A somewhat gentle introduction to differential graded commutative algebra. (English) Zbl 1315.13004
Cooper, Susan M. (ed.) et al., Connections between algebra, combinatorics, and geometry. Selected papers based on the presentations at the workshop, Regina, Canada, May 29 – June 1, 2012, the special session on interactions between algebraic geometry and commutative algebra, Regina, Canada, June 2–3, 2012 and the conference on further connections between algebra and geometry, Fargo, ND, USA, February 2–3, 2013. New York, NY: Springer (ISBN 978-1-4939-0625-3/hbk; 978-1-4939-0626-0/ebook). Springer Proceedings in Mathematics & Statistics 76, 3-99 (2014).
Summary: Differential graded (DG) commutative algebra provides powerful techniques for proving theorems about modules over commutative rings. These notes are a somewhat colloquial introduction to these techniques. In order to provide some motivation for commutative algebraists who are wondering about the benefits of learning and using these techniques, we present them in the context of a recent result of Nasseh and Sather-Wagstaff. These notes were used for the course “differential graded commutative algebra” that was part of the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29–June 1, 2012.
For the entire collection see [Zbl 1291.13001].
For the entire collection see [Zbl 1291.13001].
MSC:
| 13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |
| 13N15 | Derivations and commutative rings |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
References:
| [1] | Apassov, D.: Homological dimensions over differential graded rings. In: Complexes and Differential Graded Modules. Ph.D. thesis, Lund University, pp. 25-39 (1999) |
| [2] | Artin, M., Bertin, J.E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M., Serre, J.-P.: Schémas en groupes. Fasc. 1: Exposés 1 à 4. In: Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques, vol. 1963. Institut des Hautes Études Scientifiques, Paris (1963/1964). MR 0207702 (34 #7517) · Zbl 0212.52801 |
| [3] | Auslander, M., Ding, S., Solberg, Ø.: Liftings and weak liftings of modules. J. Algebra 156, 273-397 (1993). MR 94d:16007 · Zbl 0778.13007 |
| [4] | Avramov, L.L.: Infinite free resolutions. In: Six Lectures on Commutative Algebra (Bellaterra, 1996). Progress in Mathematics, vol. 166, pp. 1-118. Birkhäuser, Basel (1998). MR 99m:13022 · Zbl 0934.13008 |
| [5] | Avramov, L.L., Foxby, H.-B.: Ring homomorphisms and finite Gorenstein dimension. Proc. Lond. Math. Soc. 75(2), 241-270 (1997). MR 98d:13014 · Zbl 0901.13011 |
| [6] | Avramov, L.L., Halperin, S.: Through the looking glass: a dictionary between rational homotopy theory and local algebra. In: Algebra, Algebraic Topology and Their Interactions (Stockholm, 1983). Lecture Notes in Mathematics, vol. 1183, pp. 1-27. Springer, Berlin (1986). MR 846435 (87k:55015) · Zbl 0588.13010 |
| [7] | Avramov, L.L., Foxby, H.-B., Lescot, J.: Bass series of local ring homomorphisms of finite flat dimension. Trans. Am. Math. Soc. 335(2), 497-523 (1993). MR 93d:13026 · Zbl 0770.13007 |
| [8] | Avramov, L.L., Foxby, H.-B., Herzog, B.: Structure of local homomorphisms. J. Algebra 164, 124-145 (1994). MR 95f:13029 · Zbl 0798.13002 |
| [9] | Avramov, L.L., Gasharov, V.N., Peeva, I.V.: Complete intersection dimension. Inst. Hautes Études Sci. Publ. Math. 86, 67-114 (1997). MR 1608565 (99c:13033) · Zbl 0918.13008 |
| [10] | Avramov, L.L., Foxby, H.-B., Halperin, S.: Differential graded homological algebra (in preparation) |
| [11] | Bruns, W., Herzog, J.: Cohen-Macaulay Rings, revised ed. Studies in Advanced Mathematics, vol. 39. University Press, Cambridge (1998). MR 1251956 (95h:13020) · Zbl 0909.13005 |
| [12] | Buchsbaum, D.A., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Am. J. Math. 99(3), 447-485 (1977). MR 0453723 (56 #11983) · Zbl 0373.13006 |
| [13] | Burch, L.: On ideals of finite homological dimension in local rings. Proc. Camb. Philos. Soc. 64, 941-948 (1968). MR 0229634 (37 #5208) · Zbl 0172.32302 |
| [14] | Christensen, L.W.: Semi-dualizing complexes and their Auslander categories. Trans. Am. Math. Soc. 353(5), 1839-1883 (2001). MR 2002a:13017 · Zbl 0969.13006 |
| [15] | Christensen, L.W., Sather-Wagstaff, S.: A Cohen-Macaulay algebra has only finitely many semidualizing modules. Math. Proc. Camb. Philos. Soc. 145(3), 601-603 (2008). MR 2464778 · Zbl 1153.13020 |
| [16] | Christensen, L.W., Sather-Wagstaff, S.: Descent via Koszul extensions. J. Algebra 322(9), 3026-3046 (2009). MR 2567408 · Zbl 1186.13024 |
