(partial) tilting module

Let A be an associative, finite-dimensional algebra over a field k. Throughout all modules are finite-dimensional.

A right A-module T is called a partial tilting module if the projective dimension of T is at most 1 (pdT1) and ExtA1(T,T)=0.

Recall that if M is an A-module, then by addM we denote the class of all A-modules which are direct sumsPlanetmathPlanetmathPlanetmath of direct summandsMathworldPlanetmath of M. Since Krull-Schmidt Theorem holds in the categoryMathworldPlanetmath of finite-dimensional A-modules, then this means, that if


for some indecomposable modulesMathworldPlanetmath Ei, then addM consists of all modules which are isomorphicPlanetmathPlanetmathPlanetmath to


for some nonnegative integers a1,,an.

A partial tiliting module T is called a tilting module if there exists a short exact sequenceMathworldPlanetmathPlanetmath


such that both T,T′′addT. Here we treat the algebraPlanetmathPlanetmathPlanetmath A is a right module via multiplication.

Note that every projective moduleMathworldPlanetmath is partial tilting. Also a projective module P is tilting if and only if every indecomposableMathworldPlanetmath direct summand of A is a direct summand of P.

Title (partial) tilting module
Canonical name partialTiltingModule
Date of creation 2013-12-11 16:01:05
Last modified on 2013-12-11 16:01:05
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Definition
Classification msc 16S99
Classification msc 20C99
Classification msc 13B99