n-Cotorsion pairs

Joint with: O. Mendoza and M.A. Pérez

Motivated by some properties satisfied by Gorenstein projective and  
Gorenstein injective modules over an Iwanaga-Gorenstein ring, we  
present the concept of left and right n-cotorsion pairs in an abelian  
category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of  
objects of $\mathcal{C}$ form a left $n$-cotorsion pair  
$(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation  
$\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for  
indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a  
resolution by objects in $\mathcal{A}$ whose syzygies have  
$\mathcal{B}$-resolution dimension at most $n-1$. This concept and its  
dual generalise the notion of complete cotorsion pairs, and has an  
appealing relation with left and right approximations, especially with  
those having the so called unique mapping property.

The main purpose of this talk is to describe several properties of  
$n$-cotorsion pairs and to establish a relation with complete  
cotorsion pairs. We also give some applications in relative  
homological algebra, that will cover the study of approximations  
associated to Gorenstein projective, Gorenstein injective and  
Gorenstein flat modules and chain complexes, as well as $m$-cluster  
tilting subcategories.