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The endomorphism ring of a [[Tilting module|tilting module]] over a finite-dimensional hereditary algebra (cf. also [[Algebra|Algebra]]; [[Endomorphism|Endomorphism]]).
 
The endomorphism ring of a [[Tilting module|tilting module]] over a finite-dimensional hereditary algebra (cf. also [[Algebra|Algebra]]; [[Endomorphism|Endomorphism]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301001.png" /> be a finite-dimensional hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301002.png" />-algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301003.png" /> some field, for example the path-algebra of some finite [[Quiver|quiver]] without oriented cycles. A finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301004.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301005.png" /> is called a tilting module if
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Let $H$ be a finite-dimensional hereditary $K$-algebra, $K$ some field, for example the path-algebra of some finite [[Quiver|quiver]] without oriented cycles. A finite-dimensional $H$-module $\square _ { H } T$ is called a tilting module if
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301006.png" />, which always is satisfied in this context;
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i) $\operatorname {p.dim} T \leq 1$, which always is satisfied in this context;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301007.png" />; and
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ii) $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$; and
  
iii) there exists a short [[Exact sequence|exact sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t1301009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010011.png" />, the category of finite direct sums of direct summands of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010012.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010013.png" /> is projective dimension. The third condition also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010014.png" /> is maximal with respect to the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010015.png" />. Note further, that a tilting module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010016.png" /> over a hereditary algebra is uniquely determined by its composition factors. Cf. also [[Tilting module|Tilting module]].
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iii) there exists a short [[Exact sequence|exact sequence]] $0 \rightarrow H \rightarrow T _ { 1 } \rightarrow T _ { 2 } \rightarrow 0$ with $T _ { 1 }$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010010.png"/> in $\operatorname{add} T$, the category of finite direct sums of direct summands of $T$. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010013.png"/> is projective dimension. The third condition also says that $T$ is maximal with respect to the property $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$. Note further, that a tilting module $T$ over a hereditary algebra is uniquely determined by its composition factors. Cf. also [[Tilting module|Tilting module]].
  
The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010017.png" /> is called a tilted algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010020.png" /> becomes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010021.png" />-bimodule (cf. also [[Bimodule|Bimodule]]).
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The algebra $B = \operatorname { End } _ { H } ( T )$ is called a tilted algebra of type $H$, and $T$ becomes an $( H , B )$-bimodule (cf. also [[Bimodule|Bimodule]]).
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010022.png" />-mod, the [[Category|category]] of finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010023.png" />-modules, the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010024.png" /> defines a torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010025.png" /> with torsion class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010026.png" /> consisting of modules, generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010027.png" /> and torsion-free class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010028.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010029.png" />-mod it defines the torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010030.png" /> with torsion class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010031.png" /> and torsion-free class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010032.png" />. The Brenner–Butler theorem says that the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010033.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010034.png" />, induce equivalences between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010036.png" />, whereas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010037.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010038.png" />, induce equivalences between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010040.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010041.png" />-mod the torsion pair is splitting, that is, any indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010042.png" />-module is either torsion or torsion-free. In this sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010043.png" />-mod has "less" indecomposable modules, and information on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010044.png" />-mod can be transferred to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010045.png" />-mod.
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In $H$-mod, the [[Category|category]] of finite-dimensional $H$-modules, the module $T$ defines a torsion pair $( \mathcal{G} , \mathcal{F} )$ with torsion class $\mathcal{G}$ consisting of modules, generated by $T$ and torsion-free class $\mathcal{F} = \{ Y : \operatorname { Hom } _ { H } ( T , Y ) = 0 \}$. In $B$-mod it defines the torsion pair $\cal ( X , Y )$ with torsion class $\chi = \{ Y : T \otimes _ { B } Y = 0 \}$ and torsion-free class $\mathcal{Y} = \{ Y : \operatorname { Tor } _ { 1 } ^ { B } ( T , Y ) = 0 \}$. The Brenner–Butler theorem says that the functors $\operatorname{Hom}_H( T , - )$, respectively $T \otimes_{ B} -$, induce equivalences between $\mathcal{G}$ and $\mathcal{Y}$, whereas $\operatorname { Ext } _ { H } ^ { 1 } ( T , - )$, respectively $\operatorname { Tor } _ { 1 } ^ { B } ( T , - )$, induce equivalences between $\mathcal{F}$ and $\mathcal{X}$. In $B$-mod the torsion pair is splitting, that is, any indecomposable $B$-module is either torsion or torsion-free. In this sense, $B$-mod has "less" indecomposable modules, and information on the category $H$-mod can be transferred to $B$-mod.
  
