The Natural Middle of a Complete Resolution

Abstract

It is widely known that minimal free resolutions of a module over a complete intersection ring have nice patterns that arise in their Betti sequences. In the late 1990's Avramov, Gasharov and Peeva defined a new class of R-modules that would exhibit similar patterns in their free resolutions. In doing so, they additionally defined the notion of critical degree for an R-module, which serves as a “flag” for when such patterns arise in the module’s Betti sequence. The main purpose of this thesis is to present an extension of critical degree to the category of totally acyclic complexes, Ktac(R), where R is a commutative Noetherian, local ring. Furthermore, we will provide an appropriate dual analogue and then look towards realizing the cohomological characterization for these notions, utilizing the original such characterization. With regard to this topic, our attention will predominantly turn towards when R is further assumed to be a complete intersection ring of the form R = Q/(f1,…, fc) where (Q, m, k) is a regular local ring and f1,…, fc a Q-regular sequence in the maximal ideal, m.

We then investigate how the critical and cocritical degrees of an R-complex may change under certain operations of R-complexes, such as translations, direct sums, and tensoring with a bounded complex. Lastly, we introduce a new invariant of R-complexes and R-modules called the critical width, or diameter, which we define to be the “distance” between the critical and cocritical degrees of an R-complex.