A curious example of triangulated-equivalent model categories which are not Quillen equivalent

Dugger, Daniel¹ ; Shipley, Brooke²

¹University of Oregon, Department of Mathematics, Eugene, OR 97403, USA
²Department of Mathematics, Statistics and Computer Science, 508 SEO (m/c 249), 851 S. Morgan St., Chicago, IL 60607-7045, USA

Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 135-166

Résumé

The paper gives a new proof that the model categories of stable modules for the rings ℤ∕p2 and ℤ∕p[ϵ]∕(ϵ2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K–theories.

Détail

DOI : 10.2140/agt.2009.9.135

Keywords: model categories, stable module category, differential graded algebras

Bibliographie

[1] J E Aisbett, E Lluis-Puebla, V Snaith, On $K_{*}(\mathbf{Z}/n)$ and $K_{*}(\mathbf{F}_q[t]/(t^2))$, Mem. Amer. Math. Soc. 57, 329 (1985)

[2] H J Baues, F Muro, Cohomologically triangulated categories. I, J. K–Theory 1 (2008) 3

[3] D Dugger, Spectral enrichments of model categories, Homology, Homotopy Appl. 8 (2006) 1

[4] D Dugger, B Shipley, $K$–theory and derived equivalences, Duke Math. J. 124 (2004) 587

[5] D Dugger, B Shipley, Postnikov extensions of ring spectra, Algebr. Geom. Topol. 6 (2006) 1785

[6] D Dugger, B Shipley, Enriched model categories and an application to additive endomorphism spectra, Theory Appl. Categ. 18 (2007) 400

[7] D Dugger, B Shipley, Topological equivalences for differential graded algebras, Adv. Math. 212 (2007) 37

[8] L Evens, E M Friedlander, On $(K_{*}(\mathbf{Z}/p^{2}\mathbf{Z})$ and related homology groups, Trans. Amer. Math. Soc. 270 (1982) 1

[9] T Geisser, On $K_3$ of Witt vectors of length two over finite fields, $K$–Theory 12 (1997) 193

[10] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)

[11] M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)

[12] B Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. $(4)$ 27 (1994) 63

[13] J Milnor, The Steenrod algebra and its dual, Ann. of Math. $(2)$ 67 (1958) 150

[14] F Muro, Triangulated categories with universal Toda bracket

[15] D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967)

[16] O Renaudin, Théories homotoiques de Quillen combinatoires et dérivateurs de Grothendieck, preprint (2006)

[17] M Schlichting, A note on $K$–theory and triangulated categories, Invent. Math. 150 (2002) 111

[18] S Schwede, B Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. $(3)$ 80 (2000) 491

[19] S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103

[20] B Shipley, $H\mathbb Z$–algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351

[21] B Toën, G Vezzosi, A remark on $K$–theory and $S$–categories, Topology 43 (2004) 765