André–Quillen spectral sequence for THH

Minasian, Vahagn

doi:10.1016/s0166-8641(02)00184-0

Cited by 17 publications

(16 citation statements)

References 3 publications

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“…It can be thought of as a structured ring spectra analog of Quillen's fundamental spectral sequence [60, 6.9] for commutative rings and the corresponding spectral sequence studied by Goerss [24, 6.2] for augmented commutative F 2 -algebras. As a special case, it recovers the spectral sequence in Minasian [58] for non-unital commutative ring spectra. Under the conditions of Theorem 1.12(b), the homotopy completion spectral sequence is a second quadrant homologically graded spectral sequence and arises from the exact couple of long exact sequences associated to the homotopy completion tower and its homotopy fibers; this is the homotopy spectral sequence of a tower of fibrations [9], reindexed as a homologically graded spectral sequence.…”

Section: Introductionmentioning

confidence: 60%

Homotopy completion and topological Quillen homology of structured ring spectra

Harper

Hess

2013

Geom. Topol.

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Abstract. Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (−1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor.By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQcompletion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes.

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Section: Introductionmentioning

confidence: 60%

Homotopy completion and topological Quillen homology of structured ring spectra

Harper

Hess

2013

Geom. Topol.

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“…Since both rows are (co)fibration sequences and the right vertical map is a weak equivalence by inductive assumption, it's enough to show that the left vertical map is also a weak equivalence. This, however, is an immediate consequence of Proposition 2.4 of [14], which states that…”

Section: Thus Bmentioning

confidence: 99%

“…Recall that I S 0 /I 2 S 0 ≃ ΣT AQ(A|R) (see e.g. [14]). Thus, to show (9), it suffices to prove that ΣT AQ(B|R) ∧ X ≃ ΣT AQ(A|R) ∧ A B ∧ X, which, in turn, is an immediate consequence of the transfer sequence of T AQ:…”

Section: Thus Bmentioning

confidence: 99%

HKR theorem for smooth S-algebras

McCarthy

Minasian

2003

Journal of Pure and Applied Algebra

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We derive anétale descent formula for topological Hochschild homology and prove a HKR theorem for smooth S-algebras.Key words: spectra with additional structure,étale descent, smooth S-algebras MCS: 55P42 IntroductionOne of the main results for computing the Hochschild homology of smooth discrete algebras is the Hochschild-Kostant-Rosenberg (HKR) theorem (e.g. see Chapter 3 of [9]), which states that for a smooth algebra k → A, the Hochschild homology coincides with differential forms:In fact this result is often used not only to compute the Hochschild homology, but also the other way around: in order to generalize some results to non-smooth (or even non-commutative) algebras one replaces the differential forms by Hochschild homology. Other applications include a comparison theorem between cyclic and de Rham homology theories.One of our objectives is to develop a topological analogue of the HKR theorem in the framework provided in [5], or more precisely, in the category of commutative of S-algebras. Recall that S-algebras are equivalent to the more traditional notion of E ∞ -ring spectra, and are a generalization to stable homotopy theory of the algebraic notion of a commutative ring. In this context, the topological André-Quillen homology of a commutative S-algebra A is the natural replacement of the module of differentials Ω 1 A|k , as it is evident from the definition of TAQ. The definitions of TAQ, as well as THH, in our context are recalled in Section 2, and we refer to [1] and Chapter IX of [5] for detailed discussion of these notions. Noting that the orbits of the n ′ th smash powers of the suspension of T AQ, (ΣT AQ(A)) ∧An /Σ n , are analogous to symmetric powers in the graded context, and therefore correspond to taking exterior powers (and thus are the analogues of the higher order modules of differentials), we state our main theorem. Theorem 1.1. (HKR) For a connective smooth S-algebra A, the natural (derivative) map T HH(A) → ΣT AQ(A) has a section in the category of A-modules which induces an equivalence of A-algebras: P A ΣT AQ(A) ≃ → T HH(A),where P is the symmetric algebra triple.

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“…We prove by induction on k. Since we intend to use the formulas (13) and (14), we have to consider both k = 2 and k = 3 for the base case.…”

Section: Functional Equations and Their Related Operads 4441mentioning

confidence: 99%

“…see [1]), the right-hand side of Equation (15) has a copy of K in degree 0, three copies of K in degree n − 1 and two copies of K in degree 2(n − 1). On the other hand by Equations (13) and (14),…”

Section: Functional Equations and Their Related Operads 4441mentioning

confidence: 99%

Functional equations and their related operads

Minasian

2005

Trans. Amer. Math. Soc.

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Abstract. Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little n-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of n-Poisson algebras in terms of the homology of configuration spaces.

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André–Quillen spectral sequence for THH

Cited by 17 publications

References 3 publications

Homotopy completion and topological Quillen homology of structured ring spectra

Homotopy completion and topological Quillen homology of structured ring spectra

HKR theorem for smooth S-algebras

Functional equations and their related operads

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