Di. 27.06.2017, Oberseminar Algebra

Thema: What the zeroth derived functors can do for you Referent: Prof. Dr. Alex Martsinkovsky (Northeastern University), Raum: ENC-B 314 , Zeit: 13:00 Uhr c.t.

Oberseminar Algebra

What the zeroth derived functors can do for you

Dienstag, 27.06.2017, 13:00 Uhr c.t., Raum: ENC-B 314,

Referent: Prof. Dr. Alex Martsinkovsky (Northeastern University)

Abstrakt:

Given an additive functor from the module category over a ring to the category of abelian groups, one can compute its derived functors. The best known examples are Tor, the left-derived functor of the tensor product, and Ext, the right-derived functor of the Hom functor. In degree zero or coincides with the tensor product and Ext coincides with Hom, so in hat degree we do not gain any new information. The situation changes dramatically when the above functors are derived on the "wrong" side. In that case, the comparison map between the original functor and its zeroth derived functor is no longer an isomorphism, which leads to the notions of injective and projective stabilizations of the functor, originally introduced by M. Auslander and M. Bridger. In this expository talk I will describe two new and surprising applications of these concepts. One of them provides a common generalization of classical torsion over commutative domains and 1-torsion (= the kernel of the canonical map from a module to its bidual) of finitely presented modules over arbitrary rings. The other application yields a notion of the cotorsion module of a module, which doesn't seem to have a classical prototype. Time permitting, we should see several dualities between the two concepts. This is joint work with Jeremy Russell.