Homological properties of quantized coordinate rings of semisimple groups

We prove that the generic quantized coordinate ring _q(G) isAuslander-regular, Cohen–Macaulay, and catenary for everyconnected semisimple Lie group G. This answers questions raisedby Brown, Lenagan, and the first author. We also prove thatunder certain hypotheses concerning the existence of normalelements, a noetherian Hopf algebra is Auslander–Gorensteinand Cohen–Macaulay. This provides a new set of positivecases for a question of Brown and the first author.     

Cohen-Macaulay Complexes and Koszul Rings

Throughout this paper k denotes a fixed commutative ground ring.A Cohen–Macaulay complex is a finite simplicial complexsatisfying a certain homological vanishing condition. Thesecomplexes have been the subject of much research; introductionscan be found in, for example, Björner, Garsia and Stanley[6] or Budach, Graw, Meinel and Waack [7]. It is known (see,for example, Cibils [8], Gerstenhaber and Schack [10]) thatthere is a strong connection between the (co)homology of anarbitrary simplicial complex and that of its incidence algebra.We show how the Cohen–Macaulay property fits into thispicture, establishing the following characterization. A pure finite simplicial complex is Cohen–Macaulay overk if and only if the incidence algebra over k of its augmentedface poset, graded in the obvious way by chain lengths, is aKoszul ring.     

Two theorems about maximal Cohen–Macaulay modules

This paper contains two theorems concerning the theory of maximal Cohen–Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen–Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen–Macaulay modules exist having ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen–Macaulay lo cal ring of finite Cohen–Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen–Macaulay local ring of finite Cohen–Macaulay type is again of finite Cohen–Macaulay type . The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of divided by has a positive limit. This result relates to work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties. Received: 6 May 2001 / Published online: 6 August 2002 Both authors were partially supported by the National Science Foundation. The second author was also partially supported by the Clay Mathematics Institute.     

Homological Properties of Fully Bounded Noetherian Rings

Let R be a fully bounded Noetherian ring of finite global dimension.Then we prove that K dim (R) gldim (R). If, in addition, Ris local, in the sense that R/J(R) is simple Artinian, thenwe prove that R is Auslander-regular and satisfies a versionof the Cohen–Macaulay property. As a consequence, we showthat a local fully bounded Noetherian ring of finite globaldimension is isomorphic to a matrix ring over a local domain,and a maximal order in its simple Artinian quotient ring.     

Hilbert Coefficients and the Depths of Associated Graded Rings

This work was motivated in part by the following general question:given an ideal I in a Cohen–Macaulay (abbreviated to CM)local ring R such that dim R/I=0, what information about I andits associated graded ring can be obtained from the Hilbertfunction and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel)function of I, we mean the function H_I(n)=(R/Iⁿ) for all n1,where denotes length. Samuel [23] showed that for large valuesof n, the function H_I(n) coincides with a polynomial P_I(n) ofdegree d=dim R. This polynomial is referred to as the Hilbert,or Hilbert–Samuel, polynomial of I. The Hilbert polynomialis often written in the form where e₀(I), ..., e_d(I) are integers uniquely determined byI. These integers are known as the Hilbert coefficients of I.     

Cohen-Macaulay Properties of Thom-Boardman Strata I: Morin's Ideal

Thom–Boardman strata ^I are fundamental tools in studyingsingularities of maps. The Zariski closures of the strata ^Iare components of the set of zeros of the ideals ^I defined by B. Morin using iterated jacobian extensions in his paper‘Calcul jacobien’ (Ann. Sci. École Norm.Sup.} 8 (1975) 1–98). In this paper, we consider the questionof when the Morin ideals ^I define Cohen–Macaulay spaces.We determine all I=(i₁...,i_k) such that ^I defines a Cohen–Macaulayspace alongthe stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.     

Integrally closed and componentwise linear ideals

In a two dimensional regular local ring integrally closed ideals have a unique factorization property and their associated graded ring is Cohen–Macaulay. In higher dimension these properties do not hold and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings R of arbitrary dimension. We identify a class of integrally closed ideals, the Goto-class G^*{\mathcal {G}^*}, which is closed under product and it has a suitable unique factorization property. Ideals in G^*{\mathcal {G}^*} have a Cohen–Macaulay associated graded ring if either they are monomial or dim R ≤ 3. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals.     

