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<p>We introduce algebraic local cohomology (ALC) for mixed modules by generalizing ALC for modules over local rings introduced by Tajima, Nakamura, and Nabeshima. We show that ALC can be used to solve membership problems in mixed modules and to compute the codimensions of these mixed modules. We provide an algorithm to compute ALCs for mixed modules and demonstrate it in the context of singularity theory. In that context, the algorithm is shown to be more efficient than existing algorithms for singularities of higher dimension but lower codimension. Moreover, our framework covers a wider range of applications—such as divergent diagrams and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>-equivalence—than do previous frameworks.</p>

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Algebraic local cohomology for mixed modules and its application to singularity theory

  • 期刊论文

  • 发表日期：

    2025年11月4日

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  • Hiroshi Teramoto,
  • Katsusuke Nabeshima

摘要

We introduce algebraic local cohomology (ALC) for mixed modules by generalizing ALC for modules over local rings introduced by Tajima, Nakamura, and Nabeshima. We show that ALC can be used to solve membership problems in mixed modules and to compute the codimensions of these mixed modules. We provide an algorithm to compute ALCs for mixed modules and demonstrate it in the context of singularity theory. In that context, the algorithm is shown to be more efficient than existing algorithms for singularities of higher dimension but lower codimension. Moreover, our framework covers a wider range of applications—such as divergent diagrams and \(\phi \) ϕ -equivalence—than do previous frameworks.