导出范畴的n-黏合及上(下)黏合
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摘要
本文将Artin代数的无界导出范畴的1-黏合和2-黏合用其子范畴的上(下)黏合来刻画.作为应用,本文阐明了导出范畴的上(下)黏合与代数的整体维数(有限维数)的有限性之间的关系,这推广了Yin和Gao (2016)的结果.设A、B和C均为域k上的代数,且DA有一个关于DB和DC的3-黏合,本文证明了A是有限维的当且仅当B和C也如此.
For Artin algebras, we characterize 1-recollements and 2-recollements of their unbounded derived categories in terms of upper(respectively lower) recollements of certain subcategories. As a result, we clarify the relationship between upper(respectively lower) recollements of derived categories and the finiteness of global(respectively finiteness) dimension of algebras, which generalizes a result of Yin and Gao(2016). Let A, B,and C be algebras over a field k, and DA admit a 3-recollement relative to DB and DC. We prove that A is finite-dimensional if and only if so are B and C.
For Artin algebras, we characterize 1-recollements and 2-recollements of their unbounded derived categories in terms of upper(respectively lower) recollements of certain subcategories. As a result, we clarify the relationship between upper(respectively lower) recollements of derived categories and the finiteness of global(respectively finiteness) dimension of algebras, which generalizes a result of Yin and Gao(2016). Let A, B,and C be algebras over a field k, and DA admit a 3-recollement relative to DB and DC. We prove that A is finite-dimensional if and only if so are B and C.
引文
1 Beilinson A A,Bernstein J,Deligne P.Faisceaux Pervers.Luminy:Soc Math France,1982
2 Qin Y Y,Han Y.Reducing homological conjectures by n-recollements.Algebr Represent Theory,2016,19:377-395
3 Angeleri Hugel L,Koenig S,Liu Q,et al.Ladders and simplicity of derived module categories.J Algebra,2017,472:15-66
4 Gao N,Psaroudakis C.Ladders of compactly generated triangulated categories and preprojective algebras.Appl Categ Structures,2018,26:657-679
5 Zhang P,Zhang Y H,Zhou G D,et al.Unbounded ladders induced by Gorenstein algebras.Colloq Math,2018,151:37-56
6 Lu M.Gorenstein defect categories of triangular matrix algebras.J Algebra,2017,480:346-367
7 Zhang P.Gorenstein-projective modules and symmetric recollements.J Algebra,2013,388:65-80
8 Gao N,Yin Y Q,Zhang P.Comparisons of left recollements.Algebr Represent Theory,2017,20:659-673
9 Yin Y Q,Gao N.Upper recollements and triangle expansions.Acta Math Sin Engl Ser,2016,32:1323-1336
10 Chen H X,Xi C C.Recollements of derived categories,III:Finitistic dimensions.J Lond Math Soc(2),2017,95:633-658
11 Konig S.Tilting complexes,perpendicular categories and recollements of derived module categories of rings.J Pure Appl Algebra,1991,73:211-232
12 Wiedemann A.On stratifications of derived module categories.Canad Math Bull,1991,34:275-280
13 Happel D.Reduction techniques for homological conjectures.Tsukuba J Math,1993,17:115-130
14 Pan S Y.Recollements and Gorenstein algebras.Int J Algebra,2013,7:829-832
15 Beilinson A A,Ginsburg V A,Schechtman V V.Koszul duality.J Geom Phys,1988,5:317-350
16 Parshall B J.Finite dimensional algebras and algebraic groups.Contemp Math,1989,82:97-114
17 Nicol′as P.On torsion torsionfree triples.PhD thesis.Murcia:Universidad de Murcia,2007
18 Bokstedt M,Neeman A.Homotopy limits in triangulated categories.Compos Math,1993,86:209-234
19 Rickard J.Morita theory for derived categories.J Lond Math Soc(2),1989,39:436-456
20 Han Y.Recollements and Hochschild theory.J Algebra,2014,397:535-547
2 Qin Y Y,Han Y.Reducing homological conjectures by n-recollements.Algebr Represent Theory,2016,19:377-395
3 Angeleri Hugel L,Koenig S,Liu Q,et al.Ladders and simplicity of derived module categories.J Algebra,2017,472:15-66
4 Gao N,Psaroudakis C.Ladders of compactly generated triangulated categories and preprojective algebras.Appl Categ Structures,2018,26:657-679
5 Zhang P,Zhang Y H,Zhou G D,et al.Unbounded ladders induced by Gorenstein algebras.Colloq Math,2018,151:37-56
6 Lu M.Gorenstein defect categories of triangular matrix algebras.J Algebra,2017,480:346-367
7 Zhang P.Gorenstein-projective modules and symmetric recollements.J Algebra,2013,388:65-80
8 Gao N,Yin Y Q,Zhang P.Comparisons of left recollements.Algebr Represent Theory,2017,20:659-673
9 Yin Y Q,Gao N.Upper recollements and triangle expansions.Acta Math Sin Engl Ser,2016,32:1323-1336
10 Chen H X,Xi C C.Recollements of derived categories,III:Finitistic dimensions.J Lond Math Soc(2),2017,95:633-658
11 Konig S.Tilting complexes,perpendicular categories and recollements of derived module categories of rings.J Pure Appl Algebra,1991,73:211-232
12 Wiedemann A.On stratifications of derived module categories.Canad Math Bull,1991,34:275-280
13 Happel D.Reduction techniques for homological conjectures.Tsukuba J Math,1993,17:115-130
14 Pan S Y.Recollements and Gorenstein algebras.Int J Algebra,2013,7:829-832
15 Beilinson A A,Ginsburg V A,Schechtman V V.Koszul duality.J Geom Phys,1988,5:317-350
16 Parshall B J.Finite dimensional algebras and algebraic groups.Contemp Math,1989,82:97-114
17 Nicol′as P.On torsion torsionfree triples.PhD thesis.Murcia:Universidad de Murcia,2007
18 Bokstedt M,Neeman A.Homotopy limits in triangulated categories.Compos Math,1993,86:209-234
19 Rickard J.Morita theory for derived categories.J Lond Math Soc(2),1989,39:436-456
20 Han Y.Recollements and Hochschild theory.J Algebra,2014,397:535-547