Cohen-Macaulay representations by Graham J. Leuschke
By Graham J. Leuschke
This booklet is a finished remedy of the illustration conception of maximal Cohen-Macaulay (MCM) modules over neighborhood earrings. This subject is on the intersection of commutative algebra, singularity idea, and representations of teams and algebras. introductory chapters deal with the Krull-Remak-Schmidt Theorem on strong point of direct-sum decompositions and its failure for modules over neighborhood earrings. Chapters 3-10 research the important challenge of classifying the jewelry with basically finitely many indecomposable MCM modules as much as isomorphism, i.e., earrings of finite CM variety. the basic material--ADE/simple singularities, the double branched disguise, Auslander-Reiten thought, and the Brauer-Thrall conjectures--is coated in actual fact and entirely. a lot of the content material hasn't ever sooner than seemed in ebook shape. Examples comprise the illustration thought of Artinian pairs and Burban-Drozd's similar building in size , an creation to the McKay correspondence from the perspective of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation concept, and a cautious therapy of nonzero attribute. the rest seven chapters current effects on bounded and countable CM sort and at the illustration idea of completely reflexive modules
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8). Letting p vary over Spec(R), we obtain exactly m = spl(R) equations that must be satisﬁed by elements of Λ. Conversely, if the bj satisfy (b ) (b ) these equations, then N := V1 1 ⊕ · · · ⊕ Vn n has constant rank on each ﬁber of Spec(R) −→ Spec(R). 8, N is extended from an R-module, say N ∼ = R ⊗R L. 19 that L ∈ +(M ), whence [b1 , . . , bn ]tr ∈ Λ. In [Kat02] Kattchee showed that, for each m, there is a ﬁnitely generated Krull monoid Λ that cannot be deﬁned by m equations. Thus no single one-dimensional local ring can realize every ﬁnitely generated Krull monoid in the form +(M ) for a ﬁnitely generated module M .
To be precise, let us say that two direct-sum decompositions M∼ = M1 ⊕ · · · ⊕ Mm and M ∼ = N1 ⊕ · · · ⊕ Nn are equivalent provided m = n and, after a permutation, Mi ∼ = Ni for each i. (We do not §1. 1 in [Wie99], with a slightly non-commutative proof. We will give a commutative proof here. 2. Theorem. Let (R, m) be a local ring, and let M be a ﬁnitely generated R-module. Then there are only ﬁnitely many isomorphism classes of indecomposable modules in addR (M ). In particular, M has, up to equivalence, only ﬁnitely many direct sum decompositions.
Under these conditions, the number of isomorphism classes of indecomposable ﬁnitely generated modules is exactly the length of R. Proof. Assuming (i), we will prove (ii) and verify the last statement. Since R is a product of ﬁnitely many local rings, we may assume that R is local, with maximal ideal m. As R is a principal ideal ring, the length of R is the least integer t such that mt = 0. Since every ﬁnitely generated R-module is a direct sum of cyclic modules, the indecomposable modules are exactly the modules R/mt , 1 t .