Rank 2 totally arithmetically Cohen–Macaulay vector bundles on Hirzebruch surfaces

Here we study the totally arithmetically Cohen–Macaulay rank 2 vector bundles on any Hirzebruch surface F _e . E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Finite Modules of Finite Injective Dimension Over a Noetherian Algebra

Let R be a commutative Noetherian ring. Let P(R) (respectively,I(R)) be the category of all finite R-modules of finite projective(respectively, injective) dimension. Sharp [9] constructed acategory equivalence between I(R) and P(R) for certain Cohen–Macaulaylocal rings R. Thus many properties about finite modules offinite projective dimension can be connected with those of finiteinjective dimension through this equivalence.     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.     

On the M?bius function of a lower Eulerian Cohen–Macaulay poset

A certain inequality is shown to hold for the values of the M?bius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen–Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley’s nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen–Macaulay meet-semilattice and Adin’s nonnegativity conjecture for the cubical h-vector of a Cohen–Macaulay cubical complex.     

Bounds For Gradient Trajectories and Geodesic Diameter of Real Algebraic Sets

Let M Rⁿ be a connected component of an algebraic set ^–1(0),where is a polynomial of degree d. Assume that M is containedin a ball of radius r. We prove that the geodesic diameter ofM is bounded by 2rv(n)d(4d–5)^n–2, where v(n) =(1/2)((n+1)/2)(n/2)^–1.This estimate is based on the bound rv(n)d(4d–5)^n–2for the length of the gradient trajectories of a linear projectionrestricted to M. 2000 Mathematics Subject Classification 32Bxx,34Cxx (primary), 32Sxx, 14P10 (secondary).     

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework.We introduce an approach through the notion of Bourbaki idealsthat allows the use of deformation theory. One can talk aboutthe (essentially unique) generic Bourbaki ideal I(E) of a moduleE which, in many situations, allows one to reduce the natureof the Rees algebra of E to that of its Bourbaki ideal I(E).Properties such as Cohen–Macaulayness, normality and beingof linear type are viewed from this perspective. The known numericalinvariants, such as the analytic spread, the reduction numberand the analytic deviation, of an ideal and its associated algebrasare considered in the case of modules. Corresponding notionsof complete intersection, almost complete intersection and equimultiplemodules are examined in some detail. Special consideration isgiven to certain modules which are fairly ubiquitous becauseinteresting vector bundles appear in this way. For these modulesone is able to estimate the reduction number and other invariantsin terms of the Buchsbaum–Rim multiplicity. 2000 MathematicsSubject Classification 13A30 (primary), 13H10, 13B21 (secondary)     

Projective resolutions associated to projections

In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal. Received: 22 February 1999     

Two Diophantine Birds with One Stone

Integer Solutions are found to the equations t²–3(a²,b², (a + b)², (a–b)²) = p², q², r², s². These lead surprisinglyto solutions to the equations u² + (c², d², (c + d)², (c –d)²) = p², q², v², w², with the same values of p and q.     

Differential Operators on some Singular Surfaces

Let X be an irreducible affine algebraic variety of dimension2 defined over an algebraically closed field of characteristiczero. Suppose that the normalisation of X is non-singular, and the natural projection : X is injective. Further, suppose that X is Cohen–Macaulay.Then the rings of differential operators D(X) and D() are Morita equivalent. Current address of S. P. Smith: Department of Mathematics, Universityof Washington, Seattle, WA 98195. USA     

The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Direct limits of Cohen–Macaulay rings

Let A be a direct limit of a direct system of Cohen–Macaulay rings. In this paper, we describe the Cohen–Macaulay property of A. Our results indicate that A is not necessarily Cohen–Macaulay. We show A is Cohen–Macaulay under various assumptions. As an application, we study Cohen–Macaulayness of non-affine normal semigroup rings.