Draft:Tau-tilting theory: Difference between revisions
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== Basic definitions == |
== Basic definitions == |
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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| |
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. |
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=== <math>\tau</math>-rigidity === |
=== <math>\tau</math>-rigidity === |
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* 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 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" /> |
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* 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 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" /> |
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* A <math>\tau</math>-rigid pair <math>(M,P)</math> is called <math>\tau</math>'''-tilting''' if <math>|M| + |P| = |A|</math>.<ref name=":0" /> |
* A <math>\tau</math>-rigid pair <math>(M,P)</math> is called <math>\tau</math>'''-tilting''' if <math>|M| + |P| = |A|</math>.<ref name=":0" /><br /> |
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In mathematics, specifically the field of representation theory of finite dimensional algebras, -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, Osamu Iyama and Idun Reiten[1] and is now a very active area of research. The name reflects the combination of Auslander–Reiten theory and tilting theory, where the represents the Auslander-Reiten translation in the module category of an algebra. Many connections of -tilting theory with other areas of mathematics have been explored in recent years.
Basic definitions
Let be an algebraically closed field and be a basic finite dimensional algebra, that is, an algebra with pairwise non-isomorphic projective modules.[2] Let denote the Auslander-Reiten translation, which is defined as , where denotes the standard duality and denotes the transposition.[2] Denote by the category of finitely presented right modules of and for a module denote by the number of non-isomorphic indecomposable direct summands of . The central notion studied in this area is the following.
-rigidity
Let be an object in and be a projective module in .
- The module is called -rigid if . In other words, there does not exist a non-zero module homomorphism from the module to its Auslander-Reiten translation .[1]
- A module -rigid module is called -tilting if .[1] In other words, the number of direct non-isomorphic indecomposable direct summands of equals that of , where we may express the algebra as a direct sum of all projective modules.[2]
- A pair of a module and a projective module is called a -rigid pair if is -rigid and .[1]
- A -rigid pair is called -tilting if .[1]
References
- ^ a b c d e Adachi, Takahide; Iyama, Osamu; Reiten, Idun (2014). "τ-tilting theory". Compositio Mathematica. 150 (3). Cambridge University Press: 415–452. doi:10.1112/S0010437X13007422.
- ^ a b c Assem, Ibrahim; Skowronski, Andrzej; Simson, Daniel (2006). Elements of the Representation Theory of Associative Algebras: Techniques of Representation Theory. London Mathematical Society Student Texts. Vol. 1. Cambridge: Cambridge University Press. ISBN 978-0-521-58423-4.