The t-structure induced by an n-tilting module

TL;DR

This paper investigates the t-structure generated by an n-tilting module in the derived category of a ring, focusing on conditions under which its heart forms a Grothendieck category, linking module properties to categorical structure.

Contribution

It characterizes when the heart of the t-structure is a Grothendieck category, showing this occurs precisely when the n-tilting module is pure projective.

Findings

Heart is a Grothendieck category iff T is pure projective.

Provides module-theoretic criteria for categorical properties.

Connects tilting theory with Grothendieck categories.

Abstract

We study the t-structure induced by an n-tilting module T in the derived category D(R) of a ring R. Our main objective is to determine when the heart of the t-structure is a Grothendieck category. We obtain characterizations in terms of properties of the module category over the endomorphism ring of T and as a main result we prove that the heart is a Grothendieck category if and only if T is a pure projective $R$-module.