Representation type of commutative Noetherian rings. I: Local wildness.

*(English)*Zbl 1019.13004From the paper: This is the first of a series of four papers describing the finitely generated modules over all commutative noetherian rings that do not have wild representation type. The first paper identifies the wild rings, in the complete local case.

We call a ring \(\Lambda\) finitely generated tame if we can describe all isomorphism classes of finitely generated \(\Lambda\)-modules. In all cases for which we cannot obtain such a description (with a possible exception involving characteristic 2), the obstruction is wild representation type; or more precisely, finite-length wildness. Informally, a commutative ring \(\Lambda\) is finite-length wild if it has a residue field \(k\) such that any description of all isomorphism classes of \(\Lambda\)-modules of finite length would have to contain a description of all isomorphism classes of finite-dimensional modules over all finite-dimensional \(k\)-algebras. (The precise definition of finite-length wild is given in subsection 2.2).

Let \((\Lambda,{\mathfrak m}, k)\) be a complete local commutative noetherian ring, and \(\mu_\Lambda ({\mathfrak m})\) the minimal number of generators of the \(\Lambda\)-module \({\mathfrak m}\). We give the spirit of our main wildness theorem [2.10]:

If \(\mu_\Lambda({\mathfrak m}) \geq 3\), then \(\Lambda\) is always finite-length wild. On the other hand, if \(\mu_\Lambda ({\mathfrak m})=1\), then \(\Lambda\) is a principal ideal ring, and its tameness was well-known long before the word “tame” came into use. Thus \(\mu_\Lambda ({\mathfrak m})=2\) is the dividing line between tameness and wildness; moreover, the vast majority of rings with \(\mu_\Lambda ({\mathfrak m})=2\) are wild. In order to make this dividing line precise, we need to define several types of rings, which we do in section 2.

We call \(\Lambda\) an artinian triad if \(\mu_\Lambda ({\mathfrak m})=3\) and \({\mathfrak m}^2=0\). These are clearly the “smallest” rings such that \(\mu_\Lambda ({\mathfrak m})=3\), and they are known to be finite-length wild. We also define a special kind of local artinian ring \(\Lambda\) of composition length 5, with \(\mu_\Lambda ({\mathfrak m})=2\), and call it a Drozd ring. We prove that these are finite-length wild in section 4.

Section 3 is devoted to the proof of our ring-theoretic dichotomy theorem [3.1], which states that every complete local ring either

(i) maps onto an artinian triad or a Drozd ring, or

(ii) is a homomorphic image of a type of ring of Krull dimension 1 that we call Dedekind-like or is an exceptional type of artinian ring that we call a Klein ring. “Dedekind-like rings” are reduced rings, satisfy \(\mu_\Lambda ({\mathfrak m})\leq 2\), and are very close to their normalization (in their total quotient ring), which is either a DVR (discrete valuation ring) or the direct sum of two DVRs. “Klein rings” have composition length 4 and satisfy \(\mu_\Lambda ({\mathfrak m})=2\), a special case being the group algebra of the Klein 4-group over a field of characteristic 2.

We call a ring \(\Lambda\) finitely generated tame if we can describe all isomorphism classes of finitely generated \(\Lambda\)-modules. In all cases for which we cannot obtain such a description (with a possible exception involving characteristic 2), the obstruction is wild representation type; or more precisely, finite-length wildness. Informally, a commutative ring \(\Lambda\) is finite-length wild if it has a residue field \(k\) such that any description of all isomorphism classes of \(\Lambda\)-modules of finite length would have to contain a description of all isomorphism classes of finite-dimensional modules over all finite-dimensional \(k\)-algebras. (The precise definition of finite-length wild is given in subsection 2.2).

Let \((\Lambda,{\mathfrak m}, k)\) be a complete local commutative noetherian ring, and \(\mu_\Lambda ({\mathfrak m})\) the minimal number of generators of the \(\Lambda\)-module \({\mathfrak m}\). We give the spirit of our main wildness theorem [2.10]:

If \(\mu_\Lambda({\mathfrak m}) \geq 3\), then \(\Lambda\) is always finite-length wild. On the other hand, if \(\mu_\Lambda ({\mathfrak m})=1\), then \(\Lambda\) is a principal ideal ring, and its tameness was well-known long before the word “tame” came into use. Thus \(\mu_\Lambda ({\mathfrak m})=2\) is the dividing line between tameness and wildness; moreover, the vast majority of rings with \(\mu_\Lambda ({\mathfrak m})=2\) are wild. In order to make this dividing line precise, we need to define several types of rings, which we do in section 2.

We call \(\Lambda\) an artinian triad if \(\mu_\Lambda ({\mathfrak m})=3\) and \({\mathfrak m}^2=0\). These are clearly the “smallest” rings such that \(\mu_\Lambda ({\mathfrak m})=3\), and they are known to be finite-length wild. We also define a special kind of local artinian ring \(\Lambda\) of composition length 5, with \(\mu_\Lambda ({\mathfrak m})=2\), and call it a Drozd ring. We prove that these are finite-length wild in section 4.

Section 3 is devoted to the proof of our ring-theoretic dichotomy theorem [3.1], which states that every complete local ring either

(i) maps onto an artinian triad or a Drozd ring, or

(ii) is a homomorphic image of a type of ring of Krull dimension 1 that we call Dedekind-like or is an exceptional type of artinian ring that we call a Klein ring. “Dedekind-like rings” are reduced rings, satisfy \(\mu_\Lambda ({\mathfrak m})\leq 2\), and are very close to their normalization (in their total quotient ring), which is either a DVR (discrete valuation ring) or the direct sum of two DVRs. “Klein rings” have composition length 4 and satisfy \(\mu_\Lambda ({\mathfrak m})=2\), a special case being the group algebra of the Klein 4-group over a field of characteristic 2.

##### MSC:

13C05 | Structure, classification theorems for modules and ideals in commutative rings |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

16G10 | Representations of associative Artinian rings |