By Graham J. Leuschke

This booklet is a complete therapy 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 distinctiveness of direct-sum decompositions and its failure for modules over neighborhood earrings. Chapters 3-10 research the important challenge of classifying the earrings with in basic terms finitely many indecomposable MCM modules as much as isomorphism, i.e., jewelry of finite CM kind. the basic material--ADE/simple singularities, the double branched disguise, Auslander-Reiten thought, 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 publication shape. Examples contain 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 remedy of nonzero attribute. the remainder seven chapters current effects on bounded and countable CM kind and at the illustration concept of completely reflexive modules

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Otherwise, we can write αβ = a + bα + cβ with a, b, c ∈ k. 14. 20. Assumption. We assume from now on that dimk (N) 1. 17 we see that s (the number of components) cannot be 2. Also, if s = 3, then, after renumbering if necessary, we have N = m1 × 0 × 0 with m1 = 0. 18. We have proved that either s = 1 or s 4. 21. Case. s = 1 (D is local). 20, K := D/N must have degree at least three over k. 18 implies that each element of K has degree at most 2 over k. Therefore K/k is not separable, char(k) = 2, α2 ∈ k for each α ∈ K, and [K : k] 4.

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. Proof. Let R be the m-adic completion of R, and write R ⊗R (n ) (n ) M = V1 1 ⊕ · · · ⊕ Vt t , where each Vi is an indecomposable R-module (a ) and each ni > 0. 9. Thus we have a well-deﬁned map j : + (M ) −→ Nt0 , taking [L] to (a1 , . . , at ).

X), where x and y range over k (n) . ) Clearly DW = V . To see that (V → W ) is indecomposable, suppose ϕ is an endomorphism of (V → W ), that is, a D-endomorphism of W carrying V into V . We write ϕ = (α, β, γ, δ, ε5 , . . , εs ), where each component is an n × n matrix over k. Since ϕ((x, 0, x, x, x, . . , x)) and ϕ((0, y, y, Hy, 0, . . , 0)) are in V , there are matrices σ, τ, ξ, η satisfying the following two equations for all x ∈ k (n) : (αx, 0, γx, δx, ε5 x, . . , εs x) = (σx, τ x, (σ + τ )x, (σ + Hτ )x, σx, .