Bockstein Cohomology of Associated Graded Rings

Abstract

Let \((A,\mathfrak {m})\) be a Cohen-Macaulay local ring of dimension d and let I be an \(\mathfrak {m}\)-primary ideal. Let G be the associated graded ring of A with respect to I and let \(\mathcal R = A[It,t^{-1}]\) be the extended Rees ring of A with respect to I. Notice t− 1 is a nonzero divisor on \(\mathcal R\) and \(\mathcal R/t^{-1}\mathcal R = G\). So, we have Bockstein operators\(\beta ^{i} {\colon } {H}^{i}_{{G}_{+}}(G)(-1) \rightarrow {H}^{i + 1}_{{G}_{+}}(G)\) for i ≥  0. Since βi+ 1(+ 1) ∘ βi = 0, we have Bockstein cohomology modules BHi(G) for i = 0,…,d. In this paper, we show that certain natural conditions on I implies vanishing of some Bockstein cohomology modules.

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Correspondence to Tony J. Puthenpurakal.

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Puthenpurakal, T.J. Bockstein Cohomology of Associated Graded Rings. Acta Math Vietnam 44, 285–306 (2019). https://doi.org/10.1007/s40306-018-00324-z

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Keywords

  • Associated graded rings
  • Rees Algebras
  • Local cohomology

Mathematics Subject Classification (2010)

  • Primary 13A30
  • Secondary 13D40
  • 13D07