From classical tilting to two-term silting
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摘要
We survey some recent results generalizing classical tilting theory to a theory of two-term silting objects. In particular, this includes a generalized Brenner-Butler theorem, and a homological characterization of algebras obtained by two-term silting from hereditary algebras.
We survey some recent results generalizing classical tilting theory to a theory of two-term silting objects. In particular, this includes a generalized Brenner-Butler theorem, and a homological characterization of algebras obtained by two-term silting from hereditary algebras.
We survey some recent results generalizing classical tilting theory to a theory of two-term silting objects. In particular, this includes a generalized Brenner-Butler theorem, and a homological characterization of algebras obtained by two-term silting from hereditary algebras.
引文
1 Adachi T,Iyama O,Reiten I.τ-tilting theory.Compos Math,2015,150:415-452
2 Aihara T,Iyama O.Silting mutation in triangulated categories.J Lond Math Soc(2),2012,85:633-668
3 Angeleri Hügel L,Marks F,Vitoria J.Silting modules.Int Math Res Not IMRN,2016,2016:1251-1284
4 Auslander M,Platzeck M,Reiten I.Coxeter functors without diagrams.Trans Amer Math Soc,1979,250:1-46
5 Bernstein I N,Gelfand I M,Ponomarev V A.Coxeter functors and Gabriel’s theorem(in Russian).Uspehi Mat Nauk,1973,28:19-33
6 Brenner S,Butler M C R.Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors.In:Representation Theory,II.Lecture Notes in Mathematics,vol.832.Berlin-New York:Springer,1980:103-169
7 Brüstle T,Yang D.Ordered exchange graphs.In:Advances in Representation Theory of Algebras.Zürich:Eur Math Soc,2013:135-193
8 Buan A B,Zhou Y.A silting theorem.J Pure Appl Algebra,2016,220:2748-2770
9 Buan A B,Zhou Y.Silted algebras.Adv Math,2016,303:859-887
10 Buan A B,Zhou Y.Endomorphism algebras of 2-term silting complexes.Algebra Represent Theory,2018,21:181-194
11 Coelho F U,Lanzilotta M A.Algebras with small homological dimension.Manuscripta Math,1999,100:1-11
12 Gabriel P.Unzerlegbare Darstellungen,I.Manuscripta Math,1972,6:71-103
13 Geigle W,Lenzing H.A class of weighted projective curves arising in representation theory of finite-dimensional algebras.In:Greuel G-M,Trautmann G,eds.Singularities,Representation of Algebras,and Vector Bundles.Lecture Notes in Mathematics,vol.1273.Berlin:Springer,1987,265-297
14 Happel D.Triangulated categories in the representation theory of finite dimensional algebras.In:London Mathematical Society Lecture Note Series,119.Cambridge:Cambridge University Press,1988
15 Happel D.Quasitilted algebras.In:CMS Conference Proceedings,vol.23.Algebras and Modules,I.1998,55-83
16 Happel D.A characterization of hereditary categories with tilting object.Invent Math,2001,144:381-298
17 Happel D,Ringel C M.Tilted algebras.Trans Amer Math Soc,1982,274:399-443
18 Happel D,Reiten I,Smal?S O.Tilting in Abelian Categories and Quasitilted Algebra.Mem Amer Math Soc,vol.120.Providence:Amer Math Soc,1996
19 Hoshino M,Kato Y,Miyachi J.On t-structures and torsion theories induced by compact objects.J Pure Appl Algebra,2002,167:15-35
20 Iyama O,J?rgensen P,Yang D.Intermediate co-t-structures,two-term silting objects,τ-tilting modules,and torsion classes.Algebra Number Theory,2014,8:2413-2431
