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I try to determine the $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ as a $\mathbb{F}_2$- vector space, in which $P_6$ is polynomial algebra $\mathbb{F}_2[x_1,x_2,\dots,x_6]$ and $A$ is Steenrod algebra.

Denote the monomial $x_1^a x_2^b x_3^c x_4^d x_5^e x_6^f$ by $(a,b,c,d,e,f)$.

I consider the monomial $(a,b,c,d,e,f)$ in 6 cases, for example, case 1: the monomial depends only on one variable.

I do not know what is the genarator of $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ and its dimension. So, I just consider case by case except any criterion for hit problem.

My question is: What is the generator of $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ and its dimension?

Can you help me to give some ideas for this problem? Thanks in advance.

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  • $\begingroup$ How is $P_6$ and $A$-module? $\endgroup$
    – Fernando Muro
    Commented Oct 7, 2013 at 11:40
  • $\begingroup$ Yes. Let $Sq^j$ be a square operation in Steenrod algebra. We have $$Sq^j(x)=\left\{\begin{matrix} x, \ \ \ \text{if $j=0$,}\\ x^2, \ \ \ \text{if $j=1$,}\\ 0. \ \ \ \text{otherwise.} \end{matrix}\right.$$ And it satisfies the Cartan fomular $$Sq^{n}(fg)=\sum_{i=0}^nSq^i(f)Sq^{n-i}(g).$$ $\endgroup$
    – user69833
    Commented Oct 7, 2013 at 11:45
  • $\begingroup$ You mean that for $x= x_1,\dots, x_6$? What's the degree of the $x_i$? $\endgroup$
    – Fernando Muro
    Commented Oct 7, 2013 at 11:47
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    $\begingroup$ $P_6$ means the cohomology of a 6-fold product of $\mathbf{R}P^\infty$ with itself. Thus it's a polynomial ring on 6 generators, each in degree 1. $\endgroup$
    – John Palmieri
    Commented Oct 7, 2013 at 14:17
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    $\begingroup$ Fernando: If $P_1=\mathbb{F}_2[x]$, then $\mathbb{F}_2 \otimes_A P_1$ is spanned by the elements $x^{2^n-1}$. So it's more complicated than that. $\endgroup$
    – John Palmieri
    Commented Oct 7, 2013 at 18:08

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