View source for Tilting theory

==Artin algebras.==
A finitely-generated [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301301.png" /> over an Artin algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301302.png" /> (cf. also [[Artinian module|Artinian module]]) is called a tilting module if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301304.png" /> and there is a short [[Exact sequence|exact sequence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301305.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301306.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301307.png" /> denotes the projective dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t1301309.png" /> is the category of finite direct sums of direct summands of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013010.png" /> (see [[Tilting module|Tilting module]]). Dually, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013011.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013012.png" /> is called a cotilting module if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013013.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013014.png" /> is a tilting module, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013015.png" /> denotes the usual duality. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013016.png" /> is a tilting module and $\Gamma=\operatorname{End}_\Lambda(T)^{\operatorname{op}}$, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013018.png" /> is a tilting module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013019.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013020.png" /> is a cotilting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013021.png" />-module.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013022.png" /> be a tilting module, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013023.png" /> be the [[Category|category]] of finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013024.png" />-modules generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013025.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013026.png" /> is a torsion class in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013027.png" /> of finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013028.png" />-modules. This yields an associated torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013030.png" />. Dually, there is associated with a cotilting module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013031.png" /> the subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013033.png" />-modules cogenerated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013034.png" />. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013035.png" /> is a torsion-free class and there is an associated torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013036.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013037.png" />.

An important feature of tilting theory is the following connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013039.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013040.png" /> for a tilting module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013041.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013042.png" /> denotes the torsion pair in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013043.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013045.png" /> the torsion pair associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013046.png" />, then there are equivalences of categories:

<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/t130130/t13013047.png" /></td> </tr></table>

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/t130130/t13013048.png" /></td> </tr></table>

(Cf. also [[Tilting functor|Tilting functor]].) In the special case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013049.png" /> is a projective generator one recovers the [[Morita equivalence|Morita equivalence]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013051.png" /> is a projective generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013052.png" />. For a general module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013053.png" />, the Artin algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013055.png" /> may be quite different, but they share many homological properties; in particular, one uses the tilting functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013057.png" /> in order to transfer properties between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013059.png" />. The transfer of information is especially useful when one already knows a lot about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013060.png" /> and when the torsion pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013061.png" /> splits, that is, when each indecomposable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013062.png" />-module is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013063.png" /> or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013064.png" />. This is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013065.png" /> is hereditary. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013066.png" /> is called a tilted algebra (cf. also [[Tilted algebra|Tilted algebra]]). Tilted algebras have played an important part in representation theory, since many questions can be reduced to this class of algebras.

Tilting theory goes back to the reflection functors introduced by I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev [[#References|[a6]]] in the early 1970s. A module-theoretic interpretation of these functors was given by M. Auslander, M.I. Platzeck and I. Reiten [[#References|[a1]]]. Further generalizations where given by S. Brenner and M.C.R. Butler [[#References|[a5]]], where the equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013067.png" /> was established. The above definitions where given by D. Happel and C.M. Ringel [[#References|[a12]]], who developed an extensive theory of tilted algebras. A good reference for the early work in tilting theory is [[#References|[a4]]].

An important theoretical development of tilting theory was the connection with derived categories established by Happel [[#References|[a10]]]. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013068.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013069.png" /> is a tilting module induces an equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013071.png" /> denotes the [[Derived category|derived category]] whose objects are the bounded complexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013072.png" />-modules.

The set of all tilting modules (up to isomorphism) over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013073.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013075.png" /> an [[Algebraically closed field|algebraically closed field]], has an interesting combinatorial structure: It is a countable [[Simplicial complex|simplicial complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013076.png" />. This complex has been investigated by L. Unger in [[#References|[a21]]] and [[#References|[a22]]], where it was proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013077.png" /> is a [[shellable complex]] provided it is finite, and that certain representation-theoretical invariants are reflected by its structure.

==Analogues and generalizations.==
There is an analogous concept of a tilting sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013078.png" /> for the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013079.png" /> of coherent sheaves of a weighted projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013080.png" /> (cf. also [[Coherent sheaf|Coherent sheaf]]) as studied in [[#References|[a9]]]. The canonical algebras introduced in [[#References|[a19]]] can be realized as endomorphism algebras of certain tilting sheaves.

To obtain a common treatment of both the class of tilted algebras and the canonical algebras, in [[#References|[a13]]] tilting theory was generalized to hereditary categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013081.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013082.png" /> is a connected Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013083.png" />-category with vanishing Yoneda functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013084.png" /> and finite-dimensional homomorphism and extension spaces. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013085.png" /> denotes an algebraically closed field. An object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013088.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013089.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013090.png" />, is called a tilting object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013091.png" />. The endomorphism algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013092.png" /> of a tilting object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013093.png" /> is called a quasi-tilted algebra. Tilted algebras and canonical algebras furnish examples for quasi-tilted algebras.

There are two types of hereditary categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013094.png" /> with tilting objects: those derived equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013095.png" /> for some finite-dimensional hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013096.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013097.png" /> and those derived equivalent to some category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013098.png" /> of coherent sheaves on a weighted projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t13013099.png" />. Two categories are called derived equivalent if their derived categories are equivalent as triangulated categories. In 2000, Happel [[#References|[a11]]] proved that these are the only possible hereditary categories with tilting object. This proved a conjecture stated, for example, in [[#References|[a17]]].

===Generalizations and applications of tilting modules.===
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130100.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130101.png" /> is called a generalized tilting module if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130103.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130104.png" /> and there is an exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130106.png" />. Generalized tilting modules were introduced in [[#References|[a16]]]. This concept was generalized to the notion of tilting complexes by J. Rickard [[#References|[a18]]], who established some "Morita theory for derived categories" . Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130107.png" /> be a ring and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130108.png" /> be the category of finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130109.png" />-modules. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130110.png" /> the category of bounded complexes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130111.png" /> modulo homotopy. A complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130112.png" /> is called a tilting complex if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130113.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130114.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130115.png" /> denotes the shift functor) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130116.png" /> generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130117.png" /> as a triangulated category. Rickard proved that two rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130119.png" /> are derived equivalent (i.e. their module categories are derived equivalent) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130120.png" /> is the endomorphism ring of a tilting complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130121.png" />.

The results mentioned above uses tilting modules/objects mainly to compare <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130123.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130124.png" /> for some tilting module/object. There are other approaches, which use tilting modules to describe subcategories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130125.png" />. Kerner [[#References|[a15]]] and W. Crawley-Boevey and Kerner [[#References|[a7]]] used tilting modules to investigate subcategories of regular modules over wild hereditary algebras.

==Quasi-hereditary algebras.==
Auslander and Reiten [[#References|[a2]]] proved that there is a one-to-one correspondence between basic generalized tilting modules and certain covariantly finite subcategories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130126.png" />. This correspondence was further investigated [[#References|[a14]]]. The Auslander–Reiten correspondence was applied to quasi-hereditary algebras by Ringel [[#References|[a20]]] and his results served as a basis for applications to Schur algebras by S. Donkin [[#References|[a8]]] and to [[Quantum groups|quantum groups]] by H.H. Andersen [[#References|[a3]]]. In dealing with quasi-hereditary algebras and highest-weight categories, the notion of a tilting module is now (2000) used in a related but deviating way, namely for all the objects or modules that have both a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130128.png" />-filtration and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130130.png" />-filtration. The isomorphism classes of the indecomposables that have both a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130131.png" />-filtration and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130130/t130130132.png" />-filtration correspond bijectively to the elements of the weight poset, and their direct sum is a tilting module in the sense considered above.

====References====
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