Totally reflexive modules with respect to a semidualizing bimodule. (English) Zbl 1281.16015

In the paper totally reflexive modules with respect to a semidualizing bimodule are investigated. Let \(R,S\) be associative rings and \(_SC_R\) be a semidualizing bimodule, i.e., both \(_SC\) and \(C_R\) admit degreewise finitely generated projective resolutions, natural homothety maps \(_SS_S\to\operatorname{Hom}_R(C,C)\) and \(_RR_R\to\operatorname{Hom}(C,C)\) are isomorphisms and \(\text{Ext}_R^i(C,C)=0=\text{Ext}_S^i(C,C)\) for all \(i\geq 1\). A finitely generated right \(R\)-module \(M_R\) is called totally \(C_R\)-reflexive if \(M_R\) admits a degreewise finitely generated \(R\)-projective resolution, the biduality map \(\delta_M^C\colon M\to\operatorname{Hom}_S(\operatorname{Hom}_R(M,C),C)\) is an \(R\)-module isomorphism, \(\operatorname{Hom}_R(M,C)\) admits a degreewise finitely generated \(S\)-projective resolution and \(\text{Ext}_R^i(M,C)=0=\text{Ext}_S^i(\operatorname{Hom}_R(M,C),C)\) for all \(i\geq 1\). Among other results, the class of totally \(C_R\)-reflexive modules is characterized (Theorem 2.3) and it is shown that every totally \(C_R\)-reflexive module with finite projective dimension is already projective (Theorem 2.11).


16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
16D20 Bimodules in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
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[1] M. Auslander, M. Mangeney, Ch. Peskine, L. Szpiro: Anneaux de Gorenstein, et Torsion en Algèbre Commutative. Ecole Normale Supérieure de Jeunes Filles, Paris, 1967. (In French.)
[2] M. Auslander, M. Bridger: Stable Module Theory. Mem. Am. Math. Soc. 94, 1969. · Zbl 0204.36402
[3] T. Araya, R. Takahashi, Y. Yoshino: Homological invariants associated to semi-dualizing bimodules. J. Math. Kyoto Univ. 45 (2005), 287–306. · Zbl 1096.16001
[4] L. W. Christensen: Gorenstein Dimensions. Lecture Notes in Mathematics 1747, Springer, Berlin, 2000.
[5] L. W. Christensen, A. Frankild, H. Holm: On Gorenstein projective, injective and flat dimensions-a functorial description with applications. J. Algebra 302 (2006), 231–279. · Zbl 1104.13008 · doi:10.1016/j.jalgebra.2005.12.007
[6] E. E. Enochs, O.M.G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2000. · Zbl 0952.13001
[7] H.-B. Foxby: Gorenstein modules and related modules. Math. Scand. 31 (1972), 267–284.
[8] E. S. Golod: G-dimension and generalized perfect ideals. Tr. Mat. Inst. Steklova 165 (1984), 62–66. · Zbl 0577.13008
[9] H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167–193. · Zbl 1050.16003 · doi:10.1016/j.jpaa.2003.11.007
[10] H. Holm, P. Jørgensen: Semi-dualizing modules and related Gorenstein homological dimensions. J. Pure Appl. Algebra 205 (2006), 423–445. · Zbl 1094.13021 · doi:10.1016/j.jpaa.2005.07.010
[11] H. Holm, P. Jørgensen: Cotorsion pairs induced by duality pairs. J. Commut. Algebra 1 (2009), 621–633. · Zbl 1184.13042 · doi:10.1216/JCA-2009-1-4-621
[12] H. Holm, D. White: Foxby equivalence over associative rings. J. Math. Kyoto Univ. 47 (2007), 781–808. · Zbl 1154.16007
[13] F. Mantese, I. Reiten: Wakamatsu tilting modules. J. Algebra 278 (2004), 532–552. · Zbl 1075.16006 · doi:10.1016/j.jalgebra.2004.03.023
[14] S. Sather-Wagstaff: Semidualizing modules and the divisor class group. Ill. J. Math. 51 (2007), 255–285. · Zbl 1127.13007
[15] S. Sather-Wagstaff, T. Sharif, D. White: AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules. Algebr. Represent. Theory 14 (2011), 403–428. · Zbl 1317.13029 · doi:10.1007/s10468-009-9195-9
[16] S. Sather-Wagstaff, T. Sharif, D. White: Tate cohomology with respect to semidualizing modules. J. Algebra 324 (2010), 2336–2368. · Zbl 1207.13009 · doi:10.1016/j.jalgebra.2010.07.007
[17] S. Sather-Wagstaff, T. Sharif, D. White: Comparison of relative cohomology theories with respect to semidualizing modules. Math. Z. 264 (2010), 571–600. · Zbl 1190.13007 · doi:10.1007/s00209-009-0480-4
[18] R. Takahashi, D. White: Homological aspects of semidualizing modules. Math. Scand. 106 (2010), 5–22. · Zbl 1193.13012
[19] W. V. Vasconcelos: Divisor Theory in Module Categories. North-Holland Mathematics Studies 14. Notas de Matematica 5. North-Holland Publishing Comp., Amsterdam, 1974.
[20] T. Wakamatsu: Tilting modules and Auslander’s Gorenstein property. J. Algebra 275 (2004), 3–39. · Zbl 1076.16006 · doi:10.1016/j.jalgebra.2003.12.008
[21] D. White: Gorenstein projective dimension with respect to a semidualizing module. J. Commut. Algebra 2 (2010), 111–137. · Zbl 1237.13029 · doi:10.1216/JCA-2010-2-1-111
[22] S. Yassemi: G-dimension. Math. Scand. 77 (1995), 161–174. · Zbl 0864.13010
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