---
tag: XXYQV6R1PR
kind: concept
name: Tilting theory
msc: [16G10, "16E35"]
sources:
  - url: https://en.wikipedia.org/wiki/Tilting_theory
same_as:
  wikidata: Q7803026
  wikipedia: Tilting theory
---

## Summary

Tilting theory compares the module categories of two algebras $A$ and $B = \End_A(T)$ by means of a tilting module $T$.
It generalizes [[Morita equivalence|RD5WKA1VK7]], which is the case where $T$ is a progenerator.
The two module categories are not equivalent in general, but they share large pieces, and by Happel's theorem their bounded [[derived categories|5ST7NVJFSM]] are equivalent.
It grew out of the [[reflection functors|MG4RNMA8AR]] of Bernstein, Gelfand and Ponomarev.

## Explanations

### L1 human

Two algebras can have different modules and still be closely related.
Tilting theory gives a way to pass from one to the other through a special module $T$, the tilting module.
Part of the modules of the first algebra correspond to part of the modules of the second, and the rest is "tilted" across.
At the level of derived categories, which keep track of complexes of modules, the two algebras look exactly the same.
This lets one transfer hard questions from a complicated algebra to a simpler one.