DOCSLIB.ORG

Chap 11: Dimension Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chap 11: Dimension Theory MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 67 11. Dimension theory The dimension of a Noetherian local ring can be defined in several ways. The main theorem is that all of these descriptions give the same number. So far we have the Krull dimension dim A which is the length of the maximal chain of prime ideals. 11.1. Poincar´eseries and Hilbert functions. Suppose that m is a maximal ideal of A and Gr(A)= A = A/m m/m2 m2/m3 n ⊕ ⊕ ⊕··· is the associated graded ring. If A is Noetherian then each An = n n+1 m /m is a finite dimensional vector space over the field k = A0 = A/m and we have the Poincar´eseries ∞ n PGr(A)(t)= t dimk An n=0 11.1.1. length function. In greater generality we consider a graded ring ∞ A = A = A A A n 0 ⊕ 1 ⊕ 2 ⊕··· n=0 so that A is Noetherian and A0 is Artinian. Suppose that ∞ M = M = M M M n 0 ⊕ 1 ⊕ 2 ⊕··· n=0 is a finitely generated graded module over A. Then each Mn is a finitely generated A0-module. Since A0 is Artinian, each Mn has finite length (Mn) and we can define the Poincar´eseries of M to be ∞ n PM (t)= (Mn)t n 0 − Recall that the length of an A0-module N is defined to be the largest integer m = (N) so that there is a filtration N = N0 N1 N2 Nm =0 ··· by A0-submodules Nn. It can be shown that any two maximal fil- trations have the same length. The subquotients Nn/Nn+1 are called composition factors of N. They are simple modules which are unique up to isomorphism and permutation of indices. Given this property it is easy to see that the length function satisfies the property that, for any short exact sequence of f.g. A0-modules 0 M M M 0 → → → → 68 MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA we have (M)=(M )+(M ). Any function on f.g. A0-modules with this property is called an additive function. Some trivial observations about the Poincar´eseries: (1) 0 PM (1) in general. (2) P ≤(1) = ≤∞(M ) < iff M = 0 for all but finitely many n. M n ∞ n (We are given that (Mn) < for all n 0.) ∞ ≥ (3) PM (1) = 0 iff M =0. 11.1.2. rational functions. Theorem 11.1. The Poincar´eseries of M is a rational function of the form f(t) PM (t)= s (1 tki ) i=1 − where f(t) is a polynomial in t with coefficients in .) Z Proof. The proof is by induction on s, the number of homogeneous generators xi of the ideal A+. If s = 0 then A = A0 and each Mn is an A-submodule of M. Since M is f.g. this implies that Mn = 0 for all but finitely many n. So n PM (t)=f(t)= (Mn)t is a polynomial in t with coefficients in Z. Now suppose that s 1 and consider the last generator xs Ak . ≥ ∈ s Multiplication by xs gives a graded map M M of degree ks.LetK be the kernel and L the cokernel of this map.→ Then, for each n 0we get an exact sequence: ≥ xs 0 Kn ks Mn ks Mn Ln 0 → − → − −→ → → where Kn ks = Mn ks =0forn<ks. Then − − (Kn ks ) (Mn ks )+(Mn) (Ln)=0 − − − − Multiply by tn n n n n (Kn ks )t (Mn ks )t + (Mn)t (Ln)t =0 − − − − Sum over all n tks P (t) tks P (t)+P (t) P (t)=0 K − M M − L So: ks PL(t) t PK (t) PM (t)= − 1 tks − Since xs annihilates K and L, these are graded modules over the graded ring A = A/(x ) whose positive grade ideal has s 1 generators s − x1, x2, , xs 1, the images of xi A. So, PL(t) and PK (t) are rational ··· − s 1∈ k functions with denominator − (1 t i ). The theorem follows. i=1 − MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 69 There is one special case when we can easily calculate f(t) by induc- tion. This is the case in which K = 0, i.e., x z = 0 for all z = 0 in M. s In this case we say that xs is M-regular. Then PM/xsM (t) PM (t)= 1 tks − Next, suppose that xs 1 is M/xsM-regular. Then − PM/xs 1xsM (t) − PM/xsM (t)= 1 tks 1 − − Definition 11.2. We say that (any sequence