F-Purity and Rational Singularity in Graded Complete Intersection Rings
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Asymptotic Behavior of the Length of Local Cohomology
ASYMPTOTIC BEHAVIOR OF THE LENGTH OF LOCAL COHOMOLOGY STEVEN DALE CUTKOSKY, HUY TAI` HA,` HEMA SRINIVASAN, AND EMANOIL THEODORESCU Abstract. Let k be a field of characteristic 0, R = k[x1, . , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. In this paper, we show that λ(H0 (R/In)) λ(Extd (R/In,R(−d))) lim m = lim R n→∞ nd n→∞ nd e(I) always exists. This limit has been shown to be for m-primary ideals I in a local Cohen Macaulay d! ring [Ki, Th, Th2], where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. Introduction Let R = k[x1, . , xd] be a polynomial ring over a field k, with graded maximal ideal m, and I ⊂ R d n a proper homogeneous ideal. We investigate the asymptotic growth of λ(ExtR(R/I ,R)) as a function of n. When R is a local Gorenstein ring and I is an m-primary ideal, then this is easily seen to be equal to λ(R/In) and hence is a polynomial in n. A theorem of Theodorescu and Kirby [Ki, Th, Th2] extends this to m-primary ideals in local Cohen Macaulay rings R. We consider homogeneous ideals in a polynomial ring which are not m-primary and show that a limit exists asymptotically although it can be irrational. -
Arxiv:1703.06832V3 [Math.AC]
REGULARITY OF FI-MODULES AND LOCAL COHOMOLOGY ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN Abstract. We resolve a conjecture of Ramos and Li that relates the regularity of an FI- module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. 1. Introduction Let S be a standard-graded polynomial ring in finitely many variables over a field k, and let M be a non-zero finitely generated graded S-module. It is a classical fact in commutative algebra that the following two quantities are equal (see [Ei, §4B]): S • The minimum integer α such that Tori (M, k) is supported in degrees ≤ α + i for all i. i • The minimum integer β such that Hm(M) is supported in degrees ≤ β − i for all i. i Here Hm is local cohomology at the irrelevant ideal m. The quantity α = β is called the (Castelnuovo–Mumford) regularity of M, and is one of the most important numerical invariants of M. In this paper, we establish the analog of the α = β identity for FI-modules. To state our result precisely, we must recall some definitions. Let FI be the category of finite sets and injections. Fix a commutative noetherian ring k. An FI-module over k is a functor from FI to the category of k-modules. We write ModFI for the category of FI-modules. We refer to [CEF] for a general introduction to FI-modules. Let M be an FI-module. Define Tor0(M) to be the FI-module that assigns to S the quotient of M(S) by the sum of the images of the M(T ), as T varies over all proper subsets of S. -
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. -
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. -
Automorphisms in Birational and Affine Geometry
Springer Proceedings in Mathematics & Statistics Ivan Cheltsov Ciro Ciliberto Hubert Flenner James McKernan Yuri G. Prokhorov Mikhail Zaidenberg Editors Automorphisms in Birational and A ne Geometry Levico Terme, Italy, October 2012 Springer Proceedings in Mathematics & Statistics Vo lu m e 7 9 For further volumes: http://www.springer.com/series/10533 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. Ivan Cheltsov • Ciro Ciliberto • Hubert Flenner • James McKernan • Yuri G. Prokhorov • Mikhail Zaidenberg Editors Automorphisms in Birational and Affine Geometry Levico Terme, Italy, October 2012 123 Editors Ivan Cheltsov Ciro Ciliberto School of Mathematics Department of Mathematics University of Edinburgh University of Rome Tor Vergata Edinburgh, United Kingdom Rome, Italy Hubert Flenner James McKernan Faculty of Mathematics Department of Mathematics Ruhr University Bochum University of California San Diego Bochum, Germany La Jolla, -
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. -
Arxiv:1802.06491V2 [Math.AC] 7 Apr 2018 R Theorem
TRACE IDEALS AND THE GORENSTEIN PROPERTY HAYDEE LINDO AND NINA PANDE Abstract. Let R be a local Noetherian commutative ring. We prove that R is an Artinian Gorenstein ring if and only if every ideal in R is a trace ideal. We discuss when the trace ideal of a module coincides with its double annihilator. 1. Introduction Let R be a ring and M an R-module. The trace ideal of M, denoted τ M (R), is the ideal generated by the homomorphic images of M in R. The theory of trace ideals has proved useful in various contexts but fundamentally the literature is dominated by two avenues of inquiry. First, given an R-module M what does its trace ideal say about M? For instance, it is known that trace ideals detect free-summands and that M is projective if and only if its trace ideal is idempotent; see [1, 7, 9, 16]. More recently, Lindo discussed the role of the trace ideal of a module in calculating the center of its endomorphism ring; see [10]. Also, Herzog, Hibi, Stamate and Ding have studied the trace ideal of the canonical module to understand deviation from the Gorenstein property in R; see [4, 8]. A second category of question asks: given a ring, what do the characteristics of its class of trace ideals imply about the ring? For example, in [6] Fontana, Huckaba and Papick characterize Noetherian domains where every trace ideal is prime; see also [6, 11, 12]. This paper addresses both of these questions when R is a local Artinian Gorenstein ring. -
Grothendieck Duality and Q-Gorenstein Morphisms 101
GROTHENDIECK DUALITY AND Q-GORENSTEIN MORPHISMS YONGNAM LEE AND NOBORU NAKAYAMA Abstract. The notions of Q-Gorenstein scheme and of Q-Gorenstein mor- phism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of Q-Gorenstein algebraic variety and of Q-Gorenstein deformation satisfying Koll´ar condition, over a field. By studies on relative S2-condition and base change properties, valuable results are proved for Q-Gorenstein morphisms, which include infinitesimal criterion, valuative criterion, Q-Gorenstein refine- ment, and so forth. Contents 1. Introduction 1 2. Serre’s Sk-condition 9 3. Relative S2-condition and flatness 24 4. Grothendieck duality 41 5. Relative canonical sheaves 63 6. Q-Gorenstein schemes 72 7. Q-Gorenstein morphisms 82 Appendix A. Some basic properties in scheme theory 105 References 108 1. Introduction arXiv:1612.01690v1 [math.AG] 6 Dec 2016 The notion of Q-Gorenstein variety is important for the minimal model theory of algebraic varieties in characteristic zero: A normal algebraic variety X defined over a field of any characteristic is said to be Q-Gorenstein if rKX is Cartier for some positive integer r, where KX stands for the canonical divisor of X. In some papers, X is additionally required to be Cohen–Macaulay. Reid used this notion essentially to define the canonical singularity in [48, Def. (1.1)], and he named the notion “Q- Gorenstein” in [49, (0.12.e)], where the Cohen–Macaulay condition is required. The notion without the Cohen–Macaulay condition appears in [24] for example.