Tilting theory

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.
• Ext¹
  _A(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,−), Ext¹
_A(T,−), −⊗_BT and Tor^B
₁(−,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.

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′=Ext¹
_A(T,−), G=−⊗_BT, and G′=Tor^B
₁(−,T). F is right adjoint to G and F′ is right adjoint to G′.

Brenner & Butler (1980) showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B. Specifically, if we define the two subcategories ${\displaystyle {\mathcal {F}}=\ker(F)}$ and ${\displaystyle {\mathcal {T}}=\ker(F')}$ of A-mod, and the two subcategories ${\displaystyle {\mathcal {X}}=\ker(G)}$ and ${\displaystyle {\mathcal {Y}}=\ker(G')}$ of B-mod, then ${\displaystyle ({\mathcal {T}},{\mathcal {F}})}$ is a torsion pair in A-mod (i.e. ${\displaystyle {\mathcal {T}}}$ and ${\displaystyle {\mathcal {F}}}$ are maximal subcategories with the property ${\displaystyle \operatorname {Hom} ({\mathcal {T}},{\mathcal {F}})=0}$; this implies that every M in A-mod admits a natural short exact sequence ${\displaystyle 0\to U\to M\to V\to 0}$ with U in ${\displaystyle {\mathcal {T}}}$ and V in ${\displaystyle {\mathcal {F}}}$) and ${\displaystyle ({\mathcal {X}},{\mathcal {Y}})}$ is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between ${\displaystyle {\mathcal {T}}}$ and ${\displaystyle {\mathcal {Y}}}$, while the restrictions of F′ and G′ yield inverse equivalences between ${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {X}}}$. (Note that these equivalences switch the order of the torsion pairs ${\displaystyle ({\mathcal {T}},{\mathcal {F}})}$ and ${\displaystyle ({\mathcal {X}},{\mathcal {Y}})}$.)

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

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

Happel (1988) and Cline, Parshall & Scott (1986) 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).

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.
• Extⁱ
  _A(T,T) = 0 for all i>0.
• There is an exact sequence ${\displaystyle 0\to A\to T_{1}\to \dots \to T_{n}\to 0}$ 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).

Rickard (1989) 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.

Happel, Reiten & Smalø (1996) 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. Happel (2001) classified the hereditary abelian categories that can appear in the above construction.

Colpi & Fuller (2007) 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 Buan et al. (2006)) and cluster tilted algebra (Buan, Marsh & Reiten (2007)) 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.