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Transactions of the American Mathematical Society

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

 
 

 

A phenomenon of reciprocity in the universal Steenrod algebra


Author: Luciano Lomonaco
Journal: Trans. Amer. Math. Soc. 330 (1992), 813-821
MSC: Primary 55S99; Secondary 18G10, 18G15, 55U99
DOI: https://doi.org/10.1090/S0002-9947-1992-1044963-2
MathSciNet review: 1044963
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Abstract: In this paper we compute the cohomology algebra of certain subalgebras $ {L_r}$ and certain quotients $ {K_s}$ of the $ \bmod\, 2$ universal Steenrod algebra $ Q$, the algebra of cohomology operations for $ {H_\infty }$-ring spectra (see $ [$M$ ]$). We prove that

$\displaystyle \operatorname{Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname{Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}}$

with $ r$, $ s$ integers and $ r \leq 1$, $ s \geq 0$. We also observe that some of the algebras $ {L_r}$, $ {K_s}$ are well known objects in stable homotopy theory and in fact our computation generalizes the fact that $ {H^{\ast} }({A_L}) \cong \Lambda ^{{\text{opp}}}$ and $ {H^{\ast} }({\Lambda ^{{\text{opp}}}}) \cong {A_L}$ (see, for instance, $ [$P$ ]$). Here $ {A_L}$ is the Steenrod algebra for simplicial restricted Lie algebras and $ \Lambda $ is the $ {E_1}$-term of the Adams spectral sequence discovered in $ [$B-S$ ]$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1044963-2
Keywords: Universal Steenrod algebra, Koszul algebras
Article copyright: © Copyright 1992 American Mathematical Society

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