# Tilting classes over commutative rings

## TL;DR

This paper classifies tilting classes over any commutative ring using sequences of Thomason subsets, extending known classifications from noetherian rings and connecting them to various homology theories.

## Contribution

It generalizes the classification of tilting classes to all commutative rings and relates these classes to multiple homology and cohomology theories, also constructing new cotilting modules.

## Key findings

- Classifies all tilting classes via Thomason subsets.
- Shows n-tilting classes relate to homology theories from finitely generated ideals.
- Constructs cotilting modules for classes of cofinite type.

## Abstract

We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri-Posp\'i\v{s}il-\v{S}\v{t}ov\'i\v{c}ek-Trlifaj. We show that the n-tilting classes can be equivalently expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: Tor_*(R/I,-), Koszul homology, \v{C}ech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of \v{S}\v{t}ov\'i\v{c}ek-Trlifaj-Herbera. Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.