. e. the set AssR(D( (k[[X1;:::;Xn]]))) . 38 H(X1;:::;Xn¡2)R 5 On the meaning of a small arithmetic rank of a given ideal . 43 5.1 An Example . 43 5.2 Criteria for ara(I) · 1 and ara(I) · 2 ......................... 44 5.3 Di®erences between the local and the graded case . 48 6 Applications . 50 6.1 Hartshorne-Lichtenbaum vanishing . 50 6.2 Generalization of an example of Hartshorne . 52 6.3 A necessary condition for set-theoretic complete intersections . 54 6.4 A generalization of local duality . 55 7 Further Topics . 57 7.1 Local Cohomology of formal schemes . 57 i 7.2 D(HI (R)) has a natural D-module structure . -
Semi-Dualizing Complexes and Their Auslander Categories
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 5, Pages 1839{1883 S 0002-9947(01)02627-7 Article electronically published on January 4, 2001 SEMI-DUALIZING COMPLEXES AND THEIR AUSLANDER CATEGORIES LARS WINTHER CHRISTENSEN Abstract. Let R be a commutative Noetherian ring. We study R{modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes \lying between" these extremes is incentive. Introduction In this paper all rings are commutative and Noetherian, in particular, R always denotes such a ring. We study R{complexes (that is, complexes of R{modules) with certain excellent duality properties. The canonical example is the ring itself, considered as a complex concentrated in degree zero. Another example is dualizing complexes for R; these were introduced by A. Grothendieck in [RAD] and have proved to be a powerful tool, as demonstrated by P. Roberts in [PRb76], and by C. Peskine and L. Szpiro in [CPsLSz73]. Modules with excellent duality properties have been studied by H.{B. Foxby in [HBF72], and in [EGl84] by E.S. Golod, who used the name suitable1 for these modules. Other complexes of the kind considered here were used tacitly by L.L. Avramov and H.{B. -
Characterizing Gorenstein Rings Using Contracting Endomorphisms
CHARACTERIZING GORENSTEIN RINGS USING CONTRACTING ENDOMORPHISMS BRITTNEY FALAHOLA AND THOMAS MARLEY Dedicated to Craig Huneke on the occasion of his 65th birthday. Abstract. We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov- Hochster-Iyengar-Yao (in the case of general contracting endomorphisms). 1. Introduction In 1969 Kunz [17] proved that a commutative Noetherian local ring of prime char- acteristic is regular if and only if some (equivalently, every) power of the Frobenius endomorphism is flat. Subsequently, the Frobenius map has been employed to great effect to study homological properties of commutative Noetherian local rings; see [20], [10], [23], [16], [4] and [3], for example. In this paper, we explore characteriza- tions of Gorenstein rings using the Frobenius map, or more generally, contracting endomorphisms. Results of this type have been obtained by Iyengar and Sather- Wagstaff [14], Goto [9], Rahmati [21], Marley [18], and others. Our primary goal is to obtain characterizations for a local ring to be Gorenstein in terms of one or more vanishings of derived functors in which one of the modules is viewed as an R-module by means of a contracting endomorphism (i.e., via restriction of scalars). A prototype of a result of this kind for regularity is given by Theorem 1.1 below. Let R be a commutative Noetherian local ring with maximal ideal m and φ : R ! R an endomorphism which is contracting, i.e., φi(m) ⊆ m2 for some i. -
Cohen-Macaulay Modules Over Gorenstein Local Rings
COHEN{MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS RYO TAKAHASHI Contents Introduction 1 1. Associated primes 1 2. Depth 3 3. Krull dimension 6 4. Cohen{Macaulay rings and Gorenstein rings 9 Appendix A. Proof of Theorem 1.12 11 Appendix B. Proof of Theorem 4.8 13 References 17 Introduction Goal. The structure theorem of (maximal) Cohen{Macaulay modules over commutative Gorenstein local rings Throughout. • (R : a commutative Noetherian ring with 1 A = k[[X; Y ]]=(XY ) • k : a field B = k[[X; Y ]]=(X2;XY ) • x := X; y := Y 1. Associated primes Def 1.1. Let I ( R be an ideal of R (1) I is a maximal ideal if there exists no ideal J of R with I ( J ( R (2) I is a prime ideal if (ab 2 I ) a 2 I or b 2 I) (3) Spec R := fPrime ideals of Rg Ex 1.2. Spec A = f(x); (y); (x; y)g; Spec B = f(x); (x; y)g Prop 1.3. I ⊆ R an ideal (1) I is prime iff R=I is an integral domain (2) I is maximal iff R=I is a field 2010 Mathematics Subject Classification. 13C14, 13C15, 13D07, 13H10. Key words and phrases. Associated prime, Cohen{Macaulay module, Cohen{Macaulay ring, Ext module, Gorenstein ring, Krull dimension. 1 2 RYO TAKAHASHI (3) Every maximal ideal is prime Throughout the rest of this section, let M be an R-module. Def 1.4. p 2 Spec R is an associated prime of M if 9 x 2 M s.t p = ann(x) AssR M := fAssociated primes of Mg Ex 1.5. -
Local Cohomology and Base Change