| [17] | Cooper, S. M.; Sather-Wagstaff, S., Multiplicities of semidualizing modules, Commun. Algebra, 41, 12, 4549-4558 (2013) · Zbl 1282.13021 · doi:10.1080/00927872.2012.705933 |
| [18] | Demazure, M., Gabriel, P.: Introduction to Algebraic Geometry and Algebraic Groups. North-Holland Mathematics Studies, vol. 39. North-Holland, Amsterdam (1980). (Translated from the French by J. Bell.) MR 563524 (82e:14001) · Zbl 0431.14015 |
| [19] | Foxby, H.-B.: Gorenstein modules and related modules. Math. Scand. 31, 267-284 (1972). MR 48 #6094 · Zbl 0272.13009 |
| [20] | Foxby, H.-B., Thorup, A.: Minimal injective resolutions under flat base change. Proc. Am. Math. Soc. 67(1), 27-31 (1977). MR 56 #11984 · Zbl 0381.13006 |
| [21] | Frankild, A., Sather-Wagstaff, S.: Reflexivity and ring homomorphisms of finite flat dimension. Commun. Algebra 35(2), 461-500 (2007). MR 2294611 · Zbl 1118.13015 |
| [22] | Gabriel, P.: Finite representation type is open. In: Proceedings of the International Conference on Representations of Algebras, Carleton University, Ottawa, ON, 1974. Paper No. 10, Carleton Mathematical Lecture Notes, No. 9, p. 23, 1974. MR 0376769 (51 #12944) |
| [23] | Golod, E.S.: G-dimension and generalized perfect ideals. Trudy Mat. Inst. Steklov. 165, 62-66 (1984). (Algebr. Geom. Appl.) MR 85m:13011 · Zbl 0577.13008 |
| [24] | Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Inst. Hautes Études Sci. Publ. Math. 20, 259 (1964). MR 0173675 (30 #3885) · Zbl 0136.15901 |
| [25] | Happel, D.: Selforthogonal modules. In: Abelian Groups and Modules (Padova, 1994). Mathematics and Its Applications, vol. 343, pp. 257-276. Kluwer Academic, Dordrecht (1995). MR 1378204 (97d:16016) · Zbl 0843.16010 |
| [26] | Herzog, J., Komplexe, auflösungen, und dualität in der lokalen algebra (1973), Regensburg: Habilitationsschrift, Regensburg |
| [27] | Hilbert, D.: über die Theorie der algebraischen Formen. Math. Ann. 36, 473-534 (1890). MR 1510634 · JFM 22.0133.01 |
| [28] | Kustin, A.R.: Gorenstein algebras of codimension four and characteristic two. Commun. Algebra 15(11), 2417-2429 (1987). MR 912779 (88j:13020) · Zbl 0638.13018 |
| [29] | Kustin, A.R., Miller, M.: Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four. Math. Z. 173(2), 171-184 (1980). MR 583384 (81j:13013) · Zbl 0419.13013 |
| [30] | Mantese, F., Reiten, I.: Wakamatsu tilting modules. J. Algebra 278(2), 532-552 (2004). MR 2071651 (2005d:16023) · Zbl 1075.16006 |
| [31] | Matsumura, H.: Commutative Ring Theory. Studies in Advanced Mathematics, 2nd edn., vol. 8. Cambridge University Press, Cambridge (1989). MR 90i:13001 · Zbl 0666.13002 |
| [32] | Nasseh, S., Sather-Wagstaff, S.: A local ring has only finitely many semidualizing complexes up to shift-isomorphism. Preprint (2012). arXiv:1201.0037 |
| [33] | Nasseh, S., Sather-Wagstaff, S.: Liftings and quasi-liftings of DG modules. J. Algebra 373, 162-182 (2013). MR 2995021 · Zbl 1281.13007 |
| [34] | Reiten, I.: The converse to a theorem of Sharp on Gorenstein modules. Proc. Am. Math. Soc. 32, 417-420 (1972). MR 0296067 (45 #5128) · Zbl 0235.13016 |
| [35] | Sather-Wagstaff, S.: Semidualizing modules and the divisor class group. Ill. J. Math. 51(1), 255-285 (2007). MR 2346197 · Zbl 1127.13007 |
| [36] | Sather-Wagstaff, S.: Complete intersection dimensions and Foxby classes. J. Pure Appl. Algebra 212(12), 2594-2611 (2008). MR 2452313 (2009h:13015) · Zbl 1156.13003 |
| [37] | Sather-Wagstaff, S.: Bass numbers and semidualizing complexes. In: Commutative Algebra and Its Applications, pp. 349-381. Walter de Gruyter, Berlin (2009). MR 2640315 · Zbl 1184.13045 |
| [38] | Sather-Wagstaff, S.: Lower bounds for the number of semidualizing complexes over a local ring. Math. Scand. 110(1), 5-17 (2012). MR 2900066 · Zbl 1238.13024 |
| [39] | Sather-Wagstaff, S.: Semidualizing modules (in preparation) · Zbl 1127.13007 |
| [40] | Sharp, R.Y.: Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings. Proc. Lond. Math. Soc. 25, 303-328 (1972). MR 0306188 (46 #5315) · Zbl 0244.13015 |
| [41] | Vasconcelos, W.V.: Divisor Theory in Module Categories. North-Holland Mathematics Studies, vol. 14. Notas de Matemática No. 53 [Notes on Mathematics, No. 53]. North-Holland, Amsterdam (1974). MR 0498530 (58 #16637) · Zbl 0296.13005 |
| [42] | Voigt, D.: Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen. Lecture Notes in Mathematics, vol. 592. Springer, Berlin (1977). Mit einer englischen Einführung. MR 0486168 (58 #5949) · Zbl 0374.14010 |
| [43] | Wakamatsu, T.: On modules with trivial self-extensions. J. Algebra 114(1), 106-114 (1988). MR 931903 (89b:16020) · Zbl 0646.16025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
