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When studying an [[Algebra|algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301101.png" />, it is sometimes convenient to consider another algebra, given for instance by the endomorphism of an appropriate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301102.png" />-module, and functors between the two module categories. For instance, this is the basis of the [[Morita equivalence|Morita equivalence]] or the construction of the so-called Auslander algebras. An important example of this strategy is given by the [[Tilting theory|tilting theory]] and the tilting functors, as now described.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301103.png" /> be a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301104.png" />-algebra, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301105.png" /> is a [[Field|field]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301106.png" /> a tilting (finitely-generated) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301107.png" />-module (cf. [[Tilting module|Tilting module]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301108.png" />. One can then assign to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t1301109.png" /> the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011013.png" />, which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011015.png" />, results first established by S. Brenner and M.C.R. Butler. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011016.png" /> and its adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011017.png" /> give an equivalence between the subcategories
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011018.png" /></td> </tr></table>
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When studying an [[Algebra|algebra]] $A$, it is sometimes convenient to consider another algebra, given for instance by the endomorphism of an appropriate $A$-module, and functors between the two module categories. For instance, this is the basis of the [[Morita equivalence|Morita equivalence]] or the construction of the so-called Auslander algebras. An important example of this strategy is given by the [[Tilting theory|tilting theory]] and the tilting functors, as now described.
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Let $A$ be a finite-dimensional $k$-algebra, where $k$ is a [[Field|field]], $T$ a tilting (finitely-generated) $A$-module (cf. [[Tilting module|Tilting module]]) and $B = \operatorname { End } _ { A } ( T )$. One can then assign to $T$ the functors $\text{Hom}_A( T , - )$, $- \otimes _ { B } T$, $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$, and $\operatorname { Tor } _ { 1 } ^ { B } ( - , T )$, which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categories $\mod A$ and $\operatorname { mod} B$, results first established by S. Brenner and M.C.R. Butler. Namely, $\text{Hom}_A( T , - )$ and its adjoint $- \otimes _ { B } T$ give an equivalence between the subcategories
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\begin{equation*} \mathcal{T} ( T _ { A } ) = \{ M _ { A } : \operatorname { Ext } _ { A } ^ { 1 } ( T , M ) = 0 \} \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011019.png" /></td> </tr></table>
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\begin{equation*} \mathcal{Y} ( T _ { A } ) = \left\{ N _ { B } : \operatorname { Tor } _ { 1 } ^ { B } ( N , T ) = 0 \right\}, \end{equation*}
  
while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011021.png" /> give an equivalence between the subcategories
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while $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ and $\operatorname { Tor } _ { 1 } ^ { B } ( - , T )$ give an equivalence between the subcategories
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011022.png" /></td> </tr></table>
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\begin{equation*} \mathcal{Y} ( T _ { A } ) = \{ N _ { B } : \operatorname { Tor } _ { 1 } ^ { B } ( N , T ) = 0 \} \end{equation*}
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011023.png" /></td> </tr></table>
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\begin{equation*} \chi ( T _ { A } ) = \left\{ N _ { B } : N \bigotimes _ { B } T = 0 \right\}. \end{equation*}
  
It is not difficult to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011025.png" /> are torsion pairs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011027.png" />, respectively. Clearly, one can now transfer information from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011028.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011029.png" />. One of the most interesting cases occurs when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011030.png" /> is a hereditary algebra and so the torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011031.png" /> splits, giving in particular that each indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011032.png" />-module is the image of an indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011033.png" />-module either by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011034.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011035.png" /> (in this case, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130110/t13011036.png" /> is called tilted, cf. also [[Tilted algebra|Tilted algebra]]).
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It is not difficult to see that $( \mathcal{T} ( T _ { A } ) , \mathcal{F} ( T _ { A } ) )$ and $( \mathcal{X} ( T _ { A } ) , \mathcal{Y} ( T _ { A } ) )$ are torsion pairs in $\mod A$ and $\operatorname { mod} B$, respectively. Clearly, one can now transfer information from $\mod A$ to $\operatorname { mod} B$. One of the most interesting cases occurs when $A$ is a hereditary algebra and so the torsion pair $( \mathcal{X} ( T _ { A } ) , \mathcal{Y} ( T _ { A } ) )$ splits, giving in particular that each indecomposable $B$-module is the image of an indecomposable $A$-module either by $\text{Hom}_A( T , - )$ or by $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ (in this case, the algebra $B$ is called tilted, cf. also [[Tilted algebra|Tilted algebra]]).
  
