By Werner Hildbert Greub, Stephen Halperin, James Van Stone

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0. Algebraic Preliminaries 11 A homomorphism p : ( X , S ,) + ( Y , 6), of differential spaces is a linear map v: X -,Y such that pdX = 6,~). It restricts to maps between the cocycle and coboundary spaces, and so induces a map, written p:H ( X ) + H( Y ) , between the cohomology spaces. Assume (X", 6) and ( X , a) are finite-dimensional differential spaces such that X*, X is a pair of dual spaces, and 6 = &a*. Then the scalar product induces a scalar product between H ( X * ) and H ( X ) . A graded differential space (X, 6 ) is a differential space together with a gradation in X , such that 6 is homogeneous of some degree.

6. Evidently restricts to isomorphisms G! maps Ef;f into HPgq(Ei,di). Hence it E ? $ Z HPSq(Ei, di). In the same way it follows that the isomorphism a&: E& 5 A$(M) (cf. sec. ' = N A:&). We close this section with a condition that forces the collapse of a spectral sequence. Proposition IV: Let (M, 6 ) be a graded filtered differential space with spectral sequence ( E i , di). Assume that, for some m, Em is evenly graded with respect to the total gradation : Ek) = 0, r odd. Then the spectral sequence collapses at the mth term.

The substitution operator is dual to the multiplication operator p ( x ) in AX defined by p(x)b= x A b E AX. b, More generally, if a E AX, then p ( a ) is the multiplication operator given by p ( a ) b = a A b. The dual operator is denoted by i ( a ) . Clearly, i(x, A . A x p ) = i ( x p ) . 0 0 i(xl), xi E X . The following result is proved in [ S ; Prop. 11, p. 1381. Proposition I: Let A c AX" be a subalgebra, stable under the operators i ( x ) , x E X . Then A = A(X* n A ) . Next, suppose X = Y @ Z .