For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010046.png" /> has global dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010047.png" /> and any indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010048.png" />-module has projective dimension or injective dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010049.png" /> (cf. also [[Dimension|Dimension]] for dimension notions). These condition characterize the more general class of quasi-tilted algebras.
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For example, $B$ has global dimension at most $2$ and any indecomposable $B$-module has projective dimension or injective dimension at most $1$ (cf. also [[Dimension|Dimension]] for dimension notions). These condition characterize the more general class of quasi-tilted algebras.
  
The indecomposable injective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010050.png" />-modules are in the torsion class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010051.png" /> and their images under the [[Tilting functor|tilting functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010052.png" /> are contained in one connected component of the Auslander–Reiten quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010054.png" />-mod (cf. also [[Quiver|Quiver]]; [[Riedtmann classification|Riedtmann classification]]), and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010055.png" /> characterizes tilted algebras. Moreover, the Auslander–Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components.
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The indecomposable injective $H$-modules are in the torsion class $\mathcal{G}$ and their images under the [[Tilting functor|tilting functor]] $\operatorname{Hom}_H( T , - )$ are contained in one connected component of the Auslander–Reiten quiver $\Gamma ( B )$ of $B$-mod (cf. also [[Quiver|Quiver]]; [[Riedtmann classification|Riedtmann classification]]), and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of $\Gamma ( B )$ characterizes tilted algebras. Moreover, the Auslander–Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010056.png" /> is a basic hereditary algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010057.png" /> is a simple projective module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010059.png" /> denotes the Auslander–Reiten translation (cf. [[Riedtmann classification|Riedtmann classification]]), is a tilting module, sometimes called APR-tilting module. The induced torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010061.png" />-mod is splitting and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010062.png" /> is the unique indecomposable module in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010063.png" />. The tilting functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010064.png" /> corresponds to the reflection functor introduced by I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev for their proof of the Gabriel theorem [[#References|[a4]]].
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If $H$ is a basic hereditary algebra and $H e$ is a simple projective module, then $T = H ( 1 - e ) \oplus \operatorname { TrD }  H e$, where $\operatorname{Tr}D$ denotes the Auslander–Reiten translation (cf. [[Riedtmann classification|Riedtmann classification]]), is a tilting module, sometimes called APR-tilting module. The induced torsion pair $( \mathcal{G} , \mathcal{F} )$ in $H$-mod is splitting and $H e$ is the unique indecomposable module in $\mathcal{F}$. The tilting functor $\operatorname{Hom}_H( T , - )$ corresponds to the reflection functor introduced by I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev for their proof of the Gabriel theorem [[#References|[a4]]].
  
If the hereditary algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010065.png" /> is representation-finite (cf. also [[Algebra of finite representation type|Algebra of finite representation type]]), then any tilted algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010066.png" /> also is representation-finite. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010067.png" /> is tame (cf. also [[Representation of an associative algebra|Representation of an associative algebra]]), then a tilted algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010068.png" /> either is tame and one-parametric, or representation-finite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre-injective direct summands simultaneously. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010069.png" /> is wild (cf. also [[Representation of an associative algebra|Representation of an associative algebra]]), then a tilted algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130100/t13010070.png" /> may be wild, or tame domestic, or representation-finite.
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If the hereditary algebra $H$ is representation-finite (cf. also [[Algebra of finite representation type|Algebra of finite representation type]]), then any tilted algebra of type $H$ also is representation-finite. If $H$ is tame (cf. also [[Representation of an associative algebra|Representation of an associative algebra]]), then a tilted algebra of type $H$ either is tame and one-parametric, or representation-finite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre-injective direct summands simultaneously. If $H$ is wild (cf. also [[Representation of an associative algebra|Representation of an associative algebra]]), then a tilted algebra of type $H$ may be wild, or tame domestic, or representation-finite.
  
 
See also [[Tilting theory|Tilting theory]].
 