 
This procedure has been generalized in several ways and it is worthwhile mentioning, for instance, its connection with derived categories (cf. also [[Derived category|Derived category]]), or the notion of quasi-tilted algebras. It has also been considered for infinitely-generated modules over arbitrary rings.
 
This procedure has been generalized in several ways and it is worthwhile mentioning, for instance, its connection with derived categories (cf. also [[Derived category|Derived category]]), or the notion of quasi-tilted algebras. It has also been considered for infinitely-generated modules over arbitrary rings.
  
 
For referenes, see also [[Tilting theory|Tilting theory]]; [[Tilted algebra|Tilted algebra]].
 
For referenes, see also [[Tilting theory|Tilting theory]]; [[Tilted algebra|Tilted algebra]].

Latest revision as of 16:45, 1 July 2020

When studying an algebra $A$, it is sometimes convenient to consider another algebra, given for instance by the endomorphism of an appropriate $A$-module, and functors between the two module categories. For instance, this is the basis of the Morita equivalence or the construction of the so-called Auslander algebras. An important example of this strategy is given by the tilting theory and the tilting functors, as now described.

Let $A$ be a finite-dimensional $k$-algebra, where $k$ is a field, $T$ a tilting (finitely-generated) $A$-module (cf. Tilting module) and $B = \operatorname { End } _ { A } ( T )$. One can then assign to $T$ the functors $\text{Hom}_A( T , - )$, $- \otimes _ { B } T$, $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$, and $\operatorname { Tor } _ { 1 } ^ { B } ( - , T )$, which are called tilting functors. The importance of considering such functors is that they give equivalences between subcategories of the module categories $\mod A$ and $\operatorname { mod} B$, results first established by S. Brenner and M.C.R. Butler. Namely, $\text{Hom}_A( T , - )$ and its adjoint $- \otimes _ { B } T$ give an equivalence between the subcategories

\begin{equation*} \mathcal{T} ( T _ { A } ) = \{ M _ { A } : \operatorname { Ext } _ { A } ^ { 1 } ( T , M ) = 0 \} \end{equation*}

and

\begin{equation*} \mathcal{Y} ( T _ { A } ) = \left\{ N _ { B } : \operatorname { Tor } _ { 1 } ^ { B } ( N , T ) = 0 \right\}, \end{equation*}

while $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ and $\operatorname { Tor } _ { 1 } ^ { B } ( - , T )$ give an equivalence between the subcategories

\begin{equation*} \mathcal{Y} ( T _ { A } ) = \{ N _ { B } : \operatorname { Tor } _ { 1 } ^ { B } ( N , T ) = 0 \} \end{equation*}

and

\begin{equation*} \chi ( T _ { A } ) = \left\{ N _ { B } : N \bigotimes _ { B } T = 0 \right\}. \end{equation*}

It is not difficult to see that $( \mathcal{T} ( T _ { A } ) , \mathcal{F} ( T _ { A } ) )$ and $( \mathcal{X} ( T _ { A } ) , \mathcal{Y} ( T _ { A } ) )$ are torsion pairs in $\mod A$ and $\operatorname { mod} B$, respectively. Clearly, one can now transfer information from $\mod A$ to $\operatorname { mod} B$. One of the most interesting cases occurs when $A$ is a hereditary algebra and so the torsion pair $( \mathcal{X} ( T _ { A } ) , \mathcal{Y} ( T _ { A } ) )$ splits, giving in particular that each indecomposable $B$-module is the image of an indecomposable $A$-module either by $\text{Hom}_A( T , - )$ or by $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ (in this case, the algebra $B$ is called tilted, cf. also Tilted algebra).

This procedure has been generalized in several ways and it is worthwhile mentioning, for instance, its connection with derived categories (cf. also Derived category), or the notion of quasi-tilted algebras. It has also been considered for infinitely-generated modules over arbitrary rings.

For referenes, see also Tilting theory; Tilted algebra.

How to Cite This Entry:
Tilting functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tilting_functor&oldid=49983
This article was adapted from an original article by Flávio Ulhoa Coelho (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article