A spectral sequence for string cohomology

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Let X be a 1-connected space with free loop space LX. We introduce two spectral sequences converging towards H*(LX;Z/p) and H*((LX)ₕT;Z/p). The E2-terms are certain non Abelian derived functors applied to H*(X;Z/p). When H*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.