CHAPTER 2 m-coalgebras 1. Preliminaries In this section we define m-structures and the coherency of such structures. DEFINITION 1.1. Let C and D be two graded Z-modules. A map of graded modules / : Ci Di+k will be said to be of degree k. REMARK. 1.1.1. For instance the differential of a chain-complex will be regarded as a degree 1 map. We will make extensive use of the Koszul Convention (see [9]) regarding signs in homological calculations: DEFINITION 1.2. If/: d —• Di,g\ C2 D2 are maps, and a 06 e Ci0C 2 (wherea is a homogeneous element), then (/ 0 g) (a 06) is defined tobe(-l) d e g ( ^ d e g ( a ) /(a) 0 9(b). Remarks. 1.2.1. This convention simplifies many of the common expressions that occur in homological algebra in particular it eliminates complicated signs that occur in these expressions. For instance the differential, 9®, of the tensor product C 0 D is just 9 c 0 l + l 0 d i ) . 1.2.2. Throughout this entire paper we will follow the convention that group- elements act on the left. Multiplication of elements of symmetric groups will be carried out accordingly i.e. (1,2, 3)* (1,4) = result of applying (1,4) first and then (1,2, 3) = (1,4, 2, 3) rather than (1,2, 3,4). 1.2.3. Let /,•, gi, i 1, 2, be maps. It isn't hard to verify that the Koszul convention implies that (h 0 9l ) o (/2 0 g2) = (-l)deg(5l)deg(/2)(/i Q f2 ® gi Q g2 ). 1.2 A. We will also follow the convention that, if / is a map between chain-complexes, df = d o f (—l)deg(/)/ o d. The compositions of a map with boundary operations will be denoted by d o / and / o d - see [9]. This convention clearly implies that d(f o g) (df) o g + (—l)des(/)/ o dg. We will call any map / with df = 0 a chain- map. We will also follow the convention that if C is a chain-complex and j : C —+ TC and l:C —* Y*~lC are, respectively, the suspension and desuspension maps, then j and 9
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