NORTHEASTERN
UNIVERSITY
MATHEMATICS
DEPARTMENT
REPRESENTATION
THEORY AND RELATED TOPICS SEMINAR
Alex Martsinkovsky
(Boston)
Direct Summands of Homological Functors
ABSTRACT: In
his La Jolla paper on coherent functors, M. Auslander described
injective objects in certain functor categories as direct summands of
the covariant functors Ext1(A, —), and conjectured that they
are all of that same form. He established that result in the case A was
of finite projective dimension. In the same volume, P. Freyd gave a
positive answer in the case the underlying abelian category has
denumerable sums. Later, Auslander gave a unifying proof of these
results, but also showed that the conjecture is not true in general. In
this talk, we give a positive answer to the conjecture in the seemingly
overlooked case when A is an object of finite length. In fact, our
result holds for noetherian objects satisfying the DCC on the images of
nested endomorphisms. Moreover, our result is established for any additive bifunctor
whose endomorphisms lift to endomorphisms of the fixed argument. That
this is the case for the Ext-functor is a consequence of the
Hilton-Rees theorem, for which we give an "instant" proof. Other
immediate applications include Hom modulo projectives, and, when the
fixed argument is restricted to finitely presented modules, the
functors Tor1(A,—).
No prior experience with coherent functors is needed for this expository talk.