Quillen stratification for the Steenrod algebra

John H. Palmieri

Review from Zentralblatt MATH:

Let $A$ be the mod 2 Steenrod algebra and let ${\cal Q}_A$ be the category whose objects are elementary sub-Hopf algebras of $A$ and whose morphisms are inclusions. The author considers the ``Quillen map'' $q_A$: $H^*(A;\bbfF_2)\to\varprojlim_{{\cal Q}_A}H^*(E;\bbfF_2)$ induced by the restriction maps $H^*(A;\bbfF_2)\to H^*(E;\bbfF_2)$. He proves the following results:

(i) There is an action of $A$ on the inverse limit $\varprojlim_{{\cal Q}_A}H^*(E;\bbfF_2)$ and, the map $q_A$ factors through the invariants $(\varprojlim_{{\cal Q}_A}H^*(E;\bbfF_2))^A$.

(ii) The ``Quillen map'' $q_A$: $H^*(A;\bbfF_2)\to(\varprojlim_{{\cal Q}_A}H^*(E;\bbfF_2))^A$ is an $F$-isomorphism $(\ker(q_A)$ and $\text {coker}(q_A)$ are nilpotent).

The author computes also the inverse limit $\varprojlim_{{\cal Q}_A}H^*(E;\bbfF_2)$, gives explicitely the action of $A$ on it and describes the algebra of invariants. This gives the mod 2 cohomology of $A$, $H^*(A;\bbfF_2)= \text {Ext}^{*,*}_A(\bbfF_2, \bbfF_2)$, modulo nilpotent elements.

Reviewed by Saïd Zarati

Keywords: Hopf algebra; $F$-isomorphism

Classification (MSC2000): 55S10 55Q45 55T15

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