[Todos CMAT] Seminario segundo semestre

Marcelo Lanzilotta marclan en cmat.edu.uy
Vie Jul 15 19:41:47 UYT 2011

En el segundo semestre realizaremos un Seminario centrado en el trabajo:

*Morita Theory for Derived Categories*
Jeremy Rickard.

Mando adjunto el artículo.

Envío este mensaje para quienes estén interesados envíen respuesta
Luego definiremos una fecha y lugar de reunión inicial.

Abajo copio parte de la Introducción del artículo.


                                   Marcelo Lanzilotta

*Over the past few .years, several mathematicians interested in the
theory of finite-dimensional algebras have developed a technique called
'tilting' .
Given one algebra, one can take the endomorphism ring of a 'tilting module'
to get
a different algebra which, although not Morita equivalent, has a similar
category. The conditions for a A- module T to be a tilting module are that T
have projective dimension at most one, Ext^1(T,T) should be zero, and there
be a short exact sequence:

*                       0 ----> A ------- > T_1 ----->T_2 ----->0;

of A-modules, T_1 and T_2 being direct summands of direct sums of copies of
*In [8], Happel shows that if an algebra F arises from an algebra A of
finite global
dimension by tilting (his proof will work even in the 'generalised' sense),
there is an equivalence (of triangulated categories) between the derived
of bounded complexes of finitely-generated modules for F and A. This
takes F, regarded as a complex concentrated in degree zero,
to T. Cline, Parshall and Scott extended this work to the case of infinite
dimension and more general rings. They also proposed that, in the light of
interest in derived categories in various branches of representation theory
(of finite-
dimensional algebras, algebraic groups and Lie algebras, for example), a
theory' for derived categories of module categories should be developed,
necessary and sufficient conditions for such equivalences to occur. It is to
this problem
that this paper is devoted. The role played by finitely-generated projective
in the classical Morita theory turns out to be played in the derived
category situation
by what we call 'tilting complexes'. These are bounded complexes of
modules satisfying various conditions; if we take this complex to be a
resolution of a tilting module, then we specialise to the case that has been
dealt with before*.
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