LOCAL COHOMOLOGY AND BASE CHANGE KAREN E. SMITH f Abstract. Let X −→ S be a morphism of Noetherian schemes, with S reduced. For any closed subscheme Z of X finite over S, let j denote the open immersion X \ Z ֒→ X. Then for any coherent sheaf F on X \ Z and any index r ≥ 1, the r sheaf f∗(R j∗F) is generically free on S and commutes with base change. We prove this by proving a related statement about local cohomology: Let R be Noetherian algebra over a Noetherian domain A, and let I ⊂ R be an ideal such that R/I is finitely generated as an A-module. Let M be a finitely generated R-module. Then r there exists a non-zero g ∈ A such that the local cohomology modules HI (M)⊗A Ag are free over Ag and for any ring map A → L factoring through Ag, we have r ∼ r HI (M) ⊗A L = HI⊗AL(M ⊗A L) for all r. 1. Introduction In his work on maps between local Picard groups, Koll´ar was led to investigate the behavior of certain cohomological functors under base change [Kol]. The following theorem directly answers a question he had posed: f Theorem 1.1. Let X → S be a morphism of Noetherian schemes, with S reduced. Suppose that Z ⊂ X is closed subscheme finite over S, and let j denote the open embedding of its complement U. Then for any coherent sheaf F on U, the sheaves r f∗(R j∗F) are generically free and commute with base change for all r ≥ 1. -
Arxiv:1802.08409V2 [Math.AC]
CORRESPONDENCE BETWEEN TRACE IDEALS AND BIRATIONAL EXTENSIONS WITH APPLICATION TO THE ANALYSIS OF THE GORENSTEIN PROPERTY OF RINGS SHIRO GOTO, RYOTARO ISOBE, AND SHINYA KUMASHIRO Abstract. Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties. Contents 1. Introduction 1 2. Correspondence between trace ideals and birational extensionsofthebasering 5 3. The case where R is a Gorenstein ring of dimension one 9 4. Modules in which every submodule is a trace module 13 5. Surjectivity of the correspondence ρ in dimension one 17 6. Anti-stable rings 21 References 25 1. Introduction This paper aims to explore the structure of (not necessarily Noetherian) commutative arXiv:1802.08409v2 [math.AC] 7 Dec 2018 rings in connection with their trace ideals. Let R be a commutative ring. For R-modules M and X, let τM,X : HomR(M,X) ⊗R M → X denote the R-linear map defined by τM,X (f ⊗ m) = f(m) for all f ∈ HomR(M,X) and m ∈ M. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Ideals Generated by Pfaffians P(P, 72) =
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Ideals Generated by Pfaffians Instituie of .lfatlrenzatics, Polish Academy of Sciences, CJropinu 12, 87-100 Torwi, Poland AND PIOTR PRAGACZ Institute of Mathematics, Nicholas Copernicus University, Chopina 12, 87-100 Ton&, Poland Communicated by D. Buchsbaum Received September 7, 1978 Influenced by the structure theorem of Buchsbaum and Eisenbud for Gorenstein ideals of depth 3, [4], we study in this paper ideals generated by Pfaffians of given order of an alternating matrix. Let ,x7 be an n by n alternating matrix (i.e., xi9 = --xji for i < j and xii = 0) with entries in a commutative ring R with identity. One can associate with X an element Pf(-k-) of R called Pfaffian of X (see [I] or [2] for the definition). For n odd det S ~-=0 and Pf(X) = 0; for n even det X is a square in R and Pf(X)z = det X. For a basis-free account and basic properties ofpfaffians we send interested readers to Chapter 2 of [4] or to [IO]. For a sequence i1 ,..., &, 1 < i r < n, the matrix obtained from X by omitting rows and columns with indices i1 ,..., ik is again alternating; we write Pf il*....ik(X) for its Pfaffian and call it the (n - k)-order Pfaffian of X. We are interested in ideals Pf,,(X) generated by all the 2p-order Pfaffians of X, 0 ,( 2p < n. After some preliminaries in Section 1 we prove in Section 2 that the height and the depth of E-“&X) are bounded by the number P(P, 72) = (12- 2p + l)(n - 2p + 2)/2. -
Gorenstein Rings Examples References
Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Gorenstein Rings Chau Chi Trung Bachelor Thesis Defense Presentation Supervisor: Dr. Tran Ngoc Hoi University of Science - Vietnam National University Ho Chi Minh City May 2018 Email: [email protected] Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Content 1 Origin 2 Aim of the Thesis 3 Structure of Minimal Injective Resolution 4 Gorenstein Rings 5 Examples 6 References Origin ∙ Grothendieck introduced the notion of Gorenstein variety in algebraic geometry. ∙ Serre made a remark that rings of finite injective dimension are just Gorenstein rings. The remark can be found in [9]. ∙ Gorenstein rings have now become a popular notion in commutative algebra and given birth to several definitions such as nearly Gorenstein rings or almost Gorenstein rings. Aim of the Thesis This thesis aims to 1 present basic results on the minimal injective resolution of a module over a Noetherian ring, 2 introduce Gorenstein rings via Bass number and 3 answer elementary questions when one inspects a type of ring (e.g. Is a subring of a Gorenstein ring Gorenstein?). Origin Aim of the Thesis Structure of Minimal Injective Resolution Gorenstein Rings Examples References Structure of Minimal Injective Resolution Unless otherwise specified, let R be a Noetherian commutative ring with 1 6= 0 and M be an R-module. Theorem (E. Matlis) Let E be a nonzero injective R-module. Then we have a direct sum ∼ decomposition E = ⊕i2I Xi in which for each i 2 I, Xi = ER(R=P ) for some P 2 Spec(R). -
