# (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 $1$ ($\mathrm{pd}T\u2a7d1$) 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 $P$.

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 |