Higher Ideal Approximation Theory

A nice generalization of the classical approximation theory, known as ideal approximation theory, is introduced and studied in [2]. In this theory, the role of the objects and subcategories in classical theory is replaced by morphisms and ideals of the category. An ideal of a category is an additive subfunctor of the Hom functor which is closed under compositions by morphisms from left and right. On the other hand a higher dimensional version of the classical homological algebra, known as higher homological algebra, is appeared in the literature, through study of the structure of $n$-cluster tilting subcategories [3].

Our aim in this talk is to introduce “ideal approximation theory” into “higher homological algebra”. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce’s Lemma and Wakamatsu’s Lemma for ideals will be studied in the context of $n$-exact categories.

The talk is based on the joint work with Somayeh Sadeghi [1].

[1] J. Asadollahi, S. Sadeghi, Higher ideal approximation theory, arXiv:2010.13203 [math.RT].

[2] X. H. Fu, P. A. Guil Asensio, I. Herzog and B. Torrecillas, Ideal approximation theory, Adv. Math., 244 (2013), 750-790.

[3] G. Jasso, n-Abelian and n-exact categories, Math. Z., 283 (2016), 703-759.