See also [[Tilting theory|Tilting theory]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I. Assem, "Tilting theory - an introduction" N. Balcerzyk (ed.) et al. (ed.) , ''Topics in Algebra'' , '''26''' , Banach Center Publ. (1990) pp. 127–180 {{MR|1171230}} {{ZBL|0726.16008}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Auslander, M.I. Platzeck, I. Reiten, "Coxeter functors without diagrams" ''Trans. Amer. Math. Soc.'' , '''250''' (1979) pp. 1–46 {{MR|0530043}} {{ZBL|0421.16016}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Brenner, M. Butler, "Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors" V. Dlab (ed.) P. Gabriel (ed.) , ''Representation Theory II. Proc. ICRA II'' , ''Lecture Notes in Mathematics'' , '''832''' , Springer (1980) pp. 103–169 {{MR|0607151}} {{ZBL|0446.16031}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, V.A. Ponomarow, "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Bongartz, "Tilted algebras" M. Auslander (ed.) E. Lluis (ed.) , ''Representations of Algebras. Proc. ICRA III'' , ''Lecture Notes in Mathematics'' , '''903''' , Springer (1981) pp. 26–38 {{MR|0654701}} {{ZBL|0478.16025}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> D. Happel, "Triangulated categories in the representation theory of finite dimensional algebras" , ''London Math. Soc. Lecture Notes'' , '''119''' , Cambridge Univ. Press (1988) {{MR|0935124}} {{ZBL|0635.16017}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Happel, C.M. Ringel, "Tilted algebras" ''Trans. Amer. Math. Soc.'' , '''274''' (1982) pp. 399–443 {{MR|0675063}} {{MR|0662711}} {{ZBL|0503.16024}} {{ZBL|0489.16025}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Happel, I. Reiten, S.O. Smalø, "Tilting in abelian categories and quasitilted algebras" ''Memoirs Amer. Math. Soc.'' , '''575''' (1996) {{MR|1327209}} {{ZBL|0849.16011}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> O. Kerner, "Tilting wild algebras" ''J. London Math. Soc.'' , '''39''' : 2 (1989) pp. 29–47 {{MR|0989917}} {{ZBL|0675.16013}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> O. Kerner, "Wild tilted algebras revisited" ''Colloq. Math.'' , '''73''' (1997) pp. 67–81 {{MR|1436951}} {{ZBL|0879.16006}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Liu, "The connected components of the Auslander–Reiten quiver of a tilted algebra" ''J. Algebra'' , '''161''' (1993) pp. 505–523 {{MR|1247369}} {{ZBL|0818.16014}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> C.M. Ringel, "The regular components of the Auslander–Reiten Quiver of a tilted algebra" ''Chinese Ann. Math. Ser. B.'' , '''9''' (1988) pp. 1–18 {{MR|0943675}} {{ZBL|0667.16024}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> H. Strauss, "On the perpendicular category of a partial tilting module" ''J. Algebra'' , '''144''' (1991) pp. 43–66 {{MR|1136894}} {{ZBL|0746.16009}} </TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> I. Assem, "Tilting theory - an introduction" N. Balcerzyk (ed.) et al. (ed.) , ''Topics in Algebra'' , '''26''' , Banach Center Publ. (1990) pp. 127–180 {{MR|1171230}} {{ZBL|0726.16008}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Auslander, M.I. Platzeck, I. Reiten, "Coxeter functors without diagrams" ''Trans. Amer. Math. Soc.'' , '''250''' (1979) pp. 1–46 {{MR|0530043}} {{ZBL|0421.16016}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S. Brenner, M. Butler, "Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors" V. Dlab (ed.) P. Gabriel (ed.) , ''Representation Theory II. Proc. ICRA II'' , ''Lecture Notes in Mathematics'' , '''832''' , Springer (1980) pp. 103–169 {{MR|0607151}} {{ZBL|0446.16031}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> I.N. Bernstein, I.M. Gelfand, V.A. Ponomarow, "Coxeter functors and Gabriel's theorem" ''Russian Math. Surveys'' , '''28''' (1973) pp. 17–32</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> K. Bongartz, "Tilted algebras" M. Auslander (ed.) E. Lluis (ed.) , ''Representations of Algebras. Proc. ICRA III'' , ''Lecture Notes in Mathematics'' , '''903''' , Springer (1981) pp. 26–38 {{MR|0654701}} {{ZBL|0478.16025}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> D. Happel, "Triangulated categories in the representation theory of finite dimensional algebras" , ''London Math. Soc. Lecture Notes'' , '''119''' , Cambridge Univ. Press (1988) {{MR|0935124}} {{ZBL|0635.16017}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> D. Happel, C.M. Ringel, "Tilted algebras" ''Trans. Amer. Math. Soc.'' , '''274''' (1982) pp. 399–443 {{MR|0675063}} {{MR|0662711}} {{ZBL|0503.16024}} {{ZBL|0489.16025}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> D. Happel, I. Reiten, S.O. Smalø, "Tilting in abelian categories and quasitilted algebras" ''Memoirs Amer. Math. Soc.'' , '''575''' (1996) {{MR|1327209}} {{ZBL|0849.16011}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> O. Kerner, "Tilting wild algebras" ''J. London Math. Soc.'' , '''39''' : 2 (1989) pp. 29–47 {{MR|0989917}} {{ZBL|0675.16013}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> O. Kerner, "Wild tilted algebras revisited" ''Colloq. Math.'' , '''73''' (1997) pp. 67–81 {{MR|1436951}} {{ZBL|0879.16006}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> S. Liu, "The connected components of the Auslander–Reiten quiver of a tilted algebra" ''J. Algebra'' , '''161''' (1993) pp. 505–523 {{MR|1247369}} {{ZBL|0818.16014}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> C.M. Ringel, "The regular components of the Auslander–Reiten Quiver of a tilted algebra" ''Chinese Ann. Math. Ser. B.'' , '''9''' (1988) pp. 1–18 {{MR|0943675}} {{ZBL|0667.16024}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> C.M. Ringel, "Tame algebras and integral quadratic forms" , ''Lecture Notes in Mathematics'' , '''1099''' , Springer (1984) {{MR|0774589}} {{ZBL|0546.16013}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> H. Strauss, "On the perpendicular category of a partial tilting module" ''J. Algebra'' , '''144''' (1991) pp. 43–66 {{MR|1136894}} {{ZBL|0746.16009}} </td></tr></table>

