Algebra seminar

Michal Hrbek (IM CAS): Mutation in commutative algebra

Abstract: The notion of a mutation traditionally belongs to the representation-theoretic setting of compact silting complexes over finite-dimensional algebras. Recently, a far-reaching generalization to general cosilting complexes in compactly generated triangulated categories was introduced by Angeleri-Hügel, Laking, Šťovíček, and Vitória. In particular, these mutations were shown to induce a bijection between Gabriel spectra of two Grothendieck hearts. In our work, we study how the mutation interacts with the topology on the two spaces, more precisely with the full support topology as studied by Prest and Burke. In the general situation, we show that any right mutation induces a piece-wise homeomorphism on two complementary subspaces induced by the mutation. In the setting of commutative algebra, we improve this considerably by showing that a right mutation induces an open bijection. We conclude with some consequences of this result.