Proposition

Lemma 12 ([Projection preserving kernel representatives]). Let (H,δ) be a finite-dimensional graded complex over a field, and let K⊂H be any graded subspace. There is a chain projection P:H→H , homotopic to the identity, whose image is a …

Lemma 12 ([Projection preserving kernel representatives]). Let (H,δ) be a finite-dimensional graded complex over a field, and let K⊂H be any graded subspace. There is a chain projection P:H→H , homotopic to the identity, whose image is a space of representatives for H(H,δ) and which satisfies P(K)⊂K∩δ.

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Lemma 11 ([Kernel support]). Let v∈_ i=1 ^nH_i be killed by B_iB_jB_k for every…Lemma 10. For every three distinct active indices i,j,k , there are actual…Proposition
Write Z=δ and B=δ . Choose a complement A_K to K∩B in K∩Z , and extend A_K to a complement A to B in Z . Choose a complement W_K to K∩Z in K . It is disjoint from Z , so extend it to a complement W to Z in H . All these choices are made degree by degree. Then H=B⊕A⊕W . Let P be projection onto A . It is a chain map, and P(K)⊂A_K⊂K∩Z . The map δ:W→B is an isomorphism with the appropriate shift of degrees. Define a degree-minus-one map s to be its inverse on B , and zero on A⊕W . Then -P=δs+sδ.
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