Lectures on Local Cohomology
Contemporary Mathematics Lectures on Local Cohomology Craig Huneke and Appendix 1 by Amelia Taylor Abstract. This article is based on five lectures the author gave during the summer school, In- teractions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Grothendieck concerning the connectedness of the spectrum of certain rings. Several applications are given, including a theorem of Fulton and Hansen concern- ing the connectedness of intersections of algebraic varieties. In an appendix written by Amelia Taylor, an another application is given to prove a theorem of Kalkbrenner and Sturmfels about the reduced initial ideals of prime ideals. Contents 1. Introduction 1 2. Local Cohomology 3 3. Injective Modules over Noetherian Rings and Matlis Duality 10 4. Cohen-Macaulay and Gorenstein rings 16 d 5. Vanishing Theorems and the Structure of Hm(R) 22 6. Vanishing Theorems II 26 7. Appendix 1: Using local cohomology to prove a result of Kalkbrenner and Sturmfels 32 8. Appendix 2: Bass numbers and Gorenstein Rings 37 References 41 1. Introduction Local cohomology was introduced by Grothendieck in the early 1960s, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization 2000 Mathematics Subject Classification. Primary 13C11, 13D45, 13H10. Key words and phrases. local cohomology, Gorenstein ring, initial ideal. The first author was supported in part by a grant from the National Science Foundation, DMS-0244405. -
Self-Tests for Freeness Over Commutative Artinian Rings
SELF-TESTS FOR FREENESS OVER COMMUTATIVE ARTINIAN RINGS LIANA M. S¸EGA Abstract. We prove that the Auslander-Reiten conjecture holds for commu- tative standard graded artinian algebras, in two situations: The first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3. introduction Conjectures of homological nature play an important role in commutative algebra and representation theory of algebras. We refer to Christensen and Holm [7] and Happel [10] for background material on a cluster of such conjectures. Originally formulated over Artin algebras, many of these statements have meaningful variants for arbitrary noetherian commutative rings. In this paper we are mainly interested in the Auslander-Reiten conjecture, stated in [2] in connection to the Nakayama conjecture. Let R be a commutative noetherian ring. A commutative version of the Auslander- Reiten conjecture is stated by Auslander, Ding and Solberg [1] as follows: If a i i finitely generated R-module M satisfies ExtR(M; M) = 0 = ExtR(M; R) for all i > 0, then M is projective. It is known that the Auslander-Reiten conjecture follows from a conjecture at- tributed to Auslander; see [7]. However, Jorgensen and S¸ega [15] found counterex- amples to Auslander's conjecture over commutative artinian rings. In this paper we prove that the Auslander-Reiten conjecture does hold in the cases where the Auslander conjecture was shown to fail. The rings constructed in [15] are standard graded artinian Koszul algebras and the modules are graded, with a linear resolution; in one of the examples the maximal ideal m of R satisfies m3 = 0 and in another example the ring is Gorenstein with m4 = 0. -
Advanced Commutative Algebra Lecture Notes
Advanced Commutative Algebra Lecture Notes Lecturer: Paul Smith; written and edited by Josh Swanson September 30, 2015 Abstract The following notes were taking during a course on Advanced Commutative Algebra at the University of Washington in Fall 2014. Please send any corrections to [email protected]. Thanks! Contents September 24th, 2014: Tensor Product Definition, Universal Property................2 September 26th, 2014: Tensor-Hom Adjunction, Right Exactness of Tensor Products, Flatness..4 September 29th, 2014: Tensors and Quotients, Support of a Module.................6 October 1st, 2014: Prime Avoidance Lemma, Associated Primes...................9 October 3rd, 2014: Associated Primes Continued, Minimal Primes, Zero-Divisors, Regular Elements 10 October 6th, 2014: Duality Theories, Injective Envelopes, and Indecomposable Injectives..... 12 October 8th, 2014: Local Cohomology Defined............................. 14 October 10th, 2014: Examples of Injectives and Injective Hulls.................... 16 October 13th, 2014: Local Cohomology and Depth through Ext; Matlis Duality........... 18 October 15th, 2014: Matlis Duality Continued............................. 20 October 17th, 2014: Local Cohomology and Injective Hulls Continued................ 22 October 20th, 2014: Auslander-Buchsbaum Formula; Nakayama's Lemma.............. 24 October 22nd, 2014: Dimension Shifting, Rees' Theorem, Depth and Regular Sequences...... 26 October 27th, 2014: Krull Dimension of Modules............................ 28 October 29th, 2014: Local Cohomology over