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16 results

Poincaré Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations
  • Dagmar M. Meyer, Larry Smith
  • Published online:
    29 January 2010
    Print publication:
    18 August 2005
  • Poincaré duality algebras originated in the work of topologists on the cohomology of closed manifolds, and Macaulay's dual systems in the study of irreducible ideals in polynomial algebras. These two ideas are tied together using basic commutative algebra involving Gorenstein algebras. Steenrod operations also originated in algebraic topology, but may best be viewed as a means of encoding the information often hidden behind the Frobenius map in characteristic p<>0. They provide a noncommutative tool to study commutative algebras over a Galois field. In this Tract the authors skilfully bring together these ideas and apply them to problems in invariant theory. A number of remarkable and unexpected interdisciplinary connections are revealed that will interest researchers in the areas of commutative algebra, invariant theory or algebraic topology.

  • HIT POLYNOMIALS AND EXCESS IN THE MOD P STEENROD ALGEBRA

  • Dagmar M. Meyer
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 44 / Issue 2 / June 2001
    Published online by Cambridge University Press:
    20 January 2009, pp. 323-350
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  • Let $p$ be an odd prime. The primary purpose of this paper is to determine the excess of the conjugates of the Steenrod operations $\mrm{P}[k;f]$, which are defined as $\mrm{P}[k;f]:=\mrm{P}(p^{k-1}f)\cdot\mrm{P}(p^{k-2}f)\cdot\cdots\cdot\mrm{P}(pf)\cdot\mrm{P}(f)$. The result is then used to obtain sufficient conditions for an element in the polynomial algebra $\mathbb{F}_p[x_1,\dots,x_s]$ to be in the image under the standard action of the Steenrod algebra. Results and methods are generalizations of previous work by Judith Silverman and by myself with Judith Silverman.

    AMS 2000 Mathematics subject classification: Primary 55S10. Secondary 55S05