Homological Techniques in Commutative Algebra

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Commutative algebra is crucial in developing the foundations of algebraic geometry. Two central topics of research in these fields are multiplicities and the homological behavior of modules. This proposal concerns questions related to these topics and especially their interplay. The first project involves studying whether Hilbert-Kunz multiplicity, an intriguing and not well-understood measure of the complexity of a singularity in positive characteristic, has an interpretation in characteristic zero where it could be simpler or clearer to interpret geometrically (some known examples show that a limit as the characteristic goes to infinity might exist and seems to be a simpler number). The second concerns the homology of Koszul complexes, and although in this case the approaches are not exclusively from the realm of complete intersection rings, many of the same ideas arise since the Koszul complex is the first step in constructing a Tate resolution. Many directions are being examined, with several new approaches to old problems studied by Huneke, Simis, Vasconcelos and many others. The third concerns an older conjecture in rational homotopy theory, a field with which the ideas from resolutions of complete intersections and from differential graded resolutions have a long history of interaction, but have not yet been used for this particular problem; preliminary results of this approach look encouraging. The fourth concerns Serre's intersection multiplicity over non-regular rings. These projects are at the center of some of the main directions in commutative algebra, but with a view towards neighboring fields of mathematics or subfields of algebra. For example, the first and main project concerns multiplicities, which are numerical measures of the complexity of a non-smooth (that is, sharp) point on a curve, surface, or higher dimensional object. The investigator is especially interested in these in an algebraic or number theoretic setting that is not directly related to the physical geometry of an object. Known results show quite mysterious behavior of these numbers and the investigator hopes to shed some light on the situation by relating the setting to a more geometric one where classical geometric techniques might be applied. This is a delicate procedure and work in progress. Likewise, the third project involves a problem that would apply ideas from her abstract field of algebra to a part of topology, a field which deals with physical spaces. In summary, although the subjects of the projects are quite varied, they hold a common theme that may not be immediately evident: Namely, the investigator proposes to apply her experience in her own area to various problems further removed from this area and not traditionally studied in this way.