Tilting theory

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 = EndA(T ). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors HomA(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 = EndA(T ). Write F = HomA(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{F}=\ker(F) and \mathcal{T}=\ker(F') of A-mod, and the two subcategories \mathcal{X}=\ker(G) and \mathcal{Y}=\ker(G') of B-mod, then (\mathcal{T},\mathcal{F}) is a torsion pair in A-mod (i.e. \mathcal{T} and \mathcal{F} are maximal subcategories with the property \operatorname{Hom}(\mathcal{T},\mathcal{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 \mathcal{T} and V in \mathcal{F}) and (\mathcal{X},\mathcal{Y}) is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between \mathcal{T} and \mathcal{Y}, while the restrictions of F′ and G′ yield inverse equivalences between \mathcal{F} and \mathcal{X}. (Note that these equivalences switch the order of the torsion pairs (\mathcal{T},\mathcal{F}) and (\mathcal{X},\mathcal{Y}).)

Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case \mathcal{T}=\operatorname{mod}-A and \mathcal{Y}=\operatorname{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 (\mathcal{X},\mathcal{Y}) splits, i.e. every indecomposable object of B-mod is either in \mathcal{X} or in \mathcal{Y}.

and showed that in general A and B are derived equivalent (i.e. the derived categories Db(A-mod) and Db(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 Ti are finite direct sums of direct summands of T. These generalized tilting modules also yield derived equivalences between A and B, where B = EndA(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