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It is proved that for a simplicial complex Sullivan's de Rham complex is definable in terms of its 0-forms. 1. Statement of the result. Sullivan [1] constructed, for every simplicial complex K, a polynomial de Rham complex AK and proved that this differential graded commutative algebra over the rationals Q contains much information about the rational homotopy type of K. We demonstrated recently [2] that this polynomial de Rham complex AK, and in fact already its algebra of polynomial 0-forms A?K, contains all the information about the whole homotopy type of K. This result suggests the possibility of recovering the whole de Rham complex AK from its algebra of 0-forms A OK, and the purpose of this note is to show that this indeed can be done. In more detail: Let (DG) be the functor which (see ?2) assigns to every commutative algebra Y (over Q) the "free-est" differential graded commutative algebra with Y in degree 0. (In algebraic geometry this is often denoted by Q0(y).) Then we have THEOREM. For every finite simplicial complex K the obvious (see ?2) map p: (DG)A OK -* AK has the properties (i) p is onto, and (ii) x E ker p if and only if, for every finitely generated polynomial algebra P and every map q: (DG)A?K-* (DG)P, one has x E ker q. The rest of this note consists of a proof of this theorem. It uses some of the results of [2] and Bousfield-Gugenheim [1]. The latter reference contains a clear account of Sullivan's theory for simplicial complexes and its extension to simplicial sets. 2. Proof of the theorem. We first recall that, for a commutative algebra Y (over Q), the "free-est" differential graded commutative algebra (DG) Y with Y in degree 0 is the tensor product of the polynomial algebra on symbols [y] for all y E Y and the exterior algebra on symbols [dy] for all y E Y, modulo the ideal generated by the relations [Y + Y'] = [y] + [y'] [d(y + y')] = [dy] + [dy'], [Y -Y'] = [Y] . [y'], [d(y *y')] = [y] [dy'] + [y'] [dy], Received by the editors June 9, 1975. AMS (MOS) subject classifications (1970). Primary 58A10, 57C05, 53C65.

Sullivan’s de Rham complex is definable in terms of its 0-forms