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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Invariant theory and the lambda algebra
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by William M. Singer PDF
Trans. Amer. Math. Soc. 280 (1983), 673-693 Request permission

Abstract:

Let $A$ be the Steenrod algebra over the field ${F_2}$. In this paper we construct for any left $A$-module $M$ a chain complex whose homology groups are isomorphic to the groups $\operatorname {Tor}_s^A({F_2},M)$. This chain complex in homological degree $s$ is built from a ring of invariants associated with the action of the linear group $G{L_s}({F_2})$ on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of $G{L_2}({F_2})$. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 673-693
  • MSC: Primary 55Q45; Secondary 55S10, 55T15, 55U10
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0716844-7
  • MathSciNet review: 716844