Cohen-Macaulay Ring Research Articles

STRUCTURE OF THE FLAT COVERS OF ARTINIAN MODULES

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form <TEX>$\prod_{p\epsilonAtt_R(A)}T-p$</TEX>, where <TEX>$Tp$</TEX> is the completion of a free <TEX>R$_{p}$</TEX>-module. Also, we construct a minimal flat resolution for R/xR-module 0: <TEX>$_AX$</TEX> from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = <TEX>$\chiA$</TEX>. This result leads to a description of the structure of a minimal flat resolution for <TEX>${H^n}_{\underline{m}}(R)$</TEX>, nth local cohomology module of R with respect to the ideal <TEX>$\underline{m}$</TEX>, over a local Cohen-Macaulay ring (R, <TEX>$\underline{m}$</TEX>) of dimension n.

Tensor products of Cohen–Macaulay rings: solution to a problem of Grothendieck

In this paper we solve a problem, originally raised by Grothendieck, on the transfer of Cohen–Macaulayness to tensor products of algebras over a field k . As a prelude to this, we investigate the grade for some specific types of ideals that play a primordial role within the ideal structure of such constructions.

The Bound of the Difference between Parameter Ideals and their Tight Closures

Let R be a Noetherian local ring with the maximal ideal m and assume that R possesses positive characteristic p > 0. For each m-primary ideal I in R let eI (R) and I∗ denote the multiplicity of R with respect to I and the tight closure of I , respectively. Here let us briefly recall the definition of tight closures. For an ideal a in R let a[q] = (a | a ∈ a)R where q = p with e ≥ 0. Let R0 = R ⋃ ∈MinR p. Let a∗ denote the set of elements x ∈ R for which there exists c ∈ R0 such that cx ∈ a[q] for all q 0. Then the set a∗ forms an ideal in R containing a, which we call the tight closure of a. The ideal a is said to be tightly closed if a∗ = a. A local ring R is called F -rational if every parameter ideal in R is tightly closed. In this paper we investigate the behavior of sup R(q ∗/q) and sup {e (R)− R(R/q∗)}, where q moves all parameter ideals in R. These two values are very closely related. In fact, they agree if R is Cohen-Macaulay. By definition of F -rationality, it immediately follows that sup R(q ∗/q) = 0 if and only if R is F -rational. The second author studied sup R(q∗/q) in [N2] and he gave some necessary conditins for sup R(q ∗/q) to be finite. On the other hand, Watanabe and Yoshida posed in [WY] the conjecture that the difference e (R) − R(R/q∗) is not negative for every prameter ideal q in R if R is unmixed. Moreover, they conjectured that e (R) − R(R/q∗) = 0 for some parameter ideal q, then R is Cohen-Macaulay and F -rational. This problem is affirmatively solved in [GN] under the condition that R is a homomorphic image of a Cohen-Macaulay ring of chracteristic p > 0 and AssR = AsshR. We further investigate the finiteness of the supremum of the difference e (R) − R(R/q∗) in this paper. The first result gives a necessary and sufficient condition for sup R(q ∗/q) to be finite. Namely,

Restricted Homological Dimensions and Cohen–Macaulayness

The classical homological dimensions—the projective, flat, and injective ones—are usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors. In this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test modules that are restricted to assure automatic finiteness over commutative Noetherian rings of finite Krull dimension. When the ring is local, we use a mixture of methods from classical commutative algebra and the theory of homological dimensions to show that vanishing of these functors reveals that the underlying ring is a Cohen–Macaulay ring—or at least close to being one.

GORENSTEIN MODULES, FINITE INDEX, AND FINITE COHEN–MACAULAY TYPE

ABSTRACT A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.

On the Depth of the Associated Graded Ring of an Ideal

Let R be a local Cohen–Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G fails to be Cohen–Macaulay. We assume that I has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers. The main theorem unifies and generalizes several known results. We also give conditions that imply the Serre properties of the blow-up rings.

Core and residual intersections of ideals

D. Rees and J. Sally defined the core of an R-ideal I as the in- tersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of inte- grally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.

