Tilting Theory

In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.

Tilting theory was motivated by the introduction of reflection functors by ; these functors were used to relate representations of two quivers. These functors were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this.

Definitions

Suppose that ''A'' is a finite-dimensional unital associative algebra over some field. A finitely-generated right ''A''- module ''T'' is called a tilting module if it has the following three properties:
*''T'' has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
*Ext(''T'',''T'' ) = 0.
*The right ''A''-module ''A'' is the kernel of a surjective morphism between finite direct sums of direct summands of ''T''.

Given such a tilting module, we define the endomorphism algebra ''B'' = End_''A''(''T'' ). This is another finite-dimensional algebra, and ''T'' is a finitely-generated left ''B''-module.
The tilting functors Hom_''A''(''T'',−), Ext(''T'',−), −⊗_''B''''T'' and Tor(−,''T'') relate the category mod-''A'' of finitely-generated right ''A''-modules to the category mod-''B'' of finitely-generated right ''B''-modules.

In practice one often considers hereditary finite-dimensional algebras ''A'' because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite-dimensional algebra is called a tilted algebra.

Facts

Suppose ''A'' is a finite-dimensional algebra, ''T'' is a tilting module over ''A'', and ''B'' = End_''A''(''T'' ). Write ''F'' = Hom_''A''(''T'',−), ''F′'' = Ext(''T'',−), ''G'' = −⊗_''B''''T'', and ''G′'' = Tor(−,''T''). ''F'' is right adjoint to ''G'' and ''F′'' is right adjoint to ''G′''.

showed that tilting functors give equivalences between certain subcategories of mod-''A'' and mod-''B''. Specifically, if we define the two subcategories $\mathcal=\ker(F)$ and $\mathcal=\ker(F')$ of ''A''-mod, and the two subcategories $\mathcal=\ker(G)$ and $\mathcal=\ker(G')$ of ''B''-mod, then $(\mathcal,\mathcal)$ is a torsion pair in ''A''-mod (i.e. $\mathcal$ and $\mathcal$ are maximal subcategories with the property $\operatorname(\mathcal,\mathcal)=0$; this implies that every ''M'' in ''A''-mod admits a natural short exact sequence $0 \to U \to M \to V \to 0$ with ''U'' in $\mathcal$ and ''V'' in $\mathcal$) and $(\mathcal,\mathcal)$ is a torsion pair in ''B''-mod. Further, the restrictions of the functors ''F'' and ''G'' yield inverse equivalences between $\mathcal$ and $\mathcal$, while the restrictions of ''F′'' and ''G′'' yield inverse equivalences between $\mathcal$ and $\mathcal$. (Note that these equivalences switch the order of the torsion pairs $(\mathcal,\mathcal)$ and $(\mathcal,\mathcal)$.)

Tilting theory may be seen as a generalization of Morita equivalence which is recovered if ''T'' is a projective generator; in that case $\mathcal=\operatorname-A$ and $\mathcal=\operatorname-B$.

If ''A'' has finite global dimension, then ''B'' also has finite global dimension, and the difference of ''F'' and ''F induces an isometry between the Grothendieck groups K₀(''A'') and K₀(''B'').

In case ''A'' is hereditary (i.e. ''B'' is a tilted algebra), the global dimension of ''B'' is at most 2, and the torsion pair $(\mathcal,\mathcal)$ splits, i.e. every indecomposable object of ''B''-mod is either in $\mathcal$ or in $\mathcal$.

and showed that in general ''A'' and ''B'' are derived equivalent (i.e. the derived categories D^b(''A''-mod) and D^b(''B''-mod) are equivalent as triangulated categories).

Generalizations and extensions

A generalized tilting module over the finite-dimensional algebra ''A'' is a right ''A''-module ''T'' with the following three properties:
*''T'' has finite projective dimension.
*Ext(''T'',''T'') = 0 for all ''i'' > 0.
*There is an exact sequence $0 \to A \to T_1 \to\dots\to T_n \to 0$ where the ''T_i'' are finite direct sums of direct summands of ''T''.
These generalized tilting modules also yield derived equivalences between ''A'' and ''B'', where ''B'' = End_''A''(''T'' ).

extended the results on derived equivalence by proving that two finite-dimensional algebras ''R'' and ''S'' are derived equivalent if and only if ''S'' is the endomorphism algebra of a "tilting complex" over ''R''. Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings ''R'' and ''S''.

defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some algebraically closed field ''k''. The endomorphism algebras of these tilting objects are the quasi-tilted algebras, a generalization of tilted algebras. The quasi-tilted algebras over ''k'' are precisely the finite-dimensional algebras over ''k'' of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. classified the hereditary abelian categories that can appear in the above construction.

defined tilting objects ''T'' in an arbitrary abelian category ''C''; their definition requires that ''C'' contain the direct sums of arbitrary (possibly infinite) numbers of copies of ''T'', so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring ''R'', they establish tilting functors that provide equivalences between a torsion pair in ''C'' and a torsion pair in ''R''-Mod, the category of ''all'' ''R''-modules.

From the theory of cluster algebras came the definition of cluster category (from ) and cluster tilted algebra () associated to a hereditary algebra ''A''. A cluster tilted algebra arises from a tilted algebra as a certain semidirect product, and the cluster category of ''A'' summarizes all the module categories of cluster tilted algebras arising from ''A''.

References

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Representation theory