A course in commutative algebra by Ash R.B.

By Ash R.B.

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Then we have additivity of length, that is, l(A1 ) − l(A2 ) + · · · + (−1)n−1 l(An ) = 0. This is probably familiar for a short exact sequence 0 → N → M → M/N → 0, where the additivity property can be expressed as l(M ) = l(N ) + l(M/N ). ) The general result is accomplished by decomposing a long exact sequence into short exact sequences. ) To see how the process works, consider an exact sequence GA f GB 0 Our first short exact sequence is g GC h GD i GE G 0. 0 → A → B → coker f → 0. Now coker f = B/ im f = B/ ker g ∼ = im g (= ker h), so our second short exact sequence is 0 → im g → C → coker g → 0.

By definition, M ∗ is the union of the Mn∗ over all n ≥ 0. Therefore M ∗ is finitely generated ∗ for some m, in other words, Mm+k = I k Mm for all k ≥ 1. over R∗ if and only if M ∗ = Mm Equivalently, the filtration {Mn } is I-stable. 6 Induced Filtrations If {Mn } is a filtration of the R-module M , and N is a submodule of M , then we have filtrations induced on N and M/N , given by Nn = N ∩ Mn and (M/N )n = (Mn + N )/N respectively. 7 Artin-Rees Lemma Let M be a finitely generated module over the Noetherian ring R, and assume that M has an I-stable filtration {Mn }, where I is an ideal of R.

Call the gi coherent if θi+1 ◦ gi+1 = gi for all i. Then the gi can be lifted to a homomorphism g from M to M . Explicitly, g(x) = (gi (x)), and the coherence of the gi forces the sequence (gi (x)) to be coherent. An inverse limit of an inverse system of rings can be constructed in a similar fashion, as coherent sequences can be multiplied componentwise, that is, (xi )(yi ) = (xi yi ). 2 Examples 1. Take R = Z, and let I be the ideal (p) where p is a fixed prime. Take Mn = Z/I n and θn+1 (a + I n+1 ) = a + I n .

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