On periodic objects and SOD's corresponding to exact transformations - Mikhail Bondarko скачать в хорошем качестве

On periodic objects and SOD's corresponding to exact transformations - Mikhail Bondarko

This is a recorded version of the following talk from our "New Directions in Group Theory and Triangulated Categories" series. To receive updates about this series, or to suggest speakers (including yourself), please email me at rudradipbiswas@gmail.com. More details about this seminar series are here - https://sites.google.com/view/ndgttc/... ------------------------------------------------------------------------ 119th Meeting of "New Directions in Group Theory and Triangulated Categories" Date: March 27, 2025; Thursday Time: 4 pm UK Speaker: Mikhail Bondarko (St. Petersburg State University) Title: On periodic objects and semi-orthogonal decompositions corresponding to exact transformations (joint work with Alexander Aizatov). Abstract: For a family T of transformations between exact endofunctors on a triangulated category C we studied the subcategory Cone(T)^{\perp} of the corresponding periodic objects (that is, the orthogonal to the class {Cone(t(X)): t ∈ T, X ∈ C}) and the adjoints to the embedding Cone(T)^{\perp} (whenever they exist). Several important examples come from (motivic) stable homotopy theory and exact tensor actions. In particular, we took T_S = {X \mapsto s.id_X, s ∈ S \subset R}, where R is a commutative ring such that C is R-linear and X runs through C; Cone(T_S)^{\perp} consists of uniquely S-divisible objects. I will generalize this description to the case where T consists of transformations corresponding to tensoring (all X ∈ C) by some morphisms between projective R-modules (under the assumption that C is closed with respect to coproducts and products of the corresponding size). In the case where C is well generated (in particular, if C is compactly generated) and T consists of transformations between endofunctors E^j and F^j that respect coproducts, we proved the existence of a semi-orthogonal decomposition s_T^{cl} whose left admissible component LA_T^{cl} equals Cone(T)^{\perp}. Respectively, there exists a left adjoint to the embedding Cone(T)^{\perp} \to C; in the case T=T_S, this functor is the R[1/S]-linearization one. This adjoint is weight-exact whenever E^j and F^j are weight-exact with respect to a smashing weight structure. The right admissible component RA_T^{cl} of s_T^{cl} is the triangulated subcategory of C generated by the aforementioned Cone(T) under coproducts; in particular, one may call RA_{T_S}^{cl} the subcategory of S-torsion objects of C. I will try to mention certain 2-functoriality of decompositions of this sort. Lastly, if the right adjoints to all E^j and F^j respect coproducts, then Cone(T)^{\perp} is right admissible as well; respectively, it is the right admissible component of yet another semi-orthogonal decomposition. ------------------------------------------------------------------------