**Research lines**

## Algebra

**Researchers**

Cleto Brasileiro Miranda Neto

Diogo Diniz Pereira da Silva e Silva

Jacqueline Fabíola Rojas Arancibia

Ricardo Burity Croccia Macedo

Ugo Bruzzo

Algebraic Geometry studies properties and classifications of algebraic varieties, as well as schemes, sheaves (and sheaf cohomology as well), characteristic classes, morphisms and bundles over them. Several numerical invariants associated with such objects are also subjects of study.

Commutative Algebra deals with the investigation of commutative rings and their ideals and modules. In particular, it studies the hierarchy of the main classes of rings and their structural, arithmetic and homological properties, as well as their invariants (such as the Krull dimension, among many others), the same being true for the study of modules over such rings. It is also the main technical tool for the local study of schemes and sheaves in algebraic geometry.

Homological Algebra investigates methods of homology and cohomology in a general context, particularly on topological spaces, Lie algebras, sheaves, groups, and (non-)commutative rings. Some of the fundamental tools are the exact sequences and complexes in general, as well as the derived functors (such as Ext and Tor). The study of modern algebraic geometry, for instance, would be practically intractable without sheaf cohomology theory.

Another line of research is the use of asymptotic methods and of the theory of representations of S_n to study codimension, cocharacters and exponents of algebras with polynomial identities. The theory Algebras with Polynomial Identities (or PI-Algebras), is a branch of Mathematics that studies the class of algebras tha satisfy a non-trivial polynomial identity. The study of PI-Algebras intertwines, in the Representability Theorem, with the Theory of Graded Rings, in the solution of A. Kemer to the problem posed by W. Specht: determine if the ideal of identities of a ring, of characteristic zero, is finitely generated as a T-ideal. In this line of research we study PI-algebras with additional structure. One of the main questions is the analogous of the Specht Problem for these algebras, and also to exhibit a finite generating set. Another line of research is the use of asymptotic methods and of the theory of representations of S_n to study codimension, cocharacters and exponents of algebras with polynomial identities.