Reduced Bass numbers, Auslanders δ-invariant and certain homological conjectures

In the category of finitely generated modules over a local Cohen-Macaulay ring with dualizing module, one can, following Auslander-Buchweitz, consider maximal Cohen-Macaulay approximations and hulls of finite injective dimension. The former give rise to y- (and δ for a Gorenstein ring) invariants, the latter to vd -invariants which are in some sense dual. This last type of invariant has, in the guise of what we call a reduced Bass number, been encountered in the study of the Canonical Element Conjecture (CEC) by Hochster. In this paper we establish the basic properties of these invariants, some known, some not. We go on to explore the connection with CEC. The latter is, through linkage, seen to imply a fact about delta invariants. This in turn shows that CEC guarantees the existence of certain maximal Cohen-Macaulay modules over a Gorenstein ring.

EQUIMULTIPLE GOOD IDEALS WITH HEIGHT 1

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m. Then we say that I is an equimultiple good ideal in A, if I contains a reduction Q = ( <TEX>$a_1$</TEX>, <TEX>$a_2$</TEX>,ㆍㆍㆍ, <TEX>$a_{s}$</TEX> ) generated by s elements in A and G(I) =(equation omitted)<TEX>$_{n 0}$</TEX> <TEX>$I^{n}$</TEX> / <TEX>$I^{n＋1}$</TEX> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = h <TEX>$t_{A}$</TEX> I and a(G(I)) denotes the a-invariant of G(I). Let <TEX>$X_{A}$</TEX><TEX>$^{s}$</TEX> denote the set of equimultiple good ideals I in A with h <TEX>$t_{A}$</TEX> I = s, R(I) = A [It] be the Rees algebra of I, and <TEX>$K_{R(I)}$</TEX> denote the canonical module of R(I). Let a I such that <TEX>$I^{n＋l}$</TEX> = a <TEX>$I^{n}$</TEX> for some n<TEX>$\geq$</TEX>0 and <TEX>$\mu$</TEX><TEX>$_{A}$</TEX>(I)<TEX>$\geq$</TEX>2, where <TEX>$\mu$</TEX><TEX>$_{A}$</TEX>(I) denotes the number of elements in a minimal system of generators of I. Assume that A/I is a Cohen-Macaulay ring. We show that the following conditions are equivalent. (1) <TEX>$K_{R(I)}$</TEX>(equation omitted)R(I)＋as graded R(I)-modules. (2) <TEX>$I^2$</TEX> = aI and aA : I<TEX>$\in$</TEX> <TEX>$X^1$</TEX><TEX>$_{A}$</TEX>._{A}$</TEX>./.

On Cohen–Macaulay Rings of Invariants

We investigate the transfer of the Cohen–Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we discuss the special case of multiplicative actions, that is, actions on group algebras k[Zn] via an action on Zn.

A length formula for the multiplicity of distinguished components of intersections

To an arbitrary ideal I in a local ring (A, m) one can associate a multiplicity j( I, A) that generalizes the classical Hilbert–Samuel multiplicity of an m -primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen–Macaulay in A and satisfies a suitable Artin–Nagata condition then our main result states that j( I, M) is given by the length of I/( x 1,…, x d−1 )+ x d I where d≔ dim A and x 1,…, x d are sufficiently generic elements of I. This generalizes the classical length formula for m -primary ideals in Cohen–Macaulay rings. Applying this to an hypersurface H in the affine space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle H c+1 with multiplicity e( jac H,C, O H,C) , where jac H is the Jacobian ideal generated by the partial derivatives of a defining equation of H.

Multiplicity and Tight Closures of Parameters

In this paper we investigate the relation between the multiplicity and the tight closure of a parameter ideals. We shall show that local rings having a parameter ideal whose multiplicity agrees with the colength of its tight closure are Cohen–Macaulay rings.

On homological dimensions

For finite modules over a local ring the general problem is considered of finding an extension of the class of modules of finite projective dimension preserving various properties. In the first section the concept of a suitable complex is introduced, which is a generalization of both a dualizing complex and a suitable module. Several properties of the dimension of modules with respect to such complexes are established. In particular, a generalization of Golod's theorem on the behaviour of GK-dimension with respect to a suitable module K under factorization by ideals of a special kind is obtained and a new form of the Avramov-Foxby conjecture on the transitivity of G-dimension is suggested. In the second section a class of modules containing modules of finite CI-dimension is considered, which has some additional properties. A dimension constructed in the third section characterizes the Cohen-Macaulay rings in precisely the same way as the class of modules of finite projective dimension characterizes regular rings and the class of modules of finite CI-dimension characterizes complete intersections.

A dimension inequality for Cohen-Macaulay rings

The recent work of Kurano and Roberts on Serre’s positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak {p}$ and $\mathfrak {q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak {n})$ such that $e(A_{\mathfrak {p}})=e(A)$ we have $\dim (A/\mathfrak {p})+\dim (A/\mathfrak {q})\leq \dim (A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak {p}$ is regular.

