Maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum☆
Introduction
Cohen–Macaulay representation theory has been studied widely and deeply for more than four decades. The theorems of Herzog [13] in the 1970s and of Buchweitz, Greuel and Schreyer [9] in the 1980s are recognized as some of the most crucial results in this long history of Cohen–Macaulay representation theory. Both are concerned with Cohen–Macaulay local rings of finite/countable -representation type, that is, Cohen–Macaulay local rings possessing finitely/infinitely-but-countably many nonisomorphic indecomposable maximal Cohen–Macaulay modules. Herzog proved that quotient singularities of dimension two have finite -representation type and that Gorenstein local rings of finite -representation type are hypersurfaces. Buchweitz, Greuel and Schreyer proved that the local hypersurfaces of finite (resp. countable) -representation type are precisely the local hypersurfaces of type with , with , and with (resp. and ).
At the beginning of this century, Huneke and Leuschke [15] proved that Cohen–Macaulay local rings of finite -representation type have isolated singularities. However, there are ample examples of Cohen–Macaulay local rings not having isolated singularities, including the local hypersurfaces of type and appearing above. Cohen–Macaulay representation theory for non-isolated singularities has been studied by many authors so far; see [2], [10], [14], [19] for instance. It should be remarked that a Cohen–Macaulay local ring with a non-isolated singularity always admits maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum. Focusing on these modules, Araya, Iima and Takahashi [1] found out that the local hypersurfaces of type and have finite -representation type, that is, there exist only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum.
In this paper, we investigate Cohen–Macaulay local rings of finite -representation type from various viewpoints. Our basic landmark is the following conjecture, which includes the converse of the result of Araya, Iima and Takahashi stated above. We shall give positive results to this conjecture.
Conjecture 1.1 Let R be a complete local Gorenstein ring of dimension d not having an isolated singularity. Then the following two conditions are equivalent. The ring R has finite -representation type. The ring R has countable -representation type.
Combining the result of Buchweitz, Greuel and Schreyer, this conjecture says that, when R is a hypersurface having an uncountable algebraically closed coefficient field of characteristic not 2, condition (2) is equivalent to R being an or singularity. In this setting, the implication holds by [1, Proposition 2.1].
From now on, we state our main results and the organization of this paper. Section 2 is devoted to a couple of preliminary definitions and lemmas, while Section 3 presents some conjectures and questions on finite/countable -representation type. Our results are stated in the later sections. In what follows, let R be a Cohen–Macaulay local ring.
In Section 4, we consider the (Zariski-)closedness and (Krull) dimension of the singular locus Sing R of R in connection with the works of Huneke and Leuschke [15], [16]. As we state above, they proved in [15] that if R has finite -representation type, then it has an isolated singularity, i.e., Sing R has dimension at most zero. Also, they showed in [16] that if R is complete or has uncountable residue field, and has countable -representation type, then Sing R has dimension at most one. In relation to these results, we prove the following theorem, whose second assertion extends the result of Huneke and Leuschke [16] from countable -representation type to countable -representation type (i.e., having infinitely but countably many nonisomorphic indecomposable maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum).
Theorem 1.2 Theorem 4.2 and Corollary 4.3 Let be a Cohen–Macaulay local ring. Suppose that R has finite -representation type. Then the singular locus Sing R is a finite set. Equivalently, it is a closed subset of Spec R with dimension at most one. Suppose that R has countable -representation type. Then the set Sing R is at most countable. It has dimension at most one if R is either complete or k is uncountable.
Furthermore, Huneke and Leuschke [16] proved that if R admits a canonical module and has countable -representation type, then the localization at each prime ideal of R has at most countable -representation type as well. We prove a result on finite -representation type in the same context.
Theorem 1.3 Theorem 4.4 Let be a Cohen–Macaulay local ring with a canonical module. Suppose that R has finite -representation type. Then has finite -representation type for all . In particular, has finite -representation type for all .
In Section 5 we provide various necessary conditions for a given Cohen–Macaulay local ring to have finite -representation type.
Theorem 1.4 Theorem 5.5 Let be a Cohen–Macaulay local ring of dimension . Let I be an ideal of R such that is maximal Cohen–Macaulay over R. Then R has infinite -representation type in each of the following cases. The ring has infinite -representation type. The set is contained in , and either has infinite -representation type or . The ideal is not -primary, has infinite -representation type, and is either Gorenstein, a domain, or analytically unramified with .
This theorem may look technical, but it actually gives rise to a lot of restrictions which having finite -representation type produces, and is used in the later sections. One concrete example where Theorem 5.5 applies is when and ; see Corollary 5.9. Here we introduce one of the applications of the above theorem. Denote by the category of maximal Cohen–Macaulay R-modules, and by the singularity category of R.
Theorem 1.5 Theorem 5.8 Let R be a Cohen–Macaulay local ring of dimension . Let I be an ideal of R with such that is maximal Cohen–Macaulay over R. Suppose that R has finite -representation type. Then one must have . If for some integer , then has dimension at most in the sense of [18]. If R is Gorenstein, then R is a hypersurface and has dimension at most in the sense of [22].
