Minimal complexes of cotorsion flat modules
DOI:
https://doi.org/10.7146/math.scand.a-110787Abstract
Let $R$ be a commutative noetherian ring. We give criteria for a complex of cotorsion flat $R$-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the structure of cotorsion flat $R$-modules. More generally, we show that any complex built from covers in every degree (or envelopes in every degree) is minimal, as well as give a partial converse to this in the context of cotorsion pairs. As an application, we show that every $R$-module is isomorphic in the derived category over $R$ to a minimal semi-flat complex of cotorsion flat $R$-modules.
References
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Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. https://doi.org/10.1112/S0024611502013527
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Iyengar, S. B., Leuschke, G. J., Leykin, A., Miller, C., Miller, E., Singh, A. K., and Walther, U., Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/gsm/087
Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. https://doi.org/10.1112/S0010437X05001375
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Matsumura, H., Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989.
Porta, M., Shaul, L., and Yekutieli, A., On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31–67. https://doi.org/10.1007/s10468-012-9385-8
Sharp, R. Y., The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356. https://doi.org/10.1007/BF01110229
Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.
Thompson, P., Cosupport computations for finitely generated modules over commutative noetherian rings, J. Algebra 511 (2018), 249–269. https://doi.org/10.1016/j.jalgebra.2018.06.014
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781139644136
Xu, J., Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), no. 2, 451–477. https://doi.org/10.1006/jabr.1995.1196
Xu, J., Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/BFb0094173
Avramov, L. L. and Foxby, H.-B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129–155. https://doi.org/10.1016/0022-4049(91)90144-Q
Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. https://doi.org/10.1112/S0024611502013527
Benson, D. J., Iyengar, S. B., and Krause, H., Colocalizing subcategories and cosupport, J. Reine Angew. Math. 673 (2012), 161–207.
Bican, L., El Bashir, R., and Enochs, E., All modules have flat covers, Bull. London Math. Soc. 33 (2001), no. 4, 385–390. https://doi.org/10.1017/S0024609301008104
Dailey, D. J., Rigidity of the Frobenius, Matlis reflexivity, and minimal flat resolutions, Ph.D. thesis, University of Nebraska-Lincoln, 2016.
Enochs, E. E., Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (1984), no. 2, 179–184. https://doi.org/10.2307/2045180
Enochs, E. E., Minimal pure injective resolutions of flat modules, J. Algebra 105 (1987), no. 2, 351–364. https://doi.org/10.1016/0021-8693(87)90200-6
Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, De Gruyter Expositions in Mathematics, vol. 30, Walter de Gruyter & Co., Berlin, 2000. https://doi.org/10.1515/9783110803662
Enochs, E. E. and Xu, J., On invariants dual to the Bass numbers, Proc. Amer. Math. Soc. 125 (1997), no. 4, 951–960. https://doi.org/10.1090/S0002-9939-97-03662-9
Fuchs, L., Algebraically compact modules over Noetherian rings, Indian J. Math. 9 (1967), 357–374 (1968).
Gruson, L. and Jensen, C. U., Dimensions cohomologiques reliées aux foncteurs $\varprojlim ^(i)$, in “Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980)'', Lecture Notes in Math., vol. 867, Springer, Berlin-New York, 1981, pp. 234--294.
Iyengar, S. B., Leuschke, G. J., Leykin, A., Miller, C., Miller, E., Singh, A. K., and Walther, U., Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/gsm/087
Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. https://doi.org/10.1112/S0010437X05001375
Matlis, E., Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.
Matsumura, H., Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989.
Porta, M., Shaul, L., and Yekutieli, A., On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31–67. https://doi.org/10.1007/s10468-012-9385-8
Sharp, R. Y., The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356. https://doi.org/10.1007/BF01110229
Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math. 65 (1988), no. 2, 121–154.
Thompson, P., Cosupport computations for finitely generated modules over commutative noetherian rings, J. Algebra 511 (2018), 249–269. https://doi.org/10.1016/j.jalgebra.2018.06.014
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781139644136
Xu, J., Minimal injective and flat resolutions of modules over Gorenstein rings, J. Algebra 175 (1995), no. 2, 451–477. https://doi.org/10.1006/jabr.1995.1196
Xu, J., Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Springer-Verlag, Berlin, 1996. https://doi.org/10.1007/BFb0094173
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Published
2019-01-13
How to Cite
Thompson, P. (2019). Minimal complexes of cotorsion flat modules. MATHEMATICA SCANDINAVICA, 124(1), 15–33. https://doi.org/10.7146/math.scand.a-110787
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