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Tilting theory

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+#redirect [[tilting theory]]
-{{AfC submission|t||ts=20221210113759|u=LesMaxos|ns=118|demo=}}<!-- Important, do not remove this line before article has been created. -->
-In [[mathematics]], specifically the [[representation theory]] of [[Dimension (vector space)|finite dimensional]] [[Algebra over a field|algebras]], <math>\tau</math>'''-tilting theory''' is a recent development, that can be viewed as a completion of [[tilting theory]] from the viewpoint of mutation. The subfield was introduced in the early 2010s by Takahide Adachi, [https://www.math.nagoya-u.ac.jp/~iyama/ Osamu Iyama] and [[Idun Reiten]]<ref name=":0">{{Cite journal |last=Adachi |first=Takahide |last2=Iyama |first2=Osamu |last3=Reiten |first3=Idun |title=τ-tilting theory |url=https://www.researchgate.net/publication/259483313_Introduction_to_tau-tilting_theory |journal=Compositio Mathematica |publisher=Cambridge University Press |publication-date=2014 |volume=150 |issue=3 |pages=415-452 |doi=10.1112/S0010437X13007422}}</ref> and is now a very active area of research. The name reflects the combination of [[Auslander–Reiten theory]] and [[tilting theory]], where the <math>\tau</math> represents the Auslander-Reiten translation in the [[Category of modules|module category]] of an algebra. Many connections of <math>\tau</math>-tilting theory with other areas of mathematics have been explored in recent years.
-== Basic definitions ==
-Let <math>K</math> be an [[algebraically closed field]] and <math>A</math> be a basic finite dimensional algebra, that is, an algebra with pairwise [[Module homomorphism|non-isomorphic]] [[Projective module|projective modules]].<ref name=":1">{{Cite book |last=Assem |first=Ibrahim |url=https://www.cambridge.org/core/books/elements-of-the-representation-theory-of-associative-algebras/AA8066B5809D0F556A540400AD3A419C |title=Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory |last2=Skowronski |first2=Andrzej |last3=Simson |first3=Daniel |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-58423-4 |series=London Mathematical Society Student Texts |volume=1 |location=Cambridge}}</ref> Let <math>\tau</math> denote the Auslander-Reiten translation, which is defined as <math>\tau = D\text{Tr}</math>, where <math>D=\text{Hom}_K(-,K)</math> denotes the [[Duality (mathematics)|standard duality]] and <math>\text{Tr}</math> denotes the transposition.<ref name=":1" /> Denote by <math>\text{mod} A</math> the [[Category (mathematics)|category]] of [[Finitely-presented module|finitely presented]] [[Module (mathematics)|right modules]] of <math>A</math> and for a module <math>M</math> denote by <math>|M|</math> the number of non-isomorphic [[Indecomposable module|indecomposable]] [[Direct sum|direct summands]] of <math>M</math>. The central notion studied in this area is the following.
-=== <math>\tau</math>-rigidity ===
-Let <math>M</math> be an object in <math>\text{mod} A</math> and <math>P</math> be a projective module in <math>\text{mod}A</math>.
-* The module <math>M</math> is called <math>\tau</math>'''-rigid''' if <math>\text{Hom}_A(M,\tau M) = 0</math>. In other words, there does not exist a non-zero [[module homomorphism]] from the module <math>M</math> to its Auslander-Reiten translation <math>\tau M</math>.<ref name=":0" />
-* A module <math>\tau</math>-rigid module <math>M</math> is called '''<math>\tau</math>-tilting''' if <math>|M| = |A|</math>.<ref name=":0" /> In other words, the number of direct non-isomorphic indecomposable direct summands of <math>M</math> equals that of <math>A</math>, where we may express the algebra as a direct sum of all projective modules.<ref name=":1" />
-* A pair <math>(M,P)</math> of a module <math>M</math> and a projective module <math>P</math> is called a <math>\tau</math>'''-rigid''' '''pair''' if <math>M</math> is <math>\tau</math>-rigid and <math>\text{Hom}_A(P,M) = 0</math>.<ref name=":0" />
-* A <math>\tau</math>-rigid pair <math>(M,P)</math> is called '''(support-)'''<math>\tau</math>'''-tilting''' if <math>|M| + |P| = |A|</math>.<ref name=":0" />
-== Connections with other areas ==
-Already in the introductory paper a few connections with other areas were given, and more have been discovered over the years.
