Torsion pairs via large silting mutation (research)
Research talk, Institut Fourier, Université Grenoble Aples, Grenoble, France
Last decade Aihara, Iyama and Reiten proved that functorially finite torsion classes of finitely generated modules over a finite-dimensional algebre are in one-to-one correspondence with its compact 2-term silting complexes, which can be mutated at any of their indecomposable summands via silting mutation–thus inducing an operation of mutation for the aforementioned torsion classes. Recent work has been directed towards extending this mutation to arbitrary torsion classes, which are known to be one-to-one with 2-term cosilting complexes, the infinite-dimensional dual of compact 2-term silting complexes. These belong to the larger class of cosilting objects, which also have an operation of cosilting mutation. In particular, it is not guaranteed that the cosilting mutation of a 2-term cosilting complex is once again a 2-term cosilting complex, and thus that it corresponds to a torsion class. In this talk, we introduce an infinite-dimensional analog of silting mutation, which we call large silting mutation, and show how applying it to the category of large injective copresentations–which is neither exact nor triangulated, but extriangulated–results in a theory of mutation for 2-term cosilting complexes, providing a reinterpretation of cosilting mutation for this class of cosilting objects while also inducing an operation of mutation for arbitrary torsion classes. We also mention how this framework can be applied to other cases, generalized, and dualized, as well as some of its current limitations in regards to mutability criteria–since, in the infinite-dimensional context, mutability is not guaranteed in general. Links: slides.