of elements of A) xk, ,xs is an M-regular sequence if this continues in the analogous way··· up to xk. When M = A, we call this a regular sequence for A. By induction we would get: PM/xkxk+1 xsM (t) PM (t)= s ··· (1 tki ) i=k − Example 11.3. Take A = M = K[x , ,x ] graded in the usual way 1 ··· d so that x1, ,xd span A1 = M1. Then x1, ,xd is an M-regular sequence and···M/x x M = K is 1-dimensional··· and 1 ··· d 1 n + d 1 P (t)= = − tn A (1 t)d d 1 − − Proof. (of the second equality) Start with 1 n 2 3 (1 t)− = t =1+t + t + t + − ··· and differentiate d 1 times to get − d n d+1 (d 1)!(1 t)− = n(n 1) (n d +2)t − − − − ··· − d n n d+1 n + d 1 n (1 t)− = t − = − t − d 1 d 1 − − ∞ nd 1 nd 2 = − − + tn (d 1)! − (d 3)! ··· n=0 − − 11.1.3. dimension using PM (t). Definition 11.4. Let d = d(M) be the smallest positive integer so that d lim(1 t) PM (t) < t 1 → − ∞ Proposition 11.5. d(M) s for all f.g. graded A-modules M. ≤ 70 MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA Proof. 1 t 1 1 − k = k 1 1 t 1+t + + t − → k as t 1. So, (1 t)s−P (t)convergesas··· t 1. → − M → Lemma 11.6. If x A is M-regular then d(M)=d(M/xM)+1. ∈ k Proposition 11.7. If A has an M-regular sequence of length r then d(M) r. ≥ In Example 11.3, d = s = r. So, d(K[x1, ,xd]) = d. The idea is that d(M) is the number of generators of A ···which act as independent variables on M. 11.1.4. Hilbert function. The Hilbert function gives a formula for (Mn) for large n. Theorem 11.8. Suppose that ki =1for i =1, 2, ,s. Then (Mn) is a polynomial of degree d(M) 1 for large n. ··· − Proof. By Theorem 11.1 we have: f(t) (M )tn = n (1 t)s − where f(t) is a polynomial. By definition of d = d(M) we have f(t)= s d (1 t) − g(t) where g(1) = 0. So, − g(t) n + d 1 (M )tn = = − tng(t) n (1 t)d d 1 − − m Suppose that g(t)= a tk = a + a t + + a tm. Then k=0 k 0 1 ··· m m n n + d 1 n+k (M )t = a − t n k d 1 k=0 − When n m we get ≥ m n k + d 1 (M )= a − − n k d 1 k=0 − (When n<mwe take the sum over 0 k n.) Since the binomial coefficient is a polynomial in n of degree≤d ≤1 with leading coefficient − 1/(d 1)! regardless of the value of k, this formula for (Mn) is a polynomial− in n of degree d 1 with leading coefficient − 1 m g(1) a = =0 (d 1)! k (d 1)! − k=0 − MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 71 Definition 11.9. The Hilbert function of M is the polynomial func- tion hM (n)=(Mn) for sufficiently large n. 11.1.5. Samuel function. For each n consider the finite sum: g(n)=(M0)+(M1)+ + (Mn 1). · − Lemma 11.10. g(n) is a polynomial of degree d(M) for sufficiently large n. Proof. For m>m0 we have d 1 (Mm)=hM (m)=c0 + c1m + + cd 1m − ··· − Therefore n 1 − g(n)= hM (m)+const m=0 n 1 k for n>m0. But, anyone who has taught Calculus knows that m−=0 m is a polynomial in n of degree k + 1. The lemma follows. Suppose that A is a Noetherian local ring with unique maximal ideal m and quotient field K = A/m.LetM be any f.g.A-module. For any n, M/mnM is a f.g. module over the Artin local ring A/mn. Therefore it has a finite length: n 2 n 1 n (M/m M) = dim (M/mM mM/m M m − M/m M) K ⊕ ⊕···⊕ Proposition 11.11. For sufficiently large n, (M/mnM) is a polyno- mial in n of degree s = dim (m/m2). ≤ K Proof. Take the associated graded ring Gr(A)=A/m m/m2 m2/m3 ⊕ ⊕ ⊕··· 2 2 3 has positive ideal Gr(A)+ = m/m m /m generated by s ele- ments of degree 1. The associated Gr⊕ (A)-module⊕··· Gr(M)=M/mM mM/m2M ⊕ ⊕··· is a f.g.Gr(A)-module with d(Gr(M)) s by Proposition 11.5. So, (M/mnM) is a polynomial of degree d≤(Gr(M)) s for sufficiently ≤ large n by the lemma above. Definition 11.12. The Samuel function for M is defined to be the polynomial function given for large n by: M n χm (n)=(M/m M) deg χM (n)=d(Gr(M)) dim (m/m2) m ≤ K 72 MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 11.2.
