ABSTRACT:
The goal of this expository talk is to prepare the audience (including
students and non-experts) for the two upcoming lectures by Jeremy
Russell.
The talk will consist of two parts. In the first
one, I will concentrate on basic definitions and results of M.
Auslander from his foundational La Jolla paper. In the second part, I will prove, in the case of length categories,
a conjecture of Auslander on injective objects in the category of
defect-zero coherent functors. In fact, in that case, the result holds
in greater generality than required by the conjecture. Using coherent
functors, I will also give a short proof of the Hilton-Rees theorem on
natural transformations between the covariant Ext functors. A similar
result will be established for the univariate Tor functors and Hom modulo an ideal.
This
talk is similar to the one I gave a year ago, but differs from it by
extra motivational details and inclusion of proofs. Handout slides can be downloaded from here