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By Steven Dale Cutkosky

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As before, £ denotes the category of CM modules over R and we write ( , ) instead of Hom#( , ). 1) DEFINITION. 1) Write the decompositions of M and N into indecomposable modules as M = £,• M{ and N = J2j Nj an< l decompose g along this decomposition as g — (gij) where gij € (Mi, Nj). Then no gij is an isomorphism. 1) above such that 92 ' 9\}- It is easy to see from the definition that (M,N) D (M,N)\ D (M,N)i 3 ... 3 (M,N)n D (M, 7V)n+i D ... is a sequence of i£-submodules and that if M and TV are indecomposable, then (M, N)i is the set of all nonsplit homomorphisms of M into N.

11) PROPOSITION. Let M be an indecomposable CM module over R which is locally free on the punctured spectrum of R. If M is not free, then there always exists an AR sequence ending in M, and the AR translation is given by r(M) = (syzdti(M))'. 10) and let J be the Jacobson radical of A. First we note that A is an Artinian ring which, of course, is local. 8) that there is an exact sequence , R) ®R M -±+ Endtf(M) —> A —• 0. 22 Chapter 3 Since Mp is a free ify-module for any prime ideal p ^ m, we observe that (q)p is an isomorphism for those p.

N ) . PROOF: Let S denote the set R — U,^Pt- which is multiplicatively closed in R. pi(S~lR) (1 < iI < n). Since # • • • > xd} from R which forms a separable system of parameters for R/y(xi,X2)... 20). i, #2,.. • , x^} is a system of parameters for R, we can consider the power series ring T = k{x\, #2,...

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An Introduction to Galois Theory [Lecture notes] by Steven Dale Cutkosky

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