# (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 $1$ ($\mathrm{pd}T\leqslant 1$) and $\mathrm{Ext}^{1}_{A}(T,T)=0$.

Recall that if $M$ is an $A$-module, then by $\mathrm{add}M$ we denote the class of all $A$-modules which are direct sums of direct summands of $M$. Since Krull-Schmidt Theorem holds in the category of finite-dimensional $A$-modules, then this means, that if

 $M=E_{1}\oplus\cdots\oplus E_{n}$

for some indecomposable modules $E_{i}$, then $\mathrm{add}M$ consists of all modules which are isomorphic to

 $E_{1}^{a_{1}}\oplus\cdots\oplus E_{n}^{a_{n}}$

for some nonnegative integers $a_{1},\ldots,a_{n}$.

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

 $0\rightarrow A\rightarrow T^{\prime}\rightarrow T^{\prime\prime}\rightarrow 0$

such that both $T^{\prime},T^{\prime\prime}\in\mathrm{add}T$. Here we treat the algebra $A$ is a right module via multiplication.

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

Title (partial) tilting module partialTiltingModule 2013-12-11 16:01:05 2013-12-11 16:01:05 joking (16130) joking (16130) 8 joking (16130) Definition msc 16S99 msc 20C99 msc 13B99