Universal coefficient theorem in triangulated categories

PIRASHVILI, TEIMURAZ; REDONDO, MARIA JULIA

Algebras and Representation Theory

SPRINGER

Año: 2008 vol. 11 p. 107 - 114

1386-923X

It is a well-known result (available in J.-L. Verdier´s thesis [Astérisque No. 239 (1996), xii+253 pp. (1997); MR1453167 (98c:18007)]) that a triangulated category which is abelian must be semi-simple. This is in some sense the ``dimension zero´´ case. The paper under review deals with a ``dimension one´´ case. More precisely, consider a (pre)-triangulated category T and an ideal I of morphisms in T, such that I does not contain the identity of any nonzero object. Then, if the quotient A= T / I is abelian, it is necessarily hereditary. Moreover, the ideal I necessarily squares to zero and can be identified with an Ext-group in A as follows: I(X,Y)=Ext_A^1(X[1],Y). Finally, T is automatically idempotent complete.