ON PL DE RHAM THEORY AND RATIONAL HOMOTOPY TYPE 51. The Simplicial Algebra v Our approach to PL De Rham theory is based on the simplicial algebra v which corresponds to the system of polynomial differential forms on the standard simplexes. In this section we construct v and deduce its basic properties. Let k be a field of characteristic 0. By an algebra over k we mean a non-negatively graded k-module A together with maps of k-modules ja:A&A A, *n:k A such that |a(A$ia) = |j(n^A) and (^(n&A) = |a(A®Ti) = A with the usual identification A&k=k^A=A. k itself is treated as an algebra, concentrated in dimension 0, so that r\ is a map of algebras. We allow the case r\-0, A=0 if Tl^O, then it is a monomorphism. As usual, we let 1 e A denote r| l and let xyeA11*"11 denote ^(x&y) for x e Am, y An. By a DG (i.e. differential graded) algebra over k we mean an algebra A over k together with maps of k-modules d:A A , m ° such that dd=0 and d(xy) = (dx)y+ (-l)mx(dy) for x Am, y e An d is called a differential. By v (p*) P^0 we denote the commutative (i.e., "skew commuta- tive") algebra generated by indeterminates tQ,t,,...,t in grading 0, dt0,dt.. ,...,dt in grading 1 and subject to the relations tn+ t,+ ... + t = 1 0 dtA + dt1 + ... + dt =0 0 1 p We can also regard v (p,*) as freely generated by t1, ...,t ,dt1,... ,dt . The submodule of grading q will be denoted by v(p,q). Note that v(0,*) = k and v(p,q) = 0 if qp. The notation is incomplete and when needed, we shall write t.(p) for t, ev(p,*). We now let d be the unique differential on the algebra v(p,*) such that d(t.) = dt. for 0ip, and we treat v(p,*) as a DG algebra from now on. One may regard v(p,*) as the algebra of Received by the editor May 23, 1975, and revised April 7, 1976. 1
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