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), { m 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 $scr T$ and an ideal $I$ of morphisms in $scr T$, such that $I$ does not contain the identity of any nonzero object. Then, if the quotient $scr{A}=scr{T}/I$ is abelian, it is necessarily hereditary. Moreover, the ideal $I$ necessarily squares to zero and can be identified with an Ext-group in $scr{A}$ as follows: $I(X,Y)={ m Ext}_{scr{A}}^1(X[1],Y)$. Finally, $scr{T}$ is automatically idempotent complete.