Local rings of bounded Cohen-Macaulay type

Let (R,m,k) be a local Cohen-Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thrall theorem for excellent rings.Comment: 13 pages, revised and correcte

Hypersurfaces of bounded Cohen--Macaulay type

Let R = k[[x_0,...,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x_0,...,x_d]]. We investigate the question of which rings of this form have bounded Cohen--Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen--Macaulay modules. As with finite Cohen--Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen--Macaulay type if and only if R is isomorphic to k[[x_0,...,x_d]]/(g+x_2^2+...+x_d^2), where g is an element of k[[x_0,x_1]] and k[[x_0,x_1]]/(g) has bounded Cohen--Macaulay type. We determine which rings of the form k[[x_0,x_1]]/(g) have bounded Cohen--Macaulay type.Comment: 16 pages, referee's suggestions and correction

FINITELY GENERATED MODULES OVER BEZOUT [email protected]

Let R be a Bezout ring (a commutative ring in which all finitely generated ideals are principal), and let M be a finitely generated R -module. We will study questions of the following sort: (A) If every localization of M can be generated by n elements, can M itself be generated by n elements? (B) If M 0 R m = Rn for some m, n, is Af necessarily free? (C) If every localization of M has an element with zero annihilator, does M itself have such an element? We will answer these and related questions for various familiar classes of Bezout rings. For example, the answer to (B) is no for general Bezout rings but yes for Hermite rings (defined below). Also, a Hermite ring is an elementary divisor ring if and only if (A) has an affirmative answer for every module M

Ascent of module structures, vanishing of Ext, and extended modules

Let (R,\m) and (S,\n) be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that \m S = \n and the induced map on residue fields R/\m \to S/\n is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if and only if \Ext^i_R(S,M)=0 for each $i\ge 1$. We say that an $S$-module $N$ is {\it extended} if there is a finitely generated $R$-module $M$ such that $N\cong S\otimes_RM$. Given a short exact sequence $0 \to N_1\to N \to N_2\to 0$ of finitely generated $S$-modules, with two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of $R$ is a direct summand of an extended module, but that the analogous result fails for the \m-adic completion.Comment: 16 pages, AMS-TeX; final version to appear in Michigan Math. J.; corrected proof of Main Theorem and made minor editorial changes; v3 has dedication to Mel Hochste

Brauer-Thrall for totally reflexive modules

Let R be a commutative noetherian local ring that is not Gorenstein. It is known that the category of totally reflexive modules over R is representation infinite, provided that it contains a non-free module. The main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation. Local rings (R,m) with m^3=0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics. For such rings we obtain conclusive results about the category of totally reflexive modules, modeled on the Brauer-Thrall conjectures. Starting from a non-free cyclic totally reflexive module, we construct a family of indecomposable totally reflexive R-modules that contains, for every n in N, a module that is minimally generated by n elements. Moreover, if the residue field R/m is algebraically closed, then we construct for every n in N an infinite family of indecomposable and pairwise non-isomorphic totally reflexive R-modules, that are all minimally generated by n elements. The modules in both families have periodic minimal free resolutions of period at most 2.Comment: Final version; 34 pp. To appear in J. Algebr

The residue fields of a zero-dimensional ring

AbstractGilmer and Heinzer have considered the question: For an indexed family of fields oK = {Kα}αgEA, under what conditions does there exist a zero-dimensional ring R (always commutative with unity) such that oK is up to isomorphism the family of residue fields {RMα}αgEA of R? If oK is the family of residue fields of a zero-dimensional ring R, then the associated bijection from the index set A to the spectrum of R (with the Zariski topology) gives A the topology of a Boolean space. The present paper considers the following question: Given a field F, a Boolean space X and a family {Kx}xgEX of extension fields of F, under what conditions does there exist a zero-dimensional F-algebra R such that oK is up to F-isomorphism the family of residue fields of R and the associated bijection from X to Spec(R) is a homeomorphism? A necessary condition is that given x in X and any finite extension E of F in Kx, there exist a neighborhood V of x and, for each y in V, an F-embedding of E into Ky. We prove several partial converses of this result, under hypotheses which allow the “straightening” of the F-embeddings to make them compatible. We give particular attention to the cases where X has only one accumulation point and where X is countable; and we provide several examples