Commutative Algebra is a fundamental branch of Mathematics. Here in Genova, the category in which we move is mainly the one of finitely generated modules over a Noetherian ring, but also coherent sheaves over a Noetherian scheme, triangulations of topological spaces, G-equivariant objects (in contexts in which a group is involved). So we do not mind, sometimes, to move around and get by on close fields like Algebraic Geometry, Combinatorics, Topology or Representation Theory.
This said, the following are some research topics that distinguish the Commutative Algebra group of Genova:
- Homological algebra (especially free resolutions, properties of the Koszul complex and local cohomology).
- Local algebra (and therefore singularity theory).
- Determinantal rings, Grassmannians, ideals generated by Pfaffians and many other objects governed by some symmetry.
- Gorenstein K-algebras.
- Koszul rings.
- Gröbner basis, the generic initial ideal and all the related computational applications (CoCoA).
- The set-theoretic definition of algebraic varieties.
- Stanley-Reisner rings, and therefore the study of the singular homology of a simplicial complex.