Nontilting

Nontilting is a term used in mathematics, specifically in the context of tilting theory in representation theory. It describes a class of modules that have a specific relationship with a tilting complex. In essence, a module is nontilting if it satisfies certain conditions related to its Ext groups with respect to a tilting complex.

More formally, let $T$ be a tilting complex over a ring $A$. A module $M$ is called nontilting with respect to $T$ if $\operatorname{Ext}^i_A(T, M) = 0$ for all $i > 0$. This condition implies that $M$ behaves "nicely" with respect to the structure imposed by the tilting complex $T$.

The concept of nontilting modules is a generalization of almost complete tilting modules. Tilting theory provides a powerful tool for studying the structure of modules over an algebra. Nontilting modules play a crucial role in understanding the derived category of modules and its properties. They are fundamental in the classification and construction of tilting objects. The study of nontilting modules is closely related to the homological algebra of the ring and the properties of the associated tilting complex.