Direct-sum behavior of modules over one-dimensional rings [chapter]

Ryan Karr, Roger Wiegand
2010 Commutative Algebra  
Let R be a reduced, one-dimensional Noetherian local ring whose integral closure R is finitely generated over R. Since R is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated R-modules are easily described, and every finitely generated R-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few
more » ... uations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules. The two conditions above were introduced by Drozd and Roȋter in a remarkable 1967 paper [12] . They proved the theorem in the special case of a ring essentially finite over Z and asserted that it is true in general. In 1978 Green and Reiner [16] gave a much more detailed proof of the theorem in this special case. In 1989 R. Wiegand [43] proved necessity of the conditions (DR) in general, and sufficiency assuming only that each residue field of R is separable over k = R/m. Since, by (DR1), the residue field growth is of degree at most 3, this completed the proof of Theorem 1.1 except in the cases where char(k) = 2 or 3. The case of characteristic 3 was handled by indirect methods in [45] , leaving only the case where k is imperfect of characteristic 2. In his 1994 Ph.D. dissertation, Nuri Cimen [6] then used explicit, and very difficult, matrix reductions to prove the remaining case of the theorem. We will sketch some of the main ingredients of the proof, though we will not touch on the matrix reductions in [16] and [6] . The pullback representation, which we describe in more generality than needed in this section, is a common theme in most of the research leading up to the proof of the theorem. For the moment, let R be any one-dimensional Noetherian ring, not necessarily local, and let R be the integral closure of R in the total quotient ring K of R. We assume that R is finitely generated as an R-module. (This assumption is no restriction: A reduced one-dimensional ring is automatically Cohen-Macaulay. If, further, R has finite Cohen-Macaulay type, then R has to be finitely generated over R (cf. [45, Lemma 1] or Proposition 1.2 below).) The conductor f := {r ∈ R | rR ⊆ R} contains a non-zerodivisor of R; therefore R/f and R/f are Artinian rings, and we have a pullback diagram The bottom line of the pullback is an example of an Artinian pair [43], by which we mean a module-finite extension A → B of commutative Artinian rings. Of course this pullback has the additional property that R/f is a principal ideal ring. Given an Artinian pair A = (A → B), one defines an A-module to be a pair V → W , where W is a finitely generated projective B-module and V is an A-submodule of W with the property that BV = W . A morphism (V 1 → W 1 ) → (V 2 → W 2 ) of A-modules is, by definition, a B-homomorphism from W 1 to W 2 that carries V 1 into V 2 . With submodules and direct sums defined in the obvious way, we get an additive category in which every object has finite length. We say A has finite representation type provided there are, up to isomorphism, only finitely many indecomposable A-modules. In the local case, the bottom line tells the whole story: Proposition 1.2 ([43, (1.9)]). Let (R, m) be a one-dimensional, reduced, Noetherian local ring with finite integral closure R. Then R has finite Cohen-
doi:10.1007/978-1-4419-6990-3_10 fatcat:c5mbbnnncjbi7lshbycjxbpeo4