By Y. Yoshino
The aim of those notes is to provide an explanation for intimately a few themes at the intersection of commutative algebra, illustration conception and singularity conception. they're in line with lectures given in Tokyo, but in addition include new examine. it's the first cohesive account of the realm and should supply an invaluable synthesis of modern examine for algebraists.
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Extra resources for Cohen Macaulay modules over Cohen Macaulay rings
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.
Cohen Macaulay modules over Cohen Macaulay rings by Y. Yoshino