Depth and the Steenrod algebra (with an appendix by J. Lannes). (English) Zbl 0874.55017
Let \(p\) be a prime number and \(\mathbb{F}_p[u_1,\dots,u_n]\) be the graded polynomial ring on \(n\) variables of degree 1 if \(p=2\) and 2 if \(p>2\). Let \(\text{GL}_n(\mathbb{F}_p)\) be the group of invertible \(n\times n\) matrices over \(\mathbb{F}_p\). It is known that the ring of invariants \(D_n=(\mathbb{F}_p[u_1,\dots,u_n])^{\text{GL}_n(\mathbb{F}_p)}\) called the Dickson algebra is a graded polynomial ring \(\mathbb{F}_p[c_1,\dots,c_n]\) on the Dickson invariants. Let \(A\) be the \(\text{mod }p\) Steenrod algebra and \(M\) be an unstable \(\mathbb{F}_p[u_1,\dots,u_n]\)-\(A\)-module which is of finite type as \(\mathbb{F}_p[u_1,\dots,u_n]\)-module. The depth of \(M\) relative to the augmentation ideal of \(\mathbb{F}_p[u_1,\dots,u_n]\) is well defined and it is the length of a maximal regular sequence on \(M\). The main result of this paper is:
Theorem 1. Let \(M\) be an unstable \(\mathbb{F}_p[u_1,\dots,u_n]\)-\(A\)-module which is of finite type as an \(\mathbb{F}_p[u_1,\dots,u_n]\)-module. Then the depth of \(M\) is the largest \(r\) such that \(\{c_1,\dots,c_r\}\) is a regular sequence on \(M\).
As a consequence of theorem 1 we get a solution of the Landweber-Stong depth conjecture for a ring of invariants over the field \(\mathbb{F}_p\).
Proposition 2. Let \(G\) be a subgroup of \(\text{GL}_n(\mathbb{F}_p)\). The depth of the ring of invariants \((\mathbb{F}_p[u_1,\dots\), \(u_n])^G\) is the largest \(r\) such that \(\{c_1,\dots,c_r\}\) is a regular sequence on \((\mathbb{F}_p[u_1,\dots,u_n])^G\).
Theorem 1. Let \(M\) be an unstable \(\mathbb{F}_p[u_1,\dots,u_n]\)-\(A\)-module which is of finite type as an \(\mathbb{F}_p[u_1,\dots,u_n]\)-module. Then the depth of \(M\) is the largest \(r\) such that \(\{c_1,\dots,c_r\}\) is a regular sequence on \(M\).
As a consequence of theorem 1 we get a solution of the Landweber-Stong depth conjecture for a ring of invariants over the field \(\mathbb{F}_p\).
Proposition 2. Let \(G\) be a subgroup of \(\text{GL}_n(\mathbb{F}_p)\). The depth of the ring of invariants \((\mathbb{F}_p[u_1,\dots\), \(u_n])^G\) is the largest \(r\) such that \(\{c_1,\dots,c_r\}\) is a regular sequence on \((\mathbb{F}_p[u_1,\dots,u_n])^G\).
Reviewer: D.Bourgiba, S.Zarati (Tunis)
MSC:
| 55S10 | Steenrod algebra |
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