Bass numbers of local cohomology modules with supports in monomial ideals

In this paper, we will study the local cohomology modules HiI(S) of a polynomial ring S = k[x1, …, xn] with supports in a (radical) monomial ideal I. When S/I is a Cohen–Macaulay ring of dimension d (more generally, if Extn−d(S/I, [wfr ]S) is Cohen–Macaulay), we can ‘visualize’ a ℤn-graded minimal injective resolution of Hn−dI(S) using Stanley–Reisner's simplicial complex of I.

On the Set of Associated Primes of a Local Cohomology Module

Assume R is a local Cohen–Macaulay ring. It is shown that AssR(HlI(R)) is finite for any ideal I and any integer l provided AssR(H2(x,y)(R)) is finite for any x, y∈R and AssR(H3(x1,x2,y)(R)) is finite for any y∈R and any regular sequence x1, x2∈R. Furthermore it is shown that AssR(HlI(R)) is always finite if dim(R)≤3. The same statement is even true for dim(R)≤4 if R is almost factorial.

Non-commutative Cohen–Macaulay Rings

We introduce a concept of Cohen–Macaulayness for left noetherian semilocal rings (and their modules) which generalizes the corresponding notion of commutative algebra and naturally applies to orders.

ALMOST COHEN–MACAULAY MODULES

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R P , R P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X 1, …, X n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.

Depth of symmetric algebras of certain ideals

We compute the depth of the symmetric algebra of certain ideals in terms of the depth of the ring modulo the ideal generated by the entries of a minimal presentation matrix. The purpose of this paper is to give a formula for the depth of the symmetric algebra S(I) =EDj>0 Sj (I) for certain ideals. We also obtain an analogous result for the symmetric algebra S(I/I2) of the conormal module (as an R/I-module). Our method is a simple application of the recent results ([3], [5]) concerning the Cohen-Macaulay properties of the blow-up rings R[It] = .j>oPI and gri(R) = EDjO IP/I+l. In particular, these blow-up rings are Cohen-Macaulay for the ideals under our consideration. It is well known that the symmetric algebra need not be Cohen-Macaulay, but it turns out that these symmetric algebras have fairly large depth in any case. Let R be a noetherian local ring with infinite residue field k, and let I be an ideal. Recall that the analytic spread of I is ?(I) = dim R[It] OR k, or equivalently is the least number s of elements a1, ..., a, in I such that jk+1 = (a,, ..., as)Ik for some k > 0; the least integer k (over all such reductions) is the reduction number r of I. Note that r = O f = n _ v(I) (the minimal number of generators of I) I is generated by analytically independent elements. We say that I satisfies Gs if v(Ip) s > ht I (respectively, and ht I + J > s). Theorem 1. Let R be a local Cohen-Macaulay ring with infinite residue field, let I be an ideal with ht I > 2, analytic spread X, reduction number r, minimally generated by n = f + 1 elements, let 0 be a minimal presentation matrix of I, and assume that I satisfies Ge and ANJ-2, and that Sj(I) _ Ii and depthfR/IJ > dimR-C+r-j for1 <j <r. Then depth S(I) = depth R/II (0) + n, depth S(I/I2) = min{depth R/I1 (0) + n, dim R}. Received by the editors June 17, 1999 and, in revised form, September 1, 1999. 1991 Mathematics Subject Classification. Primary 13A30, 13H10. (?)2000 American Mathematical Society

Rings with a local cohomology theorem and applications to cohomology rings of groups

Cohomology rings of various classes of groups have curious duality properties expressed in terms of their local cohomology (Benson and Carlson, Trans. Amer. Math. Soc. 342 (1994) 447–488; Bull. London Math. Soc. 26 (1994) 438–448; Greenlees, J. Pure Appl. Algebra 98 (1995) 151–162; Benson and Greenlees, J. Pure Appl. Algebra 122 (1997) 41–53, J. Algebra 192 (1997) 678–700; Symonds, in preparation). We formulate a purely algebraic form of this duality, and investigate its consequences. It is obvious that a Cohen–Macaulay ring of this sort is automatically Gorenstein, and that its Hilbert series therefore satisfies a functional equation, and our main result is a generalization of this to rings with depth one less than their dimension: this proves a conjecture of Benson and Greenlees (1997).