Load more

Recommended publications
  • Associated Graded Rings Derived from Integrally Closed Ideals And
    PUBLICATIONS MATHÉMATIQUES ET INFORMATIQUES DE RENNES MELVIN HOCHSTER Associated Graded Rings Derived from Integrally Closed Ideals and the Local Homological Conjectures Publications des séminaires de mathématiques et informatique de Rennes, 1980, fasci- cule S3 « Colloque d’algèbre », , p. 1-27 <http://www.numdam.org/item?id=PSMIR_1980___S3_1_0> © Département de mathématiques et informatique, université de Rennes, 1980, tous droits réservés. L’accès aux archives de la série « Publications mathématiques et informa- tiques de Rennes » implique l’accord avec les conditions générales d’utili- sation (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/ ASSOCIATED GRADED RINGS DERIVED FROM INTEGRALLY CLOSED IDEALS AND THE LOCAL HOMOLOGICAL CONJECTURES 1 2 by Melvin Hochster 1. Introduction The second and third sections of this paper can be read independently. The second section explores the properties of certain "associated graded rings", graded by the nonnegative rational numbers, and constructed using filtrations of integrally closed ideals. The properties of these rings are then exploited to show that if x^x^.-^x^ is a system of paramters of a local ring R of dimension d, d •> 3 and this system satisfies a certain mild condition (to wit, that R can be mapped
    [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]
  • The Structure Theory of Complete Local Rings
    The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings.
    [Show full text]
  • Lecture 3. Resolutions and Derived Functors (GL)
    Lecture 3. Resolutions and derived functors (GL) This lecture is intended to be a whirlwind introduction to, or review of, reso- lutions and derived functors { with tunnel vision. That is, we'll give unabashed preference to topics relevant to local cohomology, and do our best to draw a straight line between the topics we cover and our ¯nal goals. At a few points along the way, we'll be able to point generally in the direction of other topics of interest, but other than that we will do our best to be single-minded. Appendix A contains some preparatory material on injective modules and Matlis theory. In this lecture, we will cover roughly the same ground on the projective/flat side of the fence, followed by basics on projective and injective resolutions, and de¯nitions and basic properties of derived functors. Throughout this lecture, let us work over an unspeci¯ed commutative ring R with identity. Nearly everything said will apply equally well to noncommutative rings (and some statements need even less!). In terms of module theory, ¯elds are the simple objects in commutative algebra, for all their modules are free. The point of resolving a module is to measure its complexity against this standard. De¯nition 3.1. A module F over a ring R is free if it has a basis, that is, a subset B ⊆ F such that B generates F as an R-module and is linearly independent over R. It is easy to prove that a module is free if and only if it is isomorphic to a direct sum of copies of the ring.
    [Show full text]
  • CRITERIA for FLATNESS and INJECTIVITY 3 Ring of R
    CRITERIA FOR FLATNESS AND INJECTIVITY NEIL EPSTEIN AND YONGWEI YAO Abstract. Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting en- domorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity. 1. Introduction The most important classes of modules over a commutative Noetherian ring R, from a homological point of view, are the projective, flat, and injective modules. It is relatively easy to check whether a module is projective, via the well-known criterion that a module is projective if and only if it is locally free. However, flatness and injectivity are much harder to determine. R It is well-known that an R-module M is flat if and only if Tor1 (R/P,M)=0 for all prime ideals P . For special classes of modules, there are some criteria for flatness which are easier to check. For example, a finitely generated module is flat if and only if it is projective. More generally, there is the following Local Flatness Criterion, stated here in slightly simplified form (see [Mat86, Section 22] for a self-contained proof): Theorem 1.1 ([Gro61, 10.2.2]).