Revision as of 17:02, 1 July 2020

The endomorphism ring of a tilting module over a finite-dimensional hereditary algebra (cf. also Algebra; Endomorphism).

Let $H$ be a finite-dimensional hereditary $K$-algebra, $K$ some field, for example the path-algebra of some finite quiver without oriented cycles. A finite-dimensional $H$-module $\square _ { H } T$ is called a tilting module if

i) $\operatorname {p.dim} T \leq 1$, which always is satisfied in this context;

ii) $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$; and

iii) there exists a short exact sequence $0 \rightarrow H \rightarrow T _ { 1 } \rightarrow T _ { 2 } \rightarrow 0$ with $T _ { 1 }$ and in $\operatorname{add} T$, the category of finite direct sums of direct summands of $T$. Here, is projective dimension. The third condition also says that $T$ is maximal with respect to the property $\operatorname { Ext } _ { H } ^ { 1 } ( T , T ) = 0$. Note further, that a tilting module $T$ over a hereditary algebra is uniquely determined by its composition factors. Cf. also Tilting module.

The algebra $B = \operatorname { End } _ { H } ( T )$ is called a tilted algebra of type $H$, and $T$ becomes an $( H , B )$-bimodule (cf. also Bimodule).

In $H$-mod, the category of finite-dimensional $H$-modules, the module $T$ defines a torsion pair $( \mathcal{G} , \mathcal{F} )$ with torsion class $\mathcal{G}$ consisting of modules, generated by $T$ and torsion-free class $\mathcal{F} = \{ Y : \operatorname { Hom } _ { H } ( T , Y ) = 0 \}$. In $B$-mod it defines the torsion pair $\cal ( X , Y )$ with torsion class $\chi = \{ Y : T \otimes _ { B } Y = 0 \}$ and torsion-free class $\mathcal{Y} = \{ Y : \operatorname { Tor } _ { 1 } ^ { B } ( T , Y ) = 0 \}$. The Brenner–Butler theorem says that the functors $\operatorname{Hom}_H( T , - )$, respectively $T \otimes_{ B} -$, induce equivalences between $\mathcal{G}$ and $\mathcal{Y}$, whereas $\operatorname { Ext } _ { H } ^ { 1 } ( T , - )$, respectively $\operatorname { Tor } _ { 1 } ^ { B } ( T , - )$, induce equivalences between $\mathcal{F}$ and $\mathcal{X}$. In $B$-mod the torsion pair is splitting, that is, any indecomposable $B$-module is either torsion or torsion-free. In this sense, $B$-mod has "less" indecomposable modules, and information on the category $H$-mod can be transferred to $B$-mod.

For example, $B$ has global dimension at most $2$ and any indecomposable $B$-module has projective dimension or injective dimension at most $1$ (cf. also Dimension for dimension notions). These condition characterize the more general class of quasi-tilted algebras.

The indecomposable injective $H$-modules are in the torsion class $\mathcal{G}$ and their images under the tilting functor $\operatorname{Hom}_H( T , - )$ are contained in one connected component of the Auslander–Reiten quiver $\Gamma ( B )$ of $B$-mod (cf. also Quiver; Riedtmann classification), and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of $\Gamma ( B )$ characterizes tilted algebras. Moreover, the Auslander–Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components.

