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.

## Contents

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.

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*,−), Ext^{1}_{A}(*T*,−), −⊗_{B}*T* 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′* = Ext^{1}_{A}(*T*,−), *G* = −⊗_{B}*T*, 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 and of *A*-mod, and the two subcategories and of *B*-mod, then is a torsion pair in *A*-mod (i.e. and are maximal subcategories with the property ; this implies that every *M* in *A*-mod admits a natural short exact sequence with *U* in and *V* in ) and is a torsion pair in *B*-mod. Further, the restrictions of the functors *F* and *G* yield inverse equivalences between and , while the restrictions of *F′* and *G′* yield inverse equivalences between and . (Note that these equivalences switch the order of the torsion pairs and .)

Tilting theory may be seen as a generalization of Morita equivalence which is recovered if *T* is a projective generator; in that case and .

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 K_{0}(*A*) and K_{0}(*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 splits, i.e. every indecomposable object of *B*-mod is either in or in .

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.- Ext
^{i}_{A}(*T*,*T*) = 0 for all*i*> 0. - There is an exact sequence where the
*T*are finite direct sums of direct summands of_{i}*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*.

In commutative algebra, the **prime spectrum** of a ring *R* is the set of all prime ideals of *R*, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .

**Homological algebra** is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, **K-theory** is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

In mathematics, a **finitely generated module** is a module that has a finite generating set. A finitely generated module over a ring *R* may also be called a **finite R-module**,

In the mathematical fields of topology and K-theory, the **Serre–Swan theorem**, also called **Swan's theorem**, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".

In mathematics, particularly in algebra, the class of **projective modules** enlarges the class of free modules over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

In algebraic geometry, **motives** is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In mathematics, certain functors may be *derived* to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

**Mitchell's embedding theorem**, also known as the **Freyd–Mitchell theorem** or the **full embedding theorem**, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

In mathematics, the **Grothendieck group**, or **group of differences**, of a commutative monoid *M* is a certain abelian group. This abelian group is constructed from *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, especially in the fields of representation theory and module theory, a **Frobenius algebra** is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory, . Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.

In abstract algebra, **Morita equivalence** is a relationship defined between rings that preserves many ring-theoretic properties. More precisely two rings like *R*, *S* are Morita equivalent if their categories of modules are additively equivalent. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.

In mathematics, the **Cuntz algebra**, named after Joachim Cuntz, is the universal C*-algebra generated by isometries of an infinite-dimensional Hilbert space satisfying certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, is isometric to the sequence space

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

In mathematics, the formalism of ** B-admissible representations** provides constructions of full Tannakian subcategories of the category of representations of a group

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

In mathematics, the **quotient** of an abelian category by a Serre subcategory is the abelian category which, intuitively, is obtained from by ignoring all objects from . There is a canonical exact functor whose kernel is , and is in a certain sense the most general abelian category with this property.

In mathematics, the **base change theorems** relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves:

The **Beilinson–Bernstein localization** theorem is a foundational result of geometric representation theory, a part of mathematics studying the representation theory of e.g. Lie algebras using geometry.

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