The lattice of torsion classes in the category mod A of finitely generated A-modules over a finite dimensional algebra A, usually denoted by tors A, is currently receiving a lot of attention due to its connection with silting theory and with the phenomenon of mutation. Indeed, it is shown in [1] that the functorially finite torsion classes in mod A are in bijection with two-term silting complexes, and the lattice order on these torsion classes reflects silting mutation.

But tors A carries important representation-theoretic information also for torsion classes which are not functorially finite. Its Hasse quiver is labeled by bricks, i.e. by modules whose endomorphism ring is a skew field. More precisely, torsion classes are adjacent if and only if they “differ” by exactly one brick, which passes from the torsion-free class of one torsion pair to the torsion class of the other [2,3]. In recent joint work with Laking, Štovíček and Vitória we have interpreted this process in terms of mutation of two-term cosilting complexes in the unbounded derived category of A.

In the present talk, I will report on ongoing joint work with Ivo Herzog, Rosanna Laking, and Francesco Sentieri which relates cosilting mutation with an exchange of indecomposable injective objects in suitable Grothendieck categories. More precisely, we will consider the hearts of the t-structures associated to two adjacent torsion pairs. The brick which labels the corresponding arrow in the Hasse quiver gives rise to a simple object in each of these hearts. We will discuss how to determine the injective envelopes of these simple objects and investigate their role in connection with mutation.

References:

[1] T. Adachi, O. Iyama, I. Reiten, τ-tilting theory, Compos. Math. 150 (2014) 415-452.

[2] E. Barnard, A. T. Carroll, S. Zhu. Minimal inclusions of torsion classes. Algebr. Comb. 2 (2019), 879-901.

[3] L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas, Lattice theory of torsion classes, arXiv:1711.01785

Contact Peter Jørgensen (email peter.jorgensen@math.au.dk) for the link to Zoom