Y. Yoshino's Cohen Macaulay modules over Cohen Macaulay rings PDF

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Cohen Macaulay modules over Cohen Macaulay rings by Y. Yoshino PDF

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Example text

Actually if q • g were split, then g would be an isomorphism hence q would be split. If c - a were split, then so would a, both of which are absurd. Thus we have seen that h = c- a + q - g belongs to (N,E)\, proving that (p is a monomorphism. 9)'. 6) LEMMA. Let M and N be indecomposable CM modules and assume thai there is an AR sequence starting from N: 0—>N—>E—>T1 Let v! be the number of indecomposable CM modules isomorphic to M which appear in direct summands of E. Then the following equality holds: irr(iV, M) = ri.

PROOF: Let / be an isomorphism from M/x^M onto N/x^N. 15) we have a homomorphism / from M to N such that / 0 R/xR = / 0 R/xR. In particular / is epimorphic by Nakayama's lemma. Thus we obtain an exact sequence 0 —• Ker(/) —• M —• N —*• 0, where it is easily seen that Ker(/) is also a CM module. Tensoring R/xR with this sequence we have an exact sequence: 0 —> Ker(/) ®R/xR —> M 0 R/xR f ^ R N 0 R/xR —> 0. Since / 0 R/xR is an isomorphism, we see that Ker(/) 0 R/xR = 0, hence Ker(/) = 0 again by Nakayama's lemma.

And F is finitely presented if there is an exact sequence in Mod(<£) ; ( , N) —• ( , M) —> F —• 0 with M and AT in <£. To see the following is so easy that we leave its proof to the reader as an exercise. 5) Exercise. ft-module. 6) DEFINITION. We denote by mod(£) the full subcategory of Mod(£) consisting of all finitely presented functors, and call it the Auslander category of <£. 1) makes it possible to regard C as a subcategory of Mod(£). 3) shows that C is a full subcategory of mod(C) . As remarked before it can be easily seen that Mod(C) is an Abelian category.

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