(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 $1$ ($\mathrm{pd}T⩽1$) and ${\mathrm{Ext}}_A^1(T,T)=0$.

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

$$M=E_1\oplus \mathrm{\cdots }\oplus E_n$$

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

$$E_1^{a_1}\oplus \mathrm{\cdots }\oplus E_n^{a_n}$$

for some nonnegative integers $a_1,\mathrm{\dots },a_n$.

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

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

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 $P$.