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 Template:Harvs; these functors were used to relate representations of two quivers. These functors were reformulated by Template:Harvs, and generalized by Template:Harvs who introduced tilting functors. Template:Harvs 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.
• ExtTemplate:Su(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,−), ExtTemplate:Su(T,−), −⊗_BT and TorTemplate:Su(−,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′ = ExtTemplate:Su(T,−), G = −⊗_BT, and G′ = TorTemplate:Su(−,T). F is right adjoint to G and F′ is right adjoint to G′.

Template:Harvtxt showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B. Specifically, if we define the two subcategories $ℱ=\ker (F)$ and $𝒯=\ker (F^{\prime })$ of A-mod, and the two subcategories $𝒳=\ker (G)$ and $𝒴=\ker (G^{\prime })$ of B-mod, then $(𝒯,ℱ)$ is a torsion pair in A-mod (i.e. $𝒯$ and $ℱ$ are maximal subcategories with the property $\mathrm{Hom}(𝒯,ℱ)=0$; 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 $𝒯$ and V in $ℱ$) and $(𝒳,𝒴)$ is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between $𝒯$ and $𝒴$, while the restrictions of F′ and G′ yield inverse equivalences between $ℱ$ and $𝒳$. (Note that these equivalences switch the order of the torsion pairs $(𝒯,ℱ)$ and $(𝒳,𝒴)$.)

Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case $𝒯=\mathrm{mod}-A$ and $𝒴=\mathrm{mod}-B$.

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 $(𝒳,𝒴)$ splits, i.e. every indecomposable object of B-mod is either in $𝒳$ or in $𝒴$.

Template:Harvtxt and Template:Harvtxt 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.
• ExtTemplate:Su(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 ).

Template:Harvtxt 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.

Template:Harvtxt 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. Template:Harvtxt classified the hereditary abelian categories that can appear in the above construction.

Template:Harvtxt 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 Template:Harvtxt) and cluster tilted algebra (Template:Harvtxt) 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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