If $H$ is a basic hereditary algebra and $H e$ is a simple projective module, then $T = H ( 1 - e ) \oplus \operatorname { TrD } H e$, where $\operatorname{Tr}D$ denotes the Auslander–Reiten translation (cf. Riedtmann classification), is a tilting module, sometimes called APR-tilting module. The induced torsion pair $( \mathcal{G} , \mathcal{F} )$ in $H$-mod is splitting and $H e$ is the unique indecomposable module in $\mathcal{F}$. The tilting functor $\operatorname{Hom}_H( T , - )$ corresponds to the reflection functor introduced by I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev for their proof of the Gabriel theorem [a4].

If the hereditary algebra $H$ is representation-finite (cf. also Algebra of finite representation type), then any tilted algebra of type $H$ also is representation-finite. If $H$ is tame (cf. also Representation of an associative algebra), then a tilted algebra of type $H$ either is tame and one-parametric, or representation-finite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre-injective direct summands simultaneously. If $H$ is wild (cf. also Representation of an associative algebra), then a tilted algebra of type $H$ may be wild, or tame domestic, or representation-finite.

See also Tilting theory.

References

[a1] I. Assem, "Tilting theory - an introduction" N. Balcerzyk (ed.) et al. (ed.) , Topics in Algebra , 26 , Banach Center Publ. (1990) pp. 127–180 MR1171230 Zbl 0726.16008
[a2] M. Auslander, M.I. Platzeck, I. Reiten, "Coxeter functors without diagrams" Trans. Amer. Math. Soc. , 250 (1979) pp. 1–46 MR0530043 Zbl 0421.16016
[a3] S. Brenner, M. Butler, "Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors" V. Dlab (ed.) P. Gabriel (ed.) , Representation Theory II. Proc. ICRA II , Lecture Notes in Mathematics , 832 , Springer (1980) pp. 103–169 MR0607151 Zbl 0446.16031
[a4] I.N. Bernstein, I.M. Gelfand, V.A. Ponomarow, "Coxeter functors and Gabriel's theorem" Russian Math. Surveys , 28 (1973) pp. 17–32
[a5] K. Bongartz, "Tilted algebras" M. Auslander (ed.) E. Lluis (ed.) , Representations of Algebras. Proc. ICRA III , Lecture Notes in Mathematics , 903 , Springer (1981) pp. 26–38 MR0654701 Zbl 0478.16025
[a6] D. Happel, "Triangulated categories in the representation theory of finite dimensional algebras" , London Math. Soc. Lecture Notes , 119 , Cambridge Univ. Press (1988) MR0935124 Zbl 0635.16017
[a7] D. Happel, C.M. Ringel, "Tilted algebras" Trans. Amer. Math. Soc. , 274 (1982) pp. 399–443 MR0675063 MR0662711 Zbl 0503.16024 Zbl 0489.16025
[a8] D. Happel, I. Reiten, S.O. Smalø, "Tilting in abelian categories and quasitilted algebras" Memoirs Amer. Math. Soc. , 575 (1996) MR1327209 Zbl 0849.16011
[a9] O. Kerner, "Tilting wild algebras" J. London Math. Soc. , 39 : 2 (1989) pp. 29–47 MR0989917 Zbl 0675.16013
[a10] O. Kerner, "Wild tilted algebras revisited" Colloq. Math. , 73 (1997) pp. 67–81 MR1436951 Zbl 0879.16006
[a11] S. Liu, "The connected components of the Auslander–Reiten quiver of a tilted algebra" J. Algebra , 161 (1993) pp. 505–523 MR1247369 Zbl 0818.16014
[a12] C.M. Ringel, "The regular components of the Auslander–Reiten Quiver of a tilted algebra" Chinese Ann. Math. Ser. B. , 9 (1988) pp. 1–18 MR0943675 Zbl 0667.16024
[a13] C.M. Ringel, "Tame algebras and integral quadratic forms" , Lecture Notes in Mathematics , 1099 , Springer (1984) MR0774589 Zbl 0546.16013
[a14] H. Strauss, "On the perpendicular category of a partial tilting module" J. Algebra , 144 (1991) pp. 43–66 MR1136894 Zbl 0746.16009
How to Cite This Entry:
Tilted algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tilted_algebra&oldid=24132
This article was adapted from an original article by O. Kerner (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article