Abstract
Let C be a semidualizing module over a commutative noetherian ring R. We exhibit an isomorphism \(\operatorname{Tor}^{{\mathcal{F}_C}\mathcal{M}}_{i}(-,-) \cong \operatorname{Tor}^{{\mathcal{P}_C}\mathcal{M}}_{i}(-,-)\) between the bifunctors defined via C-flat and C-projective resolutions. We show how the vanishing of these functors characterizes the finiteness of \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\), and use this to give a relation between the \({{\mathcal{F}_C}\text{-}\operatorname{pd}}\) of a module and of a pure submodule. On the other hand, we show that other isomorphisms force C to be trivial.
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References
Avramov, L.L., Martsinkovsky, A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc. (3) 85, 393–440 (2002)
Butler, M.C.R., Horrocks, G.: Classes of extensions and resolutions. Philos. Trans.-R. Soc. Lond., Ser. A 254, 155–222 (1961/1962)
Christensen, L.W.: Semi-dualizing complexes and their Auslander categories. Trans. Am. Math. Soc. 353(5), 1839–1883 (2001)
Christensen, L.W., Striuli, J., Veliche, O.: Growth in the minimal injective resolution of a local ring. J. Lond. Math. Soc. (2) 81(1), 24–44 (2010)
Eilenberg, S., Moore, J.C.: Foundations of relative homological algebra. Mem. Am. Math. Soc. No. 55, 39 (1965)
Enochs, E.E., Jenda, O.M.G.: Relative homological algebra. In: de Gruyter Expositions in Mathematics, vol. 30. Walter de Gruyter & Co., Berlin (2000)
Foxby, H.-B.: Gorenstein modules and related modules. Math. Scand. 31(1972), 267–284 (1973)
Frankild, A.J., Sather-Wagstaff, S., Taylor, A.: Relations between semidualizing complexes. J. Commut. Algebra 1(3), 393–436 (2009)
Frankild, A.J., Sather-Wagstaff, S., Wiegand, R.A.: Ascent of module structures, vanishing of Ext, and extended modules. Mich. Math. J. 57, 321–337 (2008) (Special volume in honor of Melvin Hochster)
Holm, H., Jørgensen, P.: Semi-dualizing modules and related Gorenstein homological dimensions. J. Pure Appl. Algebra 205(2), 423–445 (2006)
Holm, H., Jørgensen, P.: Cotorsion pairs induced by duality pairs. J. Commut. Algebra 1(4), 621–633 (2009)
Holm, H., White, D.: Foxby equivalence over associative rings. J. Math. Kyoto Univ. 47(4), 781–808 (2007)
Sather-Wagstaff, S.: Semidualizing modules (in preparation)
Sather-Wagstaff, S.: Bass numbers and semidualizing complexes. In: Commutative Algebra and its Applications. Walter de Gruyter, Berlin, pp. 349–381 (2009)
Sather-Wagstaff, S., Sharif, T., White, D.: Comparison of relative cohomology theories with respect to semidualizing modules. Math. Z. 264(3), 571–600 (2010)
Sather-Wagstaff, S., Sharif, T., White, D.: AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules. Algebr. Represent. Theor. 14(3), 403–428 (2011)
Takahashi, R., White, D.: Homological aspects of semidualizing modules. Math. Scand. 106(1), 5–22 (2010)
Vasconcelos, W.V.: Divisor theory in module categories. North-Holland Publishing Co., Amsterdam, North-Holland Mathematics Studies, No. 14, Notas de Matemática No. 53. Notes on Mathematics, No. 53 (1974)
Warfield, R.B. Jr.: Purity and algebraic compactness for modules. Pac. J. Math. 28, 699–719 (1969)
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This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. The research of Siamak Yassemi was in part supported by a grant from IPM (No. 91130214).
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Salimi, M., Sather-Wagstaff, S., Tavasoli, E. et al. Relative Tor Functors with Respect to a Semidualizing Module. Algebr Represent Theor 17, 103–120 (2014). https://doi.org/10.1007/s10468-012-9389-4
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DOI: https://doi.org/10.1007/s10468-012-9389-4