By Bendsoe M.P.

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**Example text**

K*{a))/C*) = H p _ i ( C ) . ) ffi(C. (a)) < . 4. A-module HP(C*). cp-O = (9pccp + ( - l r - ^ C p - i . a p l i C p - ! ) . 1 shows that the elements (—l)p~1acp_i modulo boundaries constitute the range of the joint homomorphism r p . Furthermore, let (c p ,c p _i) be a cycle. 28) that (ac p ,ac p _0 = c ^ ( O , ( - l ) % ) is a boundary. This fact states the following. 5. A multiplication by a annihilates the homology groups of the chain complex C* 0 K(a). , a r _i; Q). ar;Q) T <—#o(ai, • • • ar-i; Q) 2-Ho(a±,..

2. Any AC-algebra A can be extended to a unital algebra A by the adjunction of the identity 1 to A. The algebra A, called the unital extension of A, is defined as the direct sum of ^-modules K © A provided with the multiplication (Ai,ai)(A2,a2) = (AiA2,Aia2 + A 2 ai+aia 2 ), Ai,A 2 £/C, ai,a2eA Elements of A can be written as (A, a) = Al + a, A € /C, a G A. Let us note that, if A is a unital algebra, the identity 1^ in A fails to be that in A. 5C), all associative algebras are assumed to be commutative, unless they are graded.

It is a /C-module. ) = BQ/Imdi. 1) is exact at a term Bp if HP(B*) = 0. It is an exact sequence if all homology groups are trivial. 1) is called acyclic if its homology groups Hp>o are trivial. A chain complex 5» is acyclic if there exists a homotopy operator h. This is defined as a set of module morphisms hp:Bp^Bp+1, peN, such that h p _! odp + dp+l o h p = IdB p , p&N+. It follows that, if dpbp = 0, then bp = dp+i(hpbp), and Hp>0(Bt,) = 0. 1) is said to be a chain resolution of a module B if it is acyclic and H0(B*) = B.