Homological dimensions for endomorphism algebras of Gorenstein projective modules

Zhang, Aiping ; Lei, Xueping

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 675-682

Abstract (VO)

Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B = {\rm End}_A M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2 \leq n \infty $, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\rm Hom}_A(M, E).$ As an application, the global dimension of ${\rm End}_A E$ is less than or equal to $n$.

Détail

DOI : 10.21136/CMJ.2024.0199-23

Classification : 16E10, 16G10
Keywords: finitistic dimension; Gorenstein projective module; endomorphism algebra

Bibliographie

[1] Auslander, M., Bridger, M.: Stable Module Theory. Memoirs of the American Mathematical Society 94. AMS, Providence (1969). | DOI | MR | JFM

[2] Auslander, M., Reiten, I., Smalø, S. O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, Cambridge (1995). | DOI | MR | JFM

[3] Avramov, L. L., Martsinkovsky, A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc., III. Ser. 85 (2002), 393-440. | DOI | MR | JFM

[4] Christensen, L. W.: Gorenstein Dimension. Lecture Notes in Mathematics 1747. Springer, Berlin (2000). | DOI | MR | JFM

[5] Christensen, L. W., Foxby, H.-B., Frankild, A.: Restricted homological dimensions and Cohen-Macaulayness. J. Algebra 251 (2002), 479-502. | DOI | MR | JFM

[6] Christensen, L. W., Frankild, A., Holm, H.: On Gorenstein projective, injective and flat dimensions -- a functorial description with applications. J. Algebra 302 (2006), 231-279. | DOI | MR | JFM

[7] Dlab, V., Ringel, C. M.: Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring. Proc. Am. Math. Soc. 107 (1989), 1-5. | DOI | MR | JFM

[8] Enochs, E. E., Jenda, O. M. G.: On Gorenstein injective modules. Commun. Algebra 21 (1993), 3489-3501. | DOI | MR | JFM

[9] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. | DOI | MR | JFM

[10] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. de Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2000). | DOI | MR | JFM

[11] Enochs, E. E., Jenda, O. M. G., Xu, J.: Foxby duality and Gorenstein injective and projective modules. Trans. Am. Math. Soc. 348 (1996), 3223-3234. | DOI | MR | JFM

[12] Eshraghi, H.: Representation dimension via tilting theory. J. Algebra 494 (2018), 77-91. | DOI | MR | JFM

[13] Foxby, H.-B.: Gorenstein dimensions over Cohen-Macaulay rings. Commutative Algebra Runge. Vechtaer Universitätsschriften, Cloppenburg (1994), 59-63. | JFM

[14] Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. | DOI | MR | JFM

[15] Huang, C., Huang, Z.: Gorenstein syzygy modules. J. Algebra 324 (2010), 3408-3419. | DOI | MR | JFM

[16] Huang, Z., Sun, J.: Endomorphism algebras and Igusa-Todorov algebras. Acta Math. Hung. 140 (2013), 60-70. | DOI | MR | JFM

[17] Igusa, K., Todorov, G.: On the finitistic global dimension conjecture for Artin algebras. Representations of Algebras and Related Topics Fields Institute Communications 45. AMS, Providence (2005), 201-204. | DOI | MR | JFM

[18] Li, Z.-W., Zhang, P.: Gorenstein algebras of finite Cohen-Macaulay type. Adv. Math. 223 (2010), 728-734. | DOI | MR | JFM

[19] Psaroudakis, C.: Homological theory of recollements of abelian categories. J. Algebra 398 (2014), 63-110. | DOI | MR | JFM

[20] Wei, J.: Finitistic dimension and restricted flat dimension. J. Algebra 320 (2008), 116-127. | DOI | MR | JFM

[21] Xi, C.: On the finitistic dimension conjecture. III. Related to the pair $eAe \subseteq A$. J. Algebra 319 (2008), 3666-3688. | DOI | MR | JFM

[22] Xu, D.: Homological dimensions and strongly idempotent ideals. J. Algebra 44 (2014), 175-189. | DOI | MR | JFM

[23] Zhang, A.: Endomorphism algebras of Gorenstein projective modules. J. Algebra Appl. 17 (2018), Article ID 1850177, 6 pages. | DOI | MR | JFM

[24] Zhang, A., Zhang, S.: On the finitistic dimension conjecture of Artin algebras. J. Algebra 320 (2008), 253-258. | DOI | MR | JFM

[25] Zhang, P.: Triangulated Categories and Derived Categories. Science Press, Beijing (2015), Chinese.