Abstract Goerss-Hopkins theory

Pstrągowski, Piotr; VanKoughnett, Paul

doi:10.48550/arxiv.1904.08881

Cited by 4 publications

(10 citation statements)

References 17 publications

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“…Taken together, these two results show that D ω pCq can be thought of as a "8-categorical deformation", similar to synthetic spectra, stable motives after p-completion, and Brantner's and Balderrama's 8-categories associated to algebraic theories [4], [22], [78], [79].…”

Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 60%

“…to lifting objects (respectively morphisms, homotopies, etc.) The convergence of the Goerss-Hopkins tower, that is, the extent to which it is a limit diagram of 8-categories, is more subtle and closely related to the convergence of the H-based Adams spectral sequence [79]. In §6.…”

Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 98%

“…Our general approach to Franke's conjecture follows the strategy from the second's author thesis [77]: one wants to embed C into a suitable "derived 8-category" and apply Goerss-Hopkins theory [43], [79]. In the case of Adams-type homology theories on spectra, a suitable derived 8-category is given by synthethic spectra [78].…”

Section: Adams Spectral Sequencesmentioning

confidence: 99%

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Adams spectral sequences and Franke's algebraicity conjecture

Patchkoria¹,

Pstrągowski²

2021

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To any well-behaved homology theory we associate a derived 8-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration. 7.1. The algebraic model 87 7.2. Splittings of abelian categories and the Bousfield functor 89 7.3. Bousfield adjunction 91 7.4. The truncated thread monad 94 7.5. Proof of Franke's conjecture 99 8. Applications of the conjecture 101 8.1. Modules over ring spectra 102 8.2. Chromatic algebraicity 105 8.3. Diagram 8-categories 105 Appendix A. Comodules and quotients of abelian categories 110 References 112

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Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 60%

Section: Derived 8-categories and Goerss-hopkins Theorymentioning

confidence: 98%

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Adams spectral sequences and Franke's algebraicity conjecture

Patchkoria¹,

Pstrągowski²

2021

Preprint

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“…The ∞-category of synthetic spectra, which carries a canonical smash product symmetric monoidal structure, has several excellent properties, see [Pst18], [PV21], and [BHS19].…”

Section: Theorem 161 There Is a Canonical Equivalence Of Symmetric Mo...mentioning

confidence: 99%

“…On the other hand, [Pst18] has proposed another explanation of the τ -deformation picture, by introducing the algebraic notion of synthetic spectra. These have since found other applications, for instance to asymptotic chromatic algebraicity [Pst21], Goerss-Hopkins obstruction theory [PV21], and questions of manifold geometry in [BHS19], [BHS20a], and have been expanded in scope in [PP21]. Using the same filtered module spectra technique, which we use in the p-complete setting to prove Theorem 3, we obtain in Subsection 1.7 an integral comparison with synthetic spectra.…”

Section: Introductionmentioning

confidence: 99%

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

Gregoric¹

2021

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We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic τ -deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over C with ind-coherent sheaves (in a slighly non-standard sense) on a certain spectral stack τ≥0(M or FG ). The latter is the connective cover of the non-connective spectral stack M or FG , the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chromatic homotopy theory. We also provide a geometric origin on the level of stacks for the observed τ -deformation behavior on the level of sheaves, based on a notion of extended effective Cartier divisors in spectral algebraic geometry.

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Wilson Spaces, Snaith Constructions, and Elliptic Orientations

Chatham¹,

Hahn²,

Yuan³

2019

Preprint

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We construct a canonical family of even periodic E∞-ring spectra, with exactly one member of the family for every prime p and chromatic height n. At height 1 our construction is due to Snaith, who built complex K-theory from CP ∞ . At height 2 we replace CP ∞ with a p-local retract of BU 6 , producing a new theory that orients elliptic, but not generic, height 2 Morava E-theories.In general our construction exhibits a kind of redshift, whereby BP n − 1 is used to produce a height n theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the K(n)-localization of our height n ring to work of Peterson and Westerland building E hSG ± n from K(Z, n + 1).Theorem 1.5. Suppose that E k, C is an elliptic Morava E-theory associated to a supersingular elliptic curve C over a perfect field k of characteristic 2. Then there is a natural homotopy equivalence between(1) The space of E ∞ -ring maps MSU → E k, C , and(2) The subspace of E ∞ -ring maps R → E k, C that respect the Weil pairing on C.The subspace in (2) is the union of a collection of path components in the space of E ∞ -ring homomorphisms R → E k, C . We clarify what it means to respect the Weil pairing in the more precisely stated Definition 7.5 and Theorem 7.6.Remark 1.6. There is geometric interest in E ∞ -ring maps out of MSU, since such maps represent highly structured invariants of manifolds up to bordism.Remark 1.7. Morava E-theories that are not elliptic often fail to receive even homotopy commutative ring maps from R (cf. Example 6.14). In Corollary 6.27 we determine exactly which Morava E-theories receive such homotopy ring maps.1.1. The construction at a general prime and height.Convention 1.8. For the remainder of this paper, we fix a prime number p and work in the p-local category. That is to say, all spectra and simply connected spaces are implicitly p-localized.Convention 1.9. If E is a spