#### DMCA

## Topological equivalences for differential graded algebras (2006)

### Cached

### Download Links

Venue: | Adv. Math |

Citations: | 20 - 7 self |

### Citations

353 | Categories for the working mathematician, Graduate Texts in Mathematics vol. 5, 2nd edition - Lane - 1997 |

340 | Model Categories and Their Localizations - Hirschhorn - 2003 |

284 | Symmetric spectra
- Hovey, Shipley, et al.
(Show Context)
Citation Context ... on C0.s6 DANIEL DUGGER AND BROOKE SHIPLEY 2.5. Ring spectra. There are, of course, different settings in which one can study ring spectra. We will work with the category of symmetric spectra Sp Σ of =-=[HSS]-=- with its symmetric monoidal smash product ∧ and unit S. A ring spectrum is just an S-algebra—that is to say, it is a spectrum R together with a unit S → R and an associative and unital pairing R ∧ R ... |

284 |
Morita theory for derived categories
- Rickard
- 1989
(Show Context)
Citation Context ... is a finitely-generated S-projective P such that P issTOPOLOGICAL EQUIVALENCES FOR DIFFERENTIAL GRADED ALGEBRAS 3 a strong generator and the endomorphism ring HomS(P, P ) is isomorphic to R. Rickard =-=[Ri]-=- developed an analogous criterion for when the derived categories D(R) and D(S) are equivalent triangulated categories, and this was extended in [DS1, 4.2] to the model category level. Explicitly, the... |

241 | Model categories, Mathematical surveys and monographs, - Hovey - 1998 |

231 | modules, and algebras in stable homotopy theory - Elmendorf, Kriz, et al. - 1997 |

231 | Algebras and modules in monoidal model categories.
- Schwede, Shipley
- 2000
(Show Context)
Citation Context ... products are over k. Remark 2.2. Cofibrant replacements of k-dgas play an important role in what follows. There is a functorial cofibrant replacement arising from the cofibrantly generated structure =-=[SS1]-=-, but this gives a very large model. We sketch a construction of a smaller cofibrant replacement which is useful in calculations. Suppose given C in k−DGA with HiC = 0 for i < 0. Choose generators of ... |

228 | Categories for the Working Mathematician, Graduate Texts - MacLane - 1971 |

189 | On differential graded categories, - Keller - 2006 |

152 | The homotopy theory of dg-categories and derived Morita theory,
- Toen
- 2007
(Show Context)
Citation Context ...ince the objects of G were a generating set, it follows that every object of G is itself a generator for Mod- T . A model category structure on dg-categories is constructed in [Ta], and later used in =-=[Tö]-=-. The weak equivalences are the quasi-equivalences. As remarked in [Tö, 2.3], there is a cofibrant-replacement Q ¯ R ∼ −→ ¯ R where the map on objectsTOPOLOGICAL EQUIVALENCES FOR DIFFERENTIAL GRADED ... |

123 | Deriving DG categories, - Keller - 1994 |

120 | Spectra and symmetric spectra in general model categories
- Hovey
(Show Context)
Citation Context ...ils can largely be ignored for the rest of the paper. Let ch+ be the category of non-negatively graded chain complexes. On this category one can form symmetric spectra based on the object Z[1], as in =-=[Ho2]-=-; call the corresponding category Sp Σ (ch+). Similarly one can form symmetric spectra based on simplicial abelian groups and the object � ZS 1 ; call this category Sp Σ (sAb). There are two Quillen e... |

117 | Stable model categories are categories of modules. - Schwede, Shipley - 2003 |

91 | Combinatorial model categories have presentations,
- Dugger
- 2001
(Show Context)
Citation Context ...veloped in [D2] and [DS2]. The present section summarizes what we need. A model category is called combinatorial if it is cofibrantly-generated and the underlying category is locally presentable. See =-=[D1]-=- for more information. The categories of modules over a dga and modules over a symmetric ring spectrum are both combinatorial model categories. If M is a combinatorial, stable model category then [D2]... |

83 | On the K-theory of finite algebras over Witt vectors of perfect fields, - Hesselholt, Madsen - 1997 |

65 |
Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories
- Tabuada
(Show Context)
Citation Context ...rphic in Ho (Mod- T ). Since the objects of G were a generating set, it follows that every object of G is itself a generator for Mod- T . A model category structure on dg-categories is constructed in =-=[Ta]-=-, and later used in [Tö]. The weak equivalences are the quasi-equivalences. As remarked in [Tö, 2.3], there is a cofibrant-replacement Q ¯ R ∼ −→ ¯ R where the map on objectsTOPOLOGICAL EQUIVALENCES ... |

62 | HZ-algebra spectra are differential graded algebras,
- Shipley
- 2002
(Show Context)
Citation Context ... → HB is a weak equivalence. It is also the case that H preserves homotopy limits. Unfortunately, giving a precise construction of H(−) seems to require a morass of machinery. This is accomplished in =-=[S1]-=-. We will give a brief summary, but the reader should note that these details can largely be ignored for the rest of the paper. Let ch+ be the category of non-negatively graded chain complexes. On thi... |

48 | Equivalences of monoidal model categories, - Schwede, Shipley - 2003 |

44 | B.: K-theory and derived equivalences - DUGGER, SHIPLEY |

30 |
theory of A∞ ring spectra and applications to MU - modules
- Lazarev
(Show Context)
Citation Context ... − 1}. Finally, in order to connect all this with the classification of dgas, one can prove that there is an isomorphism Der n (C, M) ∼ = Ho (k−DGA /C)(C, C ∨ Σ n M). The proof of this isomorphism in =-=[L]-=- seems to contain gaps; we are very grateful to Mike Mandell for showing us a complete proof [M]. Example 3.15. We’ll use the above machinery to determine all dgas C over Z such that H∗(C) ∼ = ΛFp (gn... |

24 | enrichments of model categories
- Dugger
(Show Context)
Citation Context ...child cohomologies. The tilting theory results are given in Section 7. To prove these, one needs to use certain invariants of stable model categories—namely, the homotopy endomorphism ring spectra of =-=[D2]-=-. These invariants are defined very abstractly and so are difficults4 DANIEL DUGGER AND BROOKE SHIPLEY to compute, but in Section 6 we state some auxiliary results simplifying things in the case of mo... |

24 |
A note on K-theory and triangulated categories
- Schlichting
(Show Context)
Citation Context ... Z/p[x1, x −1 ; dx = 0]. The verification that this is indeed a counterexample will be taken up in the paper [DS4]. The dgas R and S arise in connection with the stable module categories discussed in =-=[Sch]-=-. 8. A model category example In this section we give an example of two additive model categories which are Quillen equivalent but not additively Quillen equivalent. Let C and D be the dgas Z[e1; de =... |

20 | Moduli problems for structured ring spectra, preprint available at www.math.northwestern.edu/pgoerss. - Goerss, Hopkins - 2004 |

19 | An algebraic model for rational torus-equivariant spectra - Greenlees, Shipley - 2004 |

19 | Autour de la cohomologie de Mac Lane des corps finis - Franjou, Lannes, et al. - 1994 |

15 | Autour de la cohomologie de MacLane des corps finis. - Franjou, Lannes, et al. - 1994 |

13 | Postnikov extensions of ring spectra
- Dugger, Shipley
(Show Context)
Citation Context ...learly a surjective map π0(A)/G → π0(B). Injectivity is a very simple exercise. � Remark 3.11. A more complete proof of the above proposition and corollary can be obtained by following the methods of =-=[DS3]-=-. That paper takes place in the setting of ring spectra, but all the arguments adapt verbatim. Alternatively, using [S1] one can consider k-dgas as Hk-algebra spectra—so from that perspective the abov... |

12 |
On differential graded categories, preprint math
- Keller
(Show Context)
Citation Context ...n fact, by combining our result with Keller’s one finds that two dgas have module categories which are additively Quillen equivalent if and only if the dgas are “dg Morita equivalent” in the sense of =-=[Ke2]-=-. This is discussed in detail in Section 7.6. 1.6. Topological equivalence over fields. We do not know a general method for deciding whether two given dgas are topologically equivalent or not. The exa... |

7 | Morita Theory In Stable Homotopy Theory, Handbook on Tilting Theory - Shipley - 2006 |

6 |
Associative MU -algebras
- Goerss
(Show Context)
Citation Context ...topy types of ring spectra are. One of them, of course, is the trivial example H(ΛFp (g2p−2)). The other is the first nontrivial Postnikov section of connective Morava K-theory, P2p−2k(1) (see [A] or =-=[G]-=- for the ring structure on k(1) when p = 2). These two ring spectra are obviously not weakly equivalent, since their underlying spectra are not even weakly equivalent (the latter is not an EilenbergMa... |

5 |
Topological Hochschild homology of Z and Z/p. Unpublished preprint
- Bökstedt
(Show Context)
Citation Context ...p → HFp ∨ Σ n+1 HFp in Ho (S − Alg /HFp ). The possibilities for the k-invariant are contained in the set TDer n+1 (HFp, HFp) ∼ = THH n+2 (HFp, HFp). These THH-groups have been calculated by Bökstedt =-=[B]-=-, but see [HM, 5.2] for a published summary (those references deal with T HH homology, but one can get the cohomology groups by dualization). We have THH ∗ (Fp, Fp) ∼ = Γ[α2], a divided polynomial alg... |

2 |
A∞ obstruction theory and the strict associativity of E/I, preprint
- Angeltveit
- 2005
(Show Context)
Citation Context ...wo homotopy types of ring spectra are. One of them, of course, is the trivial example H(ΛFp (g2p−2)). The other is the first nontrivial Postnikov section of connective Morava K-theory, P2p−2k(1) (see =-=[A]-=- or [G] for the ring structure on k(1) when p = 2). These two ring spectra are obviously not weakly equivalent, since their underlying spectra are not even weakly equivalent (the latter is not an Eile... |

2 |
Enrichments of additive model categories, preprint 2006
- Dugger, Shipley
(Show Context)
Citation Context ...IPLEY to compute, but in Section 6 we state some auxiliary results simplifying things in the case of model categories enriched over ChZ. The proofs of these results are rather technical and appear in =-=[DS2]-=-. 1.9. Notation and terminology. If M and N are model categories, a Quillen pair L: M ⇄ N: R will also be referred to as a Quillen map L: M → N. The terms ‘strong monoidal-’ and ‘weak monoidal Quillen... |

2 |
Stable equivalences and DGAs
- Dugger, Shipley
(Show Context)
Citation Context ...One counterexample is R = Z〈e1, x1, x −1 ; de = p, dx = 0〉/(ex + xe = x 2 ) and S = H∗(R) = Z/p[x1, x −1 ; dx = 0]. The verification that this is indeed a counterexample will be taken up in the paper =-=[DS4]-=-. The dgas R and S arise in connection with the stable module categories discussed in [Sch]. 8. A model category example In this section we give an example of two additive model categories which are Q... |

1 |
private communication
- Mandell
(Show Context)
Citation Context ... there is an isomorphism Der n (C, M) ∼ = Ho (k−DGA /C)(C, C ∨ Σ n M). The proof of this isomorphism in [L] seems to contain gaps; we are very grateful to Mike Mandell for showing us a complete proof =-=[M]-=-. Example 3.15. We’ll use the above machinery to determine all dgas C over Z such that H∗(C) ∼ = ΛFp (gn) (an exterior algebra on a class of degree n). Such a dga has Pn−1(C) � Fp and PnC � C, so we h... |