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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 overthe mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of$\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. Thefirst, which we call the "topological complex", is defined using the orbitfiltration of $X$. The second, which we call the "algebraic complex", isdefined 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 thetheory of unstable $\mathrm{H}^{*}V$-$\mathrm{A}$-modules which is a by-productof the researches on Sullivan conjecture. There is a noteworthy overlap betweenthe topological part of our memoir and the paper "Syzygies in equivariantcohomology in positive characteristic", by Allday, Franz and Puppe, which hasjust appeared; however our techniques are quite different from theirs (the name"Steenrod" does not show up in their article).

Complexes de modules \'equivariants sur l'alg\`ebre de Steenrod associ\'es \`a un $(\mathbb{Z}/2)^{n}$-CW-complexe fini