Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $mathrm{A}$, equipped with an action of $mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the topological complex, is defined using the orbit filtration of $X$. The second, which we call the algebraic complex, is defined just in terms of the unstable $mathrm{A}$-module $mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of unstable $mathrm{H}^{*}V$-$mathrm{A}$-modules which is a by-product of the researches on Sullivan conjecture. There is a noteworthy overlap between the topological part of our memoir and the paper Syzygies in equivariant cohomology in positive characteristic, by Allday, Franz and Puppe, which has just appeared; however our techniques are quite different from theirs (the name Steenrod does not show up in their article).
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