21 Keller B,Vossieck D.Aisles in derived categories.Bull Soc Math Belg S′er A,1988,40:239-253
22 Koenig S,Yang D.Silting objects,simple-minded collections,t-structures and co-t-structures for finite-dimensional algebras.Doc Math,2014,19:403-438
23 Reiten I,Skowro′nski A.Characterizations of algebras with small homological dimensions.Adv Math,2003,179:122-154
24 Ringel C M.The canonical algebras.In:Banach Center Publcations,vol.26.Topics in Algebra,Part 1.Warsaw:PWN,1990,407-432
25 Wei J.Semi-tilting complexes.Israel J Math,2013,194:871-893
2 Aihara T,Iyama O.Silting mutation in triangulated categories.J Lond Math Soc(2),2012,85:633-668
3 Angeleri Hügel L,Marks F,Vitoria J.Silting modules.Int Math Res Not IMRN,2016,2016:1251-1284
4 Auslander M,Platzeck M,Reiten I.Coxeter functors without diagrams.Trans Amer Math Soc,1979,250:1-46
5 Bernstein I N,Gelfand I M,Ponomarev V A.Coxeter functors and Gabriel’s theorem(in Russian).Uspehi Mat Nauk,1973,28:19-33
6 Brenner S,Butler M C R.Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors.In:Representation Theory,II.Lecture Notes in Mathematics,vol.832.Berlin-New York:Springer,1980:103-169
7 Brüstle T,Yang D.Ordered exchange graphs.In:Advances in Representation Theory of Algebras.Zürich:Eur Math Soc,2013:135-193
8 Buan A B,Zhou Y.A silting theorem.J Pure Appl Algebra,2016,220:2748-2770
9 Buan A B,Zhou Y.Silted algebras.Adv Math,2016,303:859-887
10 Buan A B,Zhou Y.Endomorphism algebras of 2-term silting complexes.Algebra Represent Theory,2018,21:181-194
11 Coelho F U,Lanzilotta M A.Algebras with small homological dimension.Manuscripta Math,1999,100:1-11
12 Gabriel P.Unzerlegbare Darstellungen,I.Manuscripta Math,1972,6:71-103
13 Geigle W,Lenzing H.A class of weighted projective curves arising in representation theory of finite-dimensional algebras.In:Greuel G-M,Trautmann G,eds.Singularities,Representation of Algebras,and Vector Bundles.Lecture Notes in Mathematics,vol.1273.Berlin:Springer,1987,265-297
14 Happel D.Triangulated categories in the representation theory of finite dimensional algebras.In:London Mathematical Society Lecture Note Series,119.Cambridge:Cambridge University Press,1988
15 Happel D.Quasitilted algebras.In:CMS Conference Proceedings,vol.23.Algebras and Modules,I.1998,55-83
16 Happel D.A characterization of hereditary categories with tilting object.Invent Math,2001,144:381-298
17 Happel D,Ringel C M.Tilted algebras.Trans Amer Math Soc,1982,274:399-443
18 Happel D,Reiten I,Smal?S O.Tilting in Abelian Categories and Quasitilted Algebra.Mem Amer Math Soc,vol.120.Providence:Amer Math Soc,1996
19 Hoshino M,Kato Y,Miyachi J.On t-structures and torsion theories induced by compact objects.J Pure Appl Algebra,2002,167:15-35
20 Iyama O,J?rgensen P,Yang D.Intermediate co-t-structures,two-term silting objects,τ-tilting modules,and torsion classes.Algebra Number Theory,2014,8:2413-2431
21 Keller B,Vossieck D.Aisles in derived categories.Bull Soc Math Belg S′er A,1988,40:239-253
22 Koenig S,Yang D.Silting objects,simple-minded collections,t-structures and co-t-structures for finite-dimensional algebras.Doc Math,2014,19:403-438
23 Reiten I,Skowro′nski A.Characterizations of algebras with small homological dimensions.Adv Math,2003,179:122-154
24 Ringel C M.The canonical algebras.In:Banach Center Publcations,vol.26.Topics in Algebra,Part 1.Warsaw:PWN,1990,407-432
25 Wei J.Semi-tilting complexes.Israel J Math,2013,194:871-893