-=== Silting Theory ===
-A [[Chain complex|complex]] <math>\boldsymbol{P} = (P^i, d^i)</math>  in the bounded [[homotopy category]] <math>K^b(\text{proj} A)</math> of projective <math>A</math>-modules is a '''2-term-silting complex''' if
-* <math>\boldsymbol{P}</math> is concentrated in degrees <math>-1</math> and <math>0</math>, i.e. <math>P^i = 0</math> for all <math>i \neq 0, -1</math>;
-* <math>\text{Hom}_{K^b(\text{proj A})} (\boldsymbol{P},\boldsymbol{P}[i])=0</math> for any <math>i > 0</math>, where <math>\boldsymbol{P}[i]</math> denotes the <math>i</math>-th [[Triangulated category|shift in the triangulated category]];
-* Each object in <math>K^b(\text{proj} A)</math> can be obtained from <math>\boldsymbol{P}</math> by taking finitely many shifts, [[Mapping cone (homological algebra)|mapping cones]] and direct summands. (Without this property <math>\boldsymbol{P}</math> is called '''2-term pre-silting'''). <ref>{{Cite journal |last=Aihara |first=Takuma |last2=Iyama |first2=Osamu |orig-date=2012 |title=Silting mutation in triangulated categories |url=https://doi.org/10.1112/jlms/jdr055 |journal=Journal of the London Mathematical Society |volume=85 |issue=3 |pages=633-668}}</ref>
-There exists a [[bijection]] between <math>\tau</math>-tilting pairs <math>(M,P)</math> and two-term silting complexes. <ref name=":0" />
-=== Cluster-Tilting Theory ===
-Let <math>\mathcal{C}</math> be a <math>\text{Hom}</math>-finite [[Krull–Schmidt category|Krull–Schmidt]] 2-Calabi-Yau triangulated category.
-* An object <math>T</math> in <math>\mathcal{C}</math> is called '''rigid''' if <math>\text{Hom}_{\mathcal{C}} ( T, T[1])=0</math>.
-* An object <math>T</math> in <math>\mathcal{C}</math> is called '''cluster-tilting''' if <math>\text{add} (T) = \{ X \in \mathcal{C} : \text{Hom}_\mathcal{C} (T,X[1])=0 \}</math>, where <math>\text{add} (T)</math> is the [[full subcategory]] consisting of all direct summands of finite direct sums of copies of <math>T</math>.
-* An object <math>T</math> in <math>\mathcal{C}</math> is called '''maximal rigid''' if <math>T</math> rigid and <math>\text{add} (T) = \{ X \in \mathcal{C} : \text{Hom}_\mathcal{C} (T \oplus X, (T \oplus X)[1])=0 \}</math>.
-Assume that <math>\mathcal{C}</math> has a cluster-tilting object <math>T</math>, and let <math>A = \text{End}_{\mathcal{C}}(T)^{\text{op}}</math> be the [[Opposite ring|opposite]] [[Endomorphism ring|endomorphism algebra]] of <math>T</math> over <math>\mathcal{C}</math>. Then there is a bijection between basic rigid objects in <math>\mathcal{C}</math> and basic <math>\tau</math>-rigid objects in <math>A</math>. There is a bijection between cluster-tilting objects in <math>\mathcal{C}</math> and basic support-<math>\tau</math>-tilting objects in <math>A</math>. And there is a bijection between basic cluster-tilting objects in <math>\mathcal{C}</math> and basic <math>\tau</math>-tilting objects in <math>A</math>.
-== References ==
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-{{reflist}}