Tilting theory

It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed root-system — a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone. … For this reason, and because the word 'tilt' inflects easily, we call our functors tilting functors or simply tilts.

Brenner & Butler (1980, p. 103)

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 JosephBernšteĭn , Israel Gelfand ,andV. A. Ponomarev ( 1973 ); these functors were used to relate representations of two quivers. These functors were reformulated by MauriceAuslander , María Inés Platzeck ,and Idun Reiten  ( 1979 ), and generalized by SheilaBrennerandMichael C. R. Butler ( 1980 ) who introduced tilting functors. DieterHappelandClaus Michael Ringel ( 1982 ) 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 ¹
  _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.

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′ = 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).

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 ⁱ
  _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 allR-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.

References

• Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning, eds. (2007), Handbook of tilting theory (PDF), London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, doi:10.1017/CBO9780511735134, ISBN   978-0-521-68045-5, MR   2385175
• Assem, Ibrahim (1990). "Tilting theory–an introduction" (PDF). In Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan; Simson, Daniel; Vogel, Wolfgang (eds.). Topics in algebra, Part 1 (Warsaw, 1988). Banach Center Publications. Vol. 26. Warsaw: PWN. pp. 127–180. doi:10.4064/-26-1-127-180. MR   1171230.
• Auslander, Maurice; Platzeck, María Inés; Reiten, Idun (1979), "Coxeter functors without diagrams", Transactions of the American Mathematical Society , 250: 1–46, doi: 10.2307/1998978 , ISSN   0002-9947, JSTOR   1998978, MR   0530043
• Bernšteĭn, Iosif N.; Gelfand, Izrail M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian Mathematical Surveys, 28 (2): 17–32, Bibcode:1973RuMaS..28...17B, CiteSeerX   10.1.1.642.2527 , doi:10.1070/RM1973v028n02ABEH001526, ISSN   0042-1316, MR   0393065
• Brenner, Sheila; Butler, Michael C. R. (1980), "Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors", Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, Berlin, New York: Springer-Verlag, pp. 103–169, doi:10.1007/BFb0088461, ISBN   978-3-540-10264-9, MR   0607151
• Buan, Aslak; Marsh, Robert; Reineke, Markus; Reiten, Idun; Todorov, Gordana (2006), "Tilting theory and cluster combinatorics", Advances in Mathematics , 204 (2): 572–618, arXiv: math/0402054 , doi: 10.1016/j.aim.2005.06.003 , MR   2249625, S2CID   15318919
• Buan, Aslak; Marsh, Robert; Reiten, Idun (2007), "Cluster-tilted algebras", Transactions of the American Mathematical Society , 359 (1): 323–332, doi: 10.1090/s0002-9947-06-03879-7 , MR   2247893
• Cline, Edward; Parshall, Brian; Scott, Leonard (1986), "Derived categories and Morita theory", Algebra, 104 (2): 397–409, doi:10.1016/0021-8693(86)90224-3, MR   0866784
• Colpi, Riccardo; Fuller, Kent R. (February 2007), "Tilting Objects in Abelian Categories and Quasitilted Rings" (PDF), Transactions of the American Mathematical Society , 359 (2): 741–765, doi: 10.1090/s0002-9947-06-03909-2
• Happel, Dieter; Reiten, Idun; Smalø, Sverre O. (1996), "Tilting in abelian categories and quasitilted algebras", Memoirs of the American Mathematical Society , 575
• Happel, Dieter; Ringel, Claus Michael (1982), "Tilted algebras", Transactions of the American Mathematical Society , 274 (2): 399–443, doi: 10.2307/1999116 , ISSN   0002-9947, JSTOR   1999116, MR   0675063
• Happel, Dieter (1988), Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes Series, vol. 119, Cambridge University Press, doi:10.1017/CBO9780511629228, ISBN   9780521339223
• Happel, Dieter (2001), "A characterization of hereditary categories with tilting object", Invent. Math., 144 (2): 381–398, Bibcode:2001InMat.144..381H, doi:10.1007/s002220100135, S2CID   120437744
• Rickard, Jeremy (1989), "Morita theory for derived categories", Journal of the London Mathematical Society, 39 (2): 436–456, doi:10.1112/jlms/s2-39.3.436
• Unger, L. (2001) [1994], "Tilting theory", Encyclopedia of Mathematics , EMS Press