2. DIFFERENTIAL HOMOLOGICAL ALGEBRA In this section, we recall definitions and facts concerning differential graded modules. 2.1. DEFINITIONS. Let N = ® N1 be a Z-graded R-module (graded module for short). We define graded modules E N, Nf and T N respectively by (I^N)1 = N1_n, (N1)1 = (N"V, and (T N)1 T j = © . + + . i N19...8Np. J 1 * * * J p p , ! . . . , p Here and throughout this paper, we put 8 = ®« for brevity. Then T*N = © T N is a graded algebra by the natural multiplication. Let S*N - S N be the quotient graded algebra of T^N by the two sided ideal generated by {x S y-(-l)Pqy ® x x e Np, y Nq, p,q e Z}. If {6-J 1 , i £ n} is a homogeneous basis of N, we denote S^N by S*(6,,...,0 ) and call it the free anti-commutative graded algebra generated by 6,,...,8 . We define a graded algebra S*N by (S^N)1 = (S^N1)"1. Note that SPN can be naturally identified with the space of anti-commutative multilinear forms on NX...XN (p times). Let N, and N2 be graded modules. Their tensor product N, ® N2 i i k is regarded as a graded module by (N1 ® N2) = & = + k N^ ® N2. In the case N is a topological graded module, that is, each N is a topological vector space, the above constructions are done in the category of topological vector spaces. In particular (N~ )' is the dual space of N 1 with the strong topology, T N is equipped with the ir-topology, and SPN is the space of all the continuous anti-commutative multilinear forms on NX...XN (p times). Furthermore T N and S N denote the completions of T N and S N respectively. Similarly, if N.. and N2 are topological graded modules, N.. 8 N2 is endowed with the TT-topology and we define N, ® N2 = N ® N ? Let N be a graded module and suppose linear maps c : N1 N1 10
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