The Finitistic Dimension of Algebras with a Directed Stratification

REFERENCES

• Ágoston , I. , Happel , D. , Lukács , E. , Unger , L. ( 2000 ). Finitistic dimension of standardly stratified algebras . Comm. Algebra 28 ( 6 ): 2745 – 2752 .
   Web of Science ®Google Scholar
• Auslander , M. , Smalø , S. O. ( 1980 ). Preprojective modules over Artin algebras . J. Algebra 66 ( 1 ): 61 – 122 .
   Web of Science ®Google Scholar
• Auslander , M. ( 1974 ). Representation theory of Artin algebras. I, II. Comm. Algebra 1:177–268; ibid. 1:269–310 .
   Google Scholar
• Auslander , M. , Reiten , I. ( 1991 ). Applications of contravariantly finite subcategories . Adv. Math. 86 ( 1 ): 111 – 152 .
   Web of Science ®Google Scholar
• Cline , E. , Parshall , B. , Scott. L. ( 1988 ). Algebraic stratification in representation categories . J. Algebra 117 ( 2 ): 504 – 521 .
   Web of Science ®Google Scholar
• Cline , E. , Parshall , B. , Scott. L. ( 1988 ). Finite-dimensional algebras and highest weight categories . J. Reine Angew. Math. 391 : 85 – 99 .
   Web of Science ®Google Scholar
• Cline , E. , Parshall , B. , Scott. L. ( 1996 ). Stratifying endomorphism algebras. Mem. Amer. Math. Soc. 124(591):viii+ 119 .
   Google Scholar
• Dietrich , K. ( 2009 ). An upper bound for the finitistic dimension of an EI category algebra. arXiv:0907.2141 [math.RT] .
   Google Scholar
• Fossum , R. M. , Griffith , P. A. , Reiten , I. ( 1975 ). Homological algebra of trivial extensions of abelian categories with applications to ring theory. In: Trivial Extensions of Abelian Categories. Volume 456 of Lecture Notes in Mathematics. Berlin: Springer-Verlag .
   Google Scholar
• Fuller , K. R. , Saorín. M. (1992). On the finitistic dimension conjecture for Artinian rings. Manuscripta Math. 74(2):117–132.
   Web of Science ®Google Scholar
• Green , E. L. , Kirkman , E. , Kuzmanovich , J. ( 1991 ). Finitistic dimensions of finite-dimensional monomial algebras . J. Algebra 136 ( 1 ): 37 – 50 .
   Web of Science ®Google Scholar
• Green , E. L. , Zimmermann-Huisgen , B. ( 1991 ). Finitistic dimension of Artinian rings with vanishing radical cube . Math. Z. 206 ( 4 ): 505 – 526 .
   Web of Science ®Google Scholar
• Happel , D. ( 1993 ). Reduction techniques for homological conjectures . Tsukuba J. Math. 17 ( 1 ): 115 – 130 .
   Google Scholar
• Happel , D. , Huisgen-Zimmermann , B. ( 1999 ). Viewing finite-dimensional representations through infinite-dimensional ones . Pacific J. Math. 187 ( 1 ): 65 – 89 .
   Web of Science ®Google Scholar
• Huard , F. , Lanzilotta , M. , Mendoza , O. ( 2009 ). Finitistic dimension through infinite projective dimension . Bull. Lond. Math. Soc. 41 ( 2 ): 367 – 376 .
   Web of Science ®Google Scholar
• Igusa , K. , Todorov , G. ( 2005 ). On the finitistic global dimension conjecture for Artin algebras. In: Representations of Algebras and Related Topics. Volume 45 of Fields Inst. Commun. Providence, RI: Amer. Math. Soc., pp. 201–204 .
   Google Scholar
• Lück , W. ( 1989 ). Transformation groups and algebraic K-theory. Volume 1408 of Lecture Notes in Mathematics. Berlin: Springer-Verlag .
   Google Scholar
• Mazorchuk , V. ( 2004 ). On finitistic dimension of stratified algebras . Algebra Discrete Math. 3 ( 3 ): 77 – 88 .
   Google Scholar
• Mitchell , B. ( 1972 ). Rings with several objects . Advances in Math. 8 : 1 – 161 .
   Web of Science ®Google Scholar
• Paquette , C. ( 2009 ). Strictly stratified algebras revisited . Comm. Algebra 37 ( 8 ): 2591 – 2600 .
   Web of Science ®Google Scholar
• Webb , P. ( 2008 ). Standard stratifications of EI categories and Alperin's weight conjecture . J. Algebra 320 ( 12 ): 4073 – 4091 .
   Web of Science ®Google Scholar
• Xi , C. ( 2004 ). Erratum to: “On the finitistic dimension conjecture. I. Related to representation-finite algebras” [J. Pure Appl. Algebra (2004). 193(1–3):287–305; mr2076389] . J. Pure Appl. Algebra 202 ( 1–3 ): 325 – 328 .
   Web of Science ®Google Scholar
• Xi , C. ( 2005 ). On the finitistic dimension conjecture. In: Advances in Ring Theory. Hackensack, NJ: World Sci. Publ., pp. 282–294 .
   Google Scholar
• Xi , C. ( 2006 ). On the finitistic dimension conjecture. II. Related to finite global dimension . Adv. Math. 201 ( 1 ): 116 – 142 .
   Web of Science ®Google Scholar
• Xi , C. ( 2008 ). On the finitistic dimension conjecture. III. Related to the pair eAe ⊆ A. J. Algebra 319(9):3666–3688 .
   Google Scholar
• Xu , F. ( 2007 ). Representations of categories and their applications . J. Algebra 317 ( 1 ): 153 – 183 .
   Web of Science ®Google Scholar
• Zimmermann-Huisgen , B. ( 1991 ). Predicting syzygies over monomial relations algebras . Manuscripta Math. 70 ( 2 ): 157 – 182 .
   Web of Science ®Google Scholar
• Zimmermann-Huisgen , B. ( 1992 ). Homological domino effects and the first finitistic dimension conjecture . Invent. Math. 108 ( 2 ): 369 – 383 .
   Web of Science ®Google Scholar
• Zimmermann Huisgen , B. ( 1995 ). The finitistic dimension conjectures—A tale of 3.5 decades. In: Abelian Groups and Modules (Padova, 1994). Volume 343 of Math. Appl. Dordrecht: Kluwer Acad. Publ., pp. 501–517 .
   Google Scholar
• Communicated by D. Zacharia.
   Google Scholar