Topological equivalences for differential graded algebras. (English) Zbl 1118.55008
This intriguing paper studies the functor \(H(-)\) which assigns to each differential graded algebra (dga) \(C\) the associated Eilenberg-MacLane ring spectrum \(HC\). This functor sends quasi-isomorphic dgas to weakly equivalent ring spectra; however, it is possible for \(HC\) and \(HD\) to be weakly equivalent even though \(C\) and \(D\) are not quasi-isomorphic. The authors call such dgas \(C\) and \(D\) topologically equivalent. One explanation for this phenomenon (explored in sections 3 and 4) is that the \(k\)-invariants for the Postnikov tower of \(C\) are in Hochschild cohomology, while the \(k\)-invariants for the Postnikov tower of \(HC\) are in topological Hochschild cohomology. There is a map from Hochschild cohomology to topological Hochschild cohomology (Section 4.4), but it need not be an injection; thus a non-trivial \(k\)-invariant in dgas can become trivial in ring spectra.
The authors go on to discuss the question of when the module categories for two dgas are Quillen equivalent (which implies they have isomorphic derived categories). They give a complete answer via a topological analog of tilting theory; it involves the notion of topological equivalence. As a sidebar, the authors take real care in constructing the functor \(H(-)\). While it is intuitively obvious that it should exist, the state of the theory of ring spectra is such that it requires a certain amount of care to define.
The authors go on to discuss the question of when the module categories for two dgas are Quillen equivalent (which implies they have isomorphic derived categories). They give a complete answer via a topological analog of tilting theory; it involves the notion of topological equivalence. As a sidebar, the authors take real care in constructing the functor \(H(-)\). While it is intuitively obvious that it should exist, the state of the theory of ring spectra is such that it requires a certain amount of care to define.
Reviewer: Paul Goerss (Evanston)
MSC:
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 55S45 | Postnikov systems, \(k\)-invariants |
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
References:
| [1] | V. Angeltveit, \(A_\inftyE / I\); V. Angeltveit, \(A_\inftyE / I\) |
| [2] | M. Bökstedt, The topological Hochschild homology of \(\mathbb{Z}\mathbb{Z} / p\); M. Bökstedt, The topological Hochschild homology of \(\mathbb{Z}\mathbb{Z} / p\) |
| [3] | Dugger, D., Combinatorial model categories have presentations, Adv. Math., 164, 1, 177-201 (2001) · Zbl 1001.18001 |
| [4] | Dugger, D., Spectral enrichments of model categories, Homology Homotopy Appl., 8, 1, 1-30 (2006) · Zbl 1084.55011 |
| [5] | Dugger, D.; Shipley, B., \(K\)-theory and derived equivalences, Duke Math. J., 124, 3, 587-617 (2004) · Zbl 1056.19002 |
| [6] | Dugger, D.; Shipley, B., Enriched model categories and additive endomorphism spectra, version 1, 2006 |
| [7] | D. Dugger, B. Shipley, Postnikov extensions of ring spectra, Algebr. Geom. Topol., 2006, in press; D. Dugger, B. Shipley, Postnikov extensions of ring spectra, Algebr. Geom. Topol., 2006, in press · Zbl 1128.55007 |
| [8] | D. Dugger, B. Shipley, A curious example of two model categories and some associated differential graded algebras, in preparation; D. Dugger, B. Shipley, A curious example of two model categories and some associated differential graded algebras, in preparation · Zbl 1159.18305 |
| [9] | Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, Modules, and Algebras in Stable Homotopy Theory. With an Appendix by M. Cole, Math. Surveys Monogr., vol. 47 (1997), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, xii+249 pp · Zbl 0894.55001 |
| [10] | Franjou, V.; Lannes, J.; Schwartz, L., Autour de la cohomologie de Mac Lane des corps finis, Invent. Math., 115, 3, 513-538 (1994) · Zbl 0798.18009 |
| [11] | P.G. Goerss, Associative MU-algebras, preprint; P.G. Goerss, Associative MU-algebras, preprint |
| [12] | Greenlees, J. P.C.; Shipley, B., An algebraic model for rational torus-equivariant spectra (2004), preprint · Zbl 1294.55002 |
| [13] | Hesselholt, L.; Madsen, I., On the \(K\)-theory of finite algebras over Witt vectors of perfect fields, Topology, 36, 1, 29-101 (1997) · Zbl 0866.55002 |
| [14] | Hovey, M., Model Categories, Math. Surveys Monogr., vol. 63 (1999), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, xii+209 pp · Zbl 0909.55001 |
| [15] | Hovey, M., Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, 165, 1, 63-127 (2001) · Zbl 1008.55006 |
| [16] | Hovey, M.; Shipley, B.; Smith, J., Symmetric spectra, J. Amer. Math. Soc., 13, 149-208 (2000) · Zbl 0931.55006 |
| [17] | Keller, B., Deriving DG categories, Ann. Sci. École Norm. Sup. (4), 27, 1, 63-102 (1994) · Zbl 0799.18007 |
| [18] | Keller, B., On differential graded categories (2006), preprint · Zbl 1140.18008 |
| [19] | Lazarev, A., Homotopy theory of \(A_\infty\) ring spectra and applications to MU-modules, \(K\)-theory, 24, 3, 243-281 (2001) · Zbl 1008.55007 |
| [20] | Mac Lane, S., Categories for the Working Mathematician, Grad. Texts in Math., vol. 5 (1971), Springer: Springer New York-Berlin · Zbl 0232.18001 |
| [21] | M. Mandell, private communication; M. Mandell, private communication |
| [22] | Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2), 39, 436-456 (1989) · Zbl 0642.16034 |
| [23] | Schlichting, M., A note on \(K\)-theory and triangulated categories, Invent. Math., 150, 1, 111-116 (2002) · Zbl 1037.18007 |
| [24] | Schwede, S.; Shipley, B., Algebras and modules in monoidal model categories, Proc. London Math. Soc., 80, 491-511 (2000) · Zbl 1026.18004 |
| [25] | Schwede, S.; Shipley, B., Stable model categories are categories of modules, Topology, 42, 103-153 (2003) · Zbl 1013.55005 |
| [26] | Schwede, S.; Shipley, B., Equivalences of monoidal model categories, Algebr. Geom. Topol., 3, 287-334 (2003) · Zbl 1028.55013 |
| [27] | Shipley, B., \(H Z\)-algebra spectra are differential graded algebras (2004), preprint |
| [28] | Shipley, B., Morita theory in stable homotopy theory, (Huegel, L.; Happel, D.; Krause, H., Handbook on Tilting Theory. Handbook on Tilting Theory, London Math. Soc. Lecture Note Ser., vol. 332 (2006)), 13 pp. · Zbl 1350.55022 |
| [29] | Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris, 340, 1, 15-19 (2005) · Zbl 1060.18010 |
| [30] | Töen, B., The homotopy theory of dg-categories and derived Morita theory (2004), preprint · Zbl 1118.18010 |
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.
