Monoids of modules over rings of infinite Cohen-Macaulay type

Baeth, Nicholas R.; Saccon, Silvia

doi:10.1216/jca-2012-4-3-297

Cited by 4 publications

(9 citation statements)

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“…We now return to studying sets of lengths. We note that the elasticity of certain monoids of modules were studied in [7] and [8], but that in Section 6 we will provide results which generalize these results to larger classes of Krull monoids. In addition, we will fine tune these results by also computing the refined elasticities.…”

Section: Arithmetical Preliminariesmentioning

confidence: 98%

“…The past decade has seen a new semigroup-theoretical approach. This approach was first introduced by Facchini and Wiegand [29] and has been used by several authors (for example, see [3], [4], [7], [8], [19], [22], [25], [26], [27], [28], [29], [46], [49], and [57]). Let R be a ring and let C be a class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms.…”

Section: Introductionmentioning

confidence: 99%

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Monoids of modules and arithmetic of direct-sum decompositions

Baeth

Geroldinger

2014

Pacific J. Math.

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Abstract. Let R be a (possibly noncommutative) ring and let C be a class of finitely generated (right) R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) of isomorphism classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup encodes all possible information about direct sum decompositions of modules in C. If the endomorphism ring of each module in C is semilocal, then V(C) is a Krull monoid. Although this fact was observed nearly a decade ago, the focus of study thus far has been on ring-and module-theoretic conditions enforcing that V(C) is Krull. If V(C) is Krull, its arithmetic depends only on the class group of V(C) and the set of classes containing prime divisors. In this paper we provide the first systematic treatment to study the direct-sum decompositions of modules using methods from Factorization Theory of Krull monoids. We do this when C is the class of finitely generated torsion-free modules over certain one-and two-dimensional commutative Noetherian local rings.

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Section: Arithmetical Preliminariesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Monoids of modules and arithmetic of direct-sum decompositions

Baeth

Geroldinger

2014

Pacific J. Math.

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“…For the complete graph K n , recall ρ(A(K n )) = n + 2 n − 2 ≥ n − 2 by Example 4.14. By Proposition 5.4 there is a Bass Ring R with ρ(T (R)) = ρ(A(K n )) ≥ n − 2; this shows (9). or (1,1).…”

Section: From Lattices To Graph Agglomerationsmentioning

confidence: 93%

“…These types of questions arose at first in the factorization theory of integral domains and monoids; we mention the recent surveys [32,33], monographs [22,28], and proceedings [2,11,12]. For monoids of modules, this perspective was pursued in [5,6,8,9,14,21] for some classes of rings; also see the survey [10]. Typically, arithmetical invariants have been studied in cancellative settings, with recent work in some noncancellative settings by Geroldinger, Fan, Kainrath, and Tringali [18,23].…”

Section: Introductionmentioning

confidence: 99%

Lattices over Bass Rings and Graph Agglomerations

Baeth

Smertnig

2021

Algebr Represent Theor

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We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

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“…From the main theorem, we can deduce the structure of the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay modules (together with [0]) when R/Q has infinite Cohen-Macaulay type for all minimal prime ideals Q of R. The following corollary is a special case of Theorem 3.15 in [5]. Recall that the splitting number q of R is defined by q = | MinSpec( R)| − | MinSpec(R)|, where MinSpec(R) and MinSpec( R) denote the set of minimal prime ideals of R and R respectively.…”

Section: Case (2d)mentioning

confidence: 99%

Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

Crabbe

Saccon

2013

Communications in Algebra

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Let (R, m, k) be a one-dimensional analytically unramified local ring with minimal prime ideals P 1 , . . . , Ps. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the monoid C(R) is determining the ranks of indecomposable maximal Cohen-Macaulay modules. Our main technical result shows that if R/P 1 has infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic indecomposable maximal Cohen-Macaulay R-modules of rank (r 1 , . . . , rs) provided r 1 ≥ r i for all i ∈ {1, . . . , s}. This result allows us to describe the monoid C(R) when R/Q has infinite Cohen-Macaulay type for every minimal prime ideal Q of the m-adic completion R.

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Monoids of modules over rings of infinite Cohen-Macaulay type

Cited by 4 publications

References 15 publications

Monoids of modules and arithmetic of direct-sum decompositions

Monoids of modules and arithmetic of direct-sum decompositions

Lattices over Bass Rings and Graph Agglomerations

Ranks of Indecomposable Modules over Rings of Infinite Cohen-Macaulay Type

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