Bounds for indecomposable torsion-free modules

Let M be a finitely generated torsion-free module over a one-dimensional reduced Noetherian ring R with finitely generated normalization. The rank of M is the tuple of vector-space dimensions of M_P over each field R_P (R localized at P), where P ranges over the minimal prime ideals of R. We assume that there exists a bound N_R on the ranks of all indecomposable finitely generated torsion-free R-modules. For such rings, what bounds and ranks occur? Partial answers to this question have been given by a plethora of authors over the past forty years. In this article we provide a final answer by giving a concise list of the ranks of indecomposable modules for R a local ring with no condition on the characteristic. We conclude that if the rank of an indecomposable module M is (r,r,…,r), then r∈{1,2,3,4,6}, even when R is not local.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Self-dual modules over local rings of curve singularities

Let C be a reduced curve singularity. C is called of finite self-dual type if there exist only finitely many isomorphism classes of indecomposable, self-dual, torsion-free modules over the local ring of C. In this paper it is shown that the singularities of finite self-dual type are those which dominate a simple plane singularity.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

Projective modules over smooth, affine varieties over Archimedean real closed fields

Let be a smooth, affine variety of dimension n≥2 over the field R of real numbers. Let P be a projective A-module of such that its nth Chern class is zero. In this set-up, Bhatwadekar-Das-Mandal showed (amongst many other results) that P?A⊕Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an Archimedean real closed field .     

Projective modules over overrings of polynomial rings and a question of Quillen

Let (R,m,K)$(R,\mathfrak{m},K)$ be a regular local ring containing a field k   such that either char k=0$k=0$ or char k=p$k=p$ and tr-deg K/_Fp?1$K/{\mathbb{F}}_p?1$. Let _g1,…,_gt$g_1,\dots ,g_t$ be regular parameters of R   which are linearly independent modulo m²${\mathfrak{m}}^2$. Let A=_Rg1?gt[_Y1,…,_Ym,_f1(_l1)⁻¹,…,_fn(_ln)⁻¹]$A=R_{g_1?g_t}[Y_1,\dots ,Y_m,f_1{(l_1)}^{-1},\dots ,f_n{(l_n)}^{-1}]$, where _fi(T)∈k[T]$f_i(T)\in k[T]$ and _li=_ai1Y1+?+_aimYm$l_i=a_{i1}{Y}_1+?+a_{im}{Y}_m$ with (_ai1,…,_aim)∈k^m−(0)$(a_{i1},\dots ,a_{im})\in k^m-(0)$. Then every projective A-module of rank ?t   is free. Laurent polynomial case _fi(_li)=_Yi$f_i(l_i)=Y_i$ of this result is due to Popescu.     

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal m$\mathfrak{m}$ satisfies m^s≠0=m^s+1${\mathfrak{m}}^s\ne 0={\mathfrak{m}}^{s+1}$ and _rankR/m(m/m²)=e${\mathrm{rank}}_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=e$. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3$s\ne 3$, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3$s=3$ examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.     

Totally reflexive modules constructed from smooth projective curves of genus <Emphasis Type="Italic">g</Emphasis> ≥ 2

In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it. Received: 2 May 2006     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

On equations defining Veronese rings

For a standard graded noetherian algebra S that is of weakly linear type, the defining equations of the Veronesian subrings S^(d) are described explicitly, for d sufficiently large. Received: 19 January 2005     

Correction to “Tensor products of modules and the rigidity of Tor”, Math. Annalen, 299 (1994), 449–476

This note makes a correction to the paper “Tensor products of modules and the ridigity of Tor”, a correction which is needed due to an incorrect convention for the depth of the zero module.     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.     

Families of artinian and low dimensional determinantal rings

Let $\mathrm{GradAlg}(H)$ be the scheme parameterizing graded quotients of $R=k[x_0,\dots ,x_n]$ with Hilbert function H (it is a subscheme of the Hilbert scheme of ${\mathbb{P}}^n$ if we restrict to quotients of positive dimension, see definition below). A graded quotient $A=R/I$ of codimension c is called standard determinantal if the ideal I can be generated by the $t\times t$ minors of a homogeneous $t\times (t+c?1)$ matrix $(f_{ij})$. Given integers $a_0\le a_1\le ...\le a_{t+c?2}$ and $b_1\le ...\le b_t$, we denote by $W_s(\underset{\_}{b};\underset{\_}{a})?\mathrm{GradAlg}(H)$ the stratum of determinantal rings where $f_{ij}\in R$ are homogeneous of degrees $a_j?b_i$.In this paper we extend previous results on the dimension and codimension of $W_s(\underset{\_}{b};\underset{\_}{a})$ in $\mathrm{GradAlg}(H)$ to artinian determinantal rings, and we show that $\mathrm{GradAlg}(H)$ is generically smooth along $W_s(\underset{\_}{b};\underset{\_}{a})$ under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of ${\mathbb{P}}^n$ is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of $\mathrm{GradAlg}(H)$.     

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z-graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).