There are folklore conjectures that a Gorenstein local ring of countable -representation type is a hypersurface, and that, for a Cohen–Macaulay local ring R of countable -representation type, has dimension at most one. The above theorem gives partial answers to the variants of these folklore conjectures for finite -representation type.
In Section 6, we prove the following, which characterizes the Gorenstein rings or finite -representation type not having an isolated singularity in the dimension 1 case. This theorem has the consequence of answering Conjecture 1.1 in the affirmative when R has an uncountable algebraically closed coefficient field of characteristic not equal to 2.
Theorem 1.6 Theorem 6.1 Let R be a homomorphic image of a regular local ring. Suppose that R does not have an isolated singularity but is Gorenstein. If , the following are equivalent. The ring R has finite -representation type. There exist a regular local ring S and a regular system of parameters such that R is isomorphic to or .
When either of these two conditions holds, the ring R has countable -representation type.
In Section 7, we explore the higher-dimensional case, that is, we try to understand the Cohen–Macaulay local rings R of finite -representation type in the case where . We prove the following two results in this section.
Theorem 1.7 Corollary 7.9 Let R be a complete local hypersurface of dimension which is not an integral domain. Suppose that R has finite -representation type. Then one has , and there exist a regular local ring S and elements with such that and have finite -representation type and is an integral domain of dimension 1.
Theorem 1.8 Corollary 7.11, Corollary 7.12 Let R be a 2-dimensional non-normal Cohen–Macaulay complete local domain. Suppose that R has finite -representation type. Then the integral closure of R has finite -representation type. If R is Gorenstein, then R is a hypersurface.
The former theorem gives a strong restriction of the structure of a hypersurface of finite -representation type which is not an integral domain. The latter theorem supports the conjecture that a Gorenstein local ring of finite -representation type is a hypersurface. Note that, under the assumption of the theorem plus the assumption that R is equicharacteristic zero, the integral closure is a quotient surface singularity by the theorem of Auslander [3] and Esnault [12].
Section snippets
Preliminaries
This section is devoted to stating our conventions, and to recalling the definitions of the notions which repeatedly appear in this paper.
Convention 2.1 Throughout this paper, unless otherwise specified, we adopt the following convention. Rings are commutative and noetherian, and modules are finitely generated. Subcategories are full and strict (i.e., closed under isomorphism). Subscripts and superscripts are often omitted unless there is a risk of confusion. An identity matrix of suitable size is denoted by E
Conjectures and questions
In this section, we present several conjectures and questions which we deal with in later sections. First of all, let us give several definitions of representation types, including that of finite -representation type, which is the main subject of this paper.
Definition 3.1 Let R be a Cohen–Macaulay ring. By we denote the subcategory of consisting of modules that are locally free on the punctured spectrum of R, and set1
The closedness and dimension of the singular locus
In this section, we discuss the structure of the singular locus of a Cohen–Macaulay local ring of finite -representation type. First, we consider what the finiteness of the singular locus means.
Lemma 4.1 Let R be a local ring with maximal ideal . The following are equivalent. Sing R is a finite set. Sing R is a closed subset of Spec R in the Zariski topology, and has dimension at most one.
Proof (2)⇒(1): We find an ideal I of R such that . As Sing R has dimension at most one, so does the local ring
Necessary conditions for finite CM+-representation type
In this section, we explore necessary conditions for a Cohen–Macaulay local ring to have finite -representation type. For this purpose we begin with stating and showing a couple of lemmas.
Lemma 5.1 Let R be a local ring. The subcategory of consisting of periodic modules is closed under finite direct sums: if the R-modules are periodic, then so is . Let be an exact sequence in . Let and be integers. If for all with , then .
Proof (1) is
The one-dimensional hypersurfaces of finite CM+-representation type
The purpose of this section is to prove the following theorem.
Theorem 6.1 Let R be a homomorphic image of a regular local ring. Suppose that R does not have an isolated singularity but is Gorenstein. If , then the following are equivalent. The ring R has finite -representation type. There exist a regular local ring S and a regular system of parameters such that R is isomorphic to or .
When either of these two conditions holds, the ring R has countable -representation type.
In
On the higher-dimensional case
In this section, we explore the higher-dimensional case: we consider Cohen–Macaulay local rings R with and having finite -representation type. In particular, we give various results supporting Conjecture 1.1. We begin with presenting an example by using a result obtained in Section 4.
Example 7.1 Let S be a regular local ring with a regular system of parameters . Then has infinite -representation type.
Proof Let be an ideal of R. Then in R, and
Acknowledgements
The authors are deeply indebted to Hailong Dao for asking them whether there exists a Cohen–Macaulay local ring of finite -representation type other than the hypersurfaces of type and . In fact, this question gave the authors a strong motivation for this work. We are also grateful to two anonymous referees whose suggestions greatly improved the paper, and we also thank Tokuji Araya for stimulating discussions. Most of this work was done during the visit of Toshinori Kobayashi to the
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2023, International Mathematics Research Notices