# Cohen-Macaulay representations by Graham J. Leuschke

By Graham J. Leuschke

This publication is a entire therapy of the illustration concept of maximal Cohen-Macaulay (MCM) modules over neighborhood jewelry. 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 distinctiveness of direct-sum decompositions and its failure for modules over neighborhood earrings. Chapters 3-10 research the significant challenge of classifying the jewelry with purely 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 hide, Auslander-Reiten concept, and the Brauer-Thrall conjectures--is coated essentially and fully. a lot of the content material hasn't ever ahead of seemed in booklet shape. Examples comprise the illustration idea of Artinian pairs and Burban-Drozd's comparable development in size , an creation to the McKay correspondence from the perspective of maximal Cohen-Macaulay modules, Auslander-Buchweitz's MCM approximation thought, and a cautious therapy of nonzero attribute. the remainder seven chapters current effects on bounded and countable CM sort and at the illustration conception of completely reflexive modules

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27 implies that f = 0. This construction is far from new. See, for example, the papers of Higman [Hig54], Heller and Reiner [HR61], and Warﬁeld [War70]. A similar construction arises in the classiﬁcation of pairs of matrices up to simultaneous equivalence (see Dieudonn´e’s discussion [Die46] of the work of Kronecker [Kro74] and Weierstrass [Wei68]). 4. Proposition. Let (S, n, k) be a CM local ring of dimension at least two, and let z be an indeterminate. Set R = S[z]/(z 2 ). Then R has indecomposable MCM modules of arbitrarily large rank.

5. 23. 2]). 11, as follows. Let C = k (r1 ) , viewed as column vectors. Deﬁne the “truncated diagonal” (r ) (r ) ∂ : C −→ W = D1 1 × · · · × Ds s by sending an element [c1 , . . , cr1 ]tr to the vector whose ith entry is [c1 , . . , cri ]tr . ) Let V be the k-subspace of W consisting of all elements ∂(u) + X∂(v) + X 3 ∂(Hv) , as u and v run over C, where X = (x, 0, . . , 0) and H is the nilpotent Jordan block with 1 on the superdiagonal and 0 elsewhere. (i) Prove that W is generated as a D-module by all elements of the form ∂(u), u ∈ C, so that in particular DV = W .

But X = 0 and X ∼ = L (else j([X]) = j([L])), and we have a contradiction to the indecomposability of L. 20), j(+(M )) contains only ﬁnitely many minimal non-zero elements, and, by what we have just shown, add(M ) has only ﬁnitely many isomorphism classes of indecomposable modules. For the last statement, let n = μR (M ), the number of elements in a minimal generating set for M , and let {N1 , . . , Nt } be a complete set of representatives for the isomorphism classes of direct summands (r ) (r ) of M .