AS Maths (Instant Revision) by Jenny Sharp, Stewart Townend

By Jenny Sharp, Stewart Townend

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M g'1 V' Z1 Xu0Y v --4. we infer that f'f Z, for Z w I TX is an automorphism. Thus (1X,f'f,g'g) is a retraction. Thus . TX g g'g is a morphism of triangles. is an isomorphism and is Z indecomposable. A similar proof shows the analogous result if we start with a sink morphism. 6 a field left k. By Let A be a finite-dimensional algebra with I over mod A we have denoted the category of finite-dimensional A-modules,by AP (reap. A1) the full subcategory of projective (resp. injective) A-modules. It is well-known and easy to see that AP 37 and are equivalent under the Nakayama functor AI where v = D HomA(-,AA), denotes the duality on mod A with respect to the base field k.

Altogether we see that the stable category nius category f' C of the Frobe- is nothing else but the homotopy category K (C,S) asso- C. Moreover, the above calculation shows that the classical shift functor coincides with the suspension functor. From the description of standard sextuples in Frobenius categories we can easily deduce the classical definition of the mapping cone Cf for a morphism f' : X. -. Y', This is the complex Cf = ((TX')1 0 Y',d' ) with differential f 29 i+1 dX f i+1 dY 0 X1+1 0 Y1 X' E C For instance if satisfies the associated truncated complex is X1 = 0 i < o for (X" = 0 induces a morphism from dX i > 1) Xi+2 0 for T X° to Yi+1 and, if X" i < o, dX, = dX X'* is for whose mapping cone X, .

For this denote by T(A), called the Q = Homk(A,k). Q A-A-bimodule structure in the obvious way: given admits an a',a" E A and q E Q, 26 then k-linear map which sends is the a'qa" Using this we can define a E A q(a"aa'). to T(A). The underlying vectorspace of T(A) = A®Q, and the multiplication is given by (a,q)(a',q') = (aa',aq' + qa') a,a' E A, q,q' E Q. for Then it is straightforward to check that is a Frobenius T(A) algebra. 2 a be an additive category with splitting idem oe = e2 E Homa(X,X) tents.

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