Sinem Odabaşı**, **Universidad de Murcia.
Date: 3th of January, 2023, Tuesday, Time: 13.00 - 14.30.
Place: Online-Join Zoom Meeting (https://zoom.us/j/99412179515?pwd=bUlLdStFaVpjMHhWb21LaFlRMWI1Zz09)
Meeting ID: 994 1217 9515 Passcode: 553266

**Abstract:** In a module category, the class FPn(R) of the so-called finitely n-presented left R-modules induces two crucial classes In(R) and Fn(Rop) of left FPn-injective and right FPn-flat Rmodules, respectively. It is known that certain homological properties of In(R) and Fn(Rop) and closure properties of FPn(R) are closely related, and these properties determine ring theoretic properties of R. The authors in [1] introduce the class FPn(G) of objects of type FPn in a Grothendieck category G as a generalization of the class FPn(R), and study certain aspects of the associated class In(G) of FPn-injective objects in G. In this talk, further homological aspects of the class In(G) will be presented. We show that the projective dimension of the class FPn(G) controls how far is In(G) from being a torsion class. Under mild conditions on G, which permit us to have ‘the external tensor product functor’, we introduce the class Fn(G) of FPn-flat objects in G. We will exhibit a close interaction between the classes In(G) and Fn(G). Further applications will be presented in Ab-valued functor categories showing n-coherency in terms of the domain category This is a joint work with Daniel Bravo, Carlos Parra and Marco Perez; see [2].

**References **

[1] Bravo, D., Gillespie, J. & Perez, M. A. (2019). Locally typeFPn and n-coherent categories. arXiv:1908.10987

[2] Bravo, D., Odabas¸ı, S., Parra, C & Perez, M. A. (2022). Torsion and Torsion-free classes from objects of finite type in Grothendieck categories. Journal of Algebra, 608, 412-444. DOI: 10.1016/j.jalgebra.2022.05.029.