    [Show full text]
  • RINGS of MINIMAL MULTIPLICITY 1. Introduction We Shall Discuss Two Theorems of S. S. Abhyankar About Rings of Minimal Multiplici
    RINGS OF MINIMAL MULTIPLICITY J. K. VERMA Abstract. In this exposition, we discuss two theorems of S. S. Abhyankar about Cohen-Macaulay local rings of minimal multiplicity and graded rings of minimal multiplicity. 1. Introduction We shall discuss two theorems of S. S. Abhyankar about rings of minimal multiplicity which he proved in his 1967 paper \Local rings of high embedding dimension" [1] which appeared in the American Journal of Mathematics. The first result gives a lower bound on the multiplicity of the maximal ideal in a Cohen-Macaulay local ring and the second result gives a lower bound on the multiplicity of a standard graded domain over an algebraically closed field. The origins of these results lie in projective geometry. Let k be an algebraically closed field and X be a projective variety. We say that it is non-degenerate if it is not contained in a hyperplane. Let I(X) be the ideal of X. It is a homogeneous ideal of the polynomial ring S = k[x0; x1; : : : ; xr]: The homogeneous coordinate ring of X is defined as R = S(X) = S=I(X): Then R is a graded ring and 1 we write it as R = ⊕n=0Rn: Here Rn = Sn=(I(X) \ Sn): The Hilbert function of R is the function HR(n) = dimk Rn: Theorem 1.1 (Hilbert-Serre). There exists a polynomial PR(x) 2 Q[x] so that for all large n; HR(n) = PR(n): The degree of PR(x) is the dimension d of X and we can write x + d x + d − 1 P (x) = e(R) − e (R) + ··· + (−1)de (R): R d 1 d − 1 d Definition 1.2.
    [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]
  • Arxiv:1906.02669V2 [Math.AC] 6 May 2020 Ti O Nw Hte H Ulne-Etncnetr Od for Holds Conjecture Auslander-Reiten Theor the [19, Whether and Rings
    THE AUSLANDER-REITEN CONJECTURE FOR CERTAIN NON-GORENSTEIN COHEN-MACAULAY RINGS SHINYA KUMASHIRO Abstract. The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay local ring R with a parameter ideal Q, the Auslander-Reiten conjecture holds for R if and only if it holds for the residue ring R/Q. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring R/Qℓ in connection with that for R, and prove the equivalence of them for the case where R is Gorenstein and ℓ ≤ dim R. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our results, we see that the Auslander-Reiten conjecture holds if there exists an Ulrich ideal whose residue ring is a complete intersection. We also explore the Auslander-Reiten conjecture for determinantal rings. 1. Introduction The Auslander-Reiten conjecture and several related conjectures are problems about the vanishing of Ext modules. As is well-known, the vanishing of cohomology plays a very important role in the study of rings and modules. For a guide to these conjectures, one can consult [8, Appendix A] and [7, 16, 24, 25]. These conjectures originate from the representation theory of algebras, and the theory of commutative ring also greatly contributes to the development of the Auslander-Reiten conjecture; see, for examples, [1, 19, 20, 21]. Let us recall the Auslander-Reiten conjecture over a commutative Noetherian ring R. i Conjecture 1.1. [3] Let M be a finitely generated R-module.
    [Show full text]
  • Integral Closures of Ideals and Rings Irena Swanson
    Integral closures of ideals and rings Irena Swanson ICTP, Trieste School on Local Rings and Local Study of Algebraic Varieties 31 May–4 June 2010 I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found in Huneke–Swanson [13], but the lectures contain also more recent material. Table of contents: Section 1: Integral closure of rings and ideals 1 Section 2: Integral closure of rings 8 Section 3: Valuation rings, Krull rings, and Rees valuations 13 Section 4: Rees algebras and integral closure 19 Section 5: Computation of integral closure 24 Bibliography 28 1 Integral closure of rings and ideals (How it arises, monomial ideals and algebras) Integral closure of a ring in an overring is a generalization of the notion of the algebraic closure of a field in an overfield: Definition 1.1 Let R be a ring and S an R-algebra containing R. An element x S is ∈ said to be integral over R if there exists an integer n and elements r1,...,rn in R such that n n 1 x + r1x − + + rn 1x + rn =0. ··· − This equation is called an equation of integral dependence of x over R (of degree n). The set of all elements of S that are integral over R is called the integral closure of R in S.
