By Graham J. Leuschke

This publication is a accomplished remedy of the illustration conception of maximal Cohen-Macaulay (MCM) modules over neighborhood earrings. This subject is on the intersection of commutative algebra, singularity conception, and representations of teams and algebras. introductory chapters deal with the Krull-Remak-Schmidt Theorem on forte of direct-sum decompositions and its failure for modules over neighborhood jewelry. Chapters 3-10 examine the valuable challenge of classifying the jewelry with in basic terms finitely many indecomposable MCM modules as much as isomorphism, i.e., earrings of finite CM style. the elemental material--ADE/simple singularities, the double branched disguise, Auslander-Reiten concept, and the Brauer-Thrall conjectures--is lined in actual fact and fully. a lot of the content material hasn't ever ahead of seemed in ebook shape. Examples comprise the illustration concept of Artinian pairs and Burban-Drozd's similar development in size , an advent to the McKay correspondence from the perspective of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation conception, and a cautious therapy of nonzero attribute. the rest seven chapters current effects on bounded and countable CM sort and at the illustration conception of completely reflexive modules

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**Extra resources for Cohen-Macaulay representations**

**Example text**

Reﬂexive) ideal J ∗∗ , where −∗ denotes the dual HomR (−, R). 2) 0 −→ F −→ M −→ J −→ 0 wherein F is a free R-module and J is an ideal of R. The following criterion for a module to be extended is Proposition 3 of [RWW99] (cf. 5)]). 15. Proposition. Let R be a two-dimensional local ring whose m-adic completion R is a normal domain. Let N be a ﬁnitely generated torsion-free R-module. Then N is extended from R if and only if cl(N ) is in the image of the natural homomorphism Φ : Cl(R) −→ Cl(R). Proof.

9, we get a local ring R and indecomposable R-modules A, B, C such that A(t) has only the obvious direct-sum decompositions for t 72, but A(73) ∼ = B ⊕ C. We deﬁne the splitting number spl(R) of a one-dimensional local ring R by spl(R) = Spec(R) − |Spec(R)| . 9 is s − 1. 12 says that every ﬁnitely generated Krull monoid deﬁned by m equations can be realized as +(M ) for some ﬁnitely generated module over a one-dimensional local ring (in fact, a domain essentially of ﬁnite type over Q) with splitting number m.

The functor (V → W ) (V → B ⊗C W ) from (A → C)-mod to (A → B)-mod is faithful and full. The functor is injective on isomorphism classes and preserves indecomposability. 11. Lemma. For a nilpotent ideal I of B, let E = A+I → BI . I V +IW W The functor (V → W ) → IW , from A-mod to E-mod, is IW surjective on isomorphism classes and reﬂects indecomposable objects. 12. Proposition. Let A → B be an Artinian pair for which either (dr1) or (dr2) fails. There is a ring C between A and B such that, with D = C/mC, we have either (i) dimk (D) 4, or (ii) D contains elements α and β such that {1, α, β} is linearly independent over k and α2 = αβ = β 2 = 0.