People

Ryo KANDA

Email kanda(@math.sci.osaka-u.ac.jp)
Research
Ring theory
Keywords Noetherian rings, spectra of abelian categories
Office Science Building B-401(Toyonaka Campus)
URL http://www.math.sci.osaka-u.ac.jp/~kanda

My research area is ring theory. A ring is an algebraic structure which has addition, subtraction, and multiplication. Typical examples are the set of integers, the set of polynomials, and the set of n-by-n matrices. I am particularly interested in noncommutative rings, whose multiplication is noncommutative.

The notion of modules plays an important role in the study of a ring. It is a generalization of vector spaces appearing in linear algebra, but the only difference in the definition is that the coefficients live in a ring, not necessarily a field such as the field of real/complex numbers. A vector space over an arbitrary coefficient field is determined by the cardinality of its basis, up to isomorphism. On the other hand, in the case of a ring, even the nature of finitely generated modules highly depends on the structure of the ring. For this reason, we can investigate the ring by looking the behavior of its modules. Especially for noncommutative rings, this approach is often clearer than looking the ring itself directly, and this leads us to deeper results.

An abelian category is a further generalization of the collection of modules. For each ring, the collection of its modules has the structure of an abelian category, and the ring can be almost recovered by the categorical structure. A similar thing holds for the abelian category consisting of coherent sheaves on an algebraic variety. Hence the notion of abelian categories is a large framework including (noncommutative) rings and (commutative) algebraic varieties. I have investigated general properties of certain classes of abelian categories, and have revealed several new properties of noncommutative rings as consequences. An advantage of this general setting is that we can consider similar problems for abelian categories which are not obtained as module categories over rings. For example, the functor category, which is the category consisting of functors from a given abelian category, is again an abelian category, and its structure reflects homological properties of the original abelian category. I expect that, by considering naive questions arising from general theory of abelian categories in a specific setting, such as the functor category, we can extend existing theories to new directions.