    [Show full text]
  • Notes on Irreducible Ideals
    BULL. AUSTRAL. MATH. SOC. I3AI5, I3H05, I 4H2O VOL. 31 (1985), 321-324. NOTES ON IRREDUCIBLE IDEALS DAVID J. SMITH Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an Af-primary ideal of a local ring (if, M) satisfying the condition that Q : M = Q + M where s is the index of Q , then all irreducible components of Q have index s . (Q is "index- unmixed" .) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples. Introduction Let i? be a commutative ring with identity. An ideal J of R is irreducible if it is not a proper intersection of any two ideals of if . A discussion of some elementary properties of irreducible ideals is found in Zariski and Samuel [4] and Grobner [!]• If i? is Noetherian then every ideal of if has an irredundant representation as a finite intersection of irreducible ideals, and every irreducible ideal is primary. It is properties of representations of primary ideals as intersections of irreducible ideals which will be discussed here. We assume henceforth that if is Noetherian. Received 25 October 1981*. Copyright Clearance Centre, Inc. Serial-fee code: OOOl*-9727/85 $A2.00 + 0.00. 32 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 24 Sep 2021 at 06:01:50, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
    [Show full text]
  • Linear Resolutions Over Koszul Complexes and Koszul Homology
    LINEAR RESOLUTIONS OVER KOSZUL COMPLEXES AND KOSZUL HOMOLOGY ALGEBRAS JOHN MYERS Abstract. Let R be a standard graded commutative algebra over a field k, let K be its Koszul complex viewed as a differential graded k-algebra, and let H be the homology algebra of K. This paper studies the interplay between homological properties of the three algebras R, K, and H. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to K (and, more generally, to any connected differential graded k-algebra) and the other, called strand- Koszulness, which applies to H. The main theoretical result is a complete description of how these Koszul properties of R, K, and H are related to each other. This result shows that strand-Koszulness of H is stronger than Koszulness of R, and we include examples of classes of algebras which have Koszul homology algebras that are strand-Koszul. Introduction Koszul complexes are classical objects of study in commutative algebra. Their structure reflects many important properties of the rings over which they are de- fined, and these reflections are often encoded in the product structure of their homology. Indeed, Koszul complexes are more than just complexes — they are, in fact, the prototypical examples of differential graded (= DG) algebras in commu- tative ring theory, and thus the homology of a Koszul complex carries an algebra structure. These homology algebras encode (among other things) the Gorenstein condition [5], the Golod condition [15], and whether or not the ring is a complete intersection [1].
    [Show full text]
  • Supplementary Notes on Commutative Algebra1
    Supplementary Notes on Commutative Algebra1 April 11, 2017 1These notes are meant as a supplement to the Lectures on Commutative Algebra, that are avail- able online at http://www.math.iitb.ac.in/∼srg/Lecnotes/AfsPuneLecNotes.pdf and are primarily meant for the students of MA 526 at IIT Bombay in Spring 2017. The contents in these notes are mainly drawn from parts of Chapters 2 and 3 of Combinatorial Topology and Algebra, RMS Lecture Notes Series No. 18, Ramanujan Math. Soc., 2012. Contents 1 Graded Rings and Modules 3 1.1 Graded Rings . 3 1.2 Graded Modules . 5 1.3 Primary Decomposition in Graded Modules . 7 2 Hilbert Functions 10 2.1 Hilbert Function of a graded module . 10 2.2 Hilbert-Samuel polynomial of a local ring . 12 3 Dimension Theory 14 3.1 The dimension of a local ring . 14 3.2 The Dimension of an Affine Algebra . 18 3.3 The Dimension of a Graded Ring . 20 References 23 2 Chapter 1 Graded Rings and Modules This chapter gives an exposition of some rudimentary aspects of graded rings and mod- ules. It may be noted that exercises are embedded within the text and not listed separately at the end of the chapter. 1.1 Graded Rings In this section, we shall study some basic facts about graded rings. The study extends to graded modules, which are discussed in the next section. The notions and results in the first two sections will allow us to discuss, in Section 1.3, some special properties of associated primes and primary decomposition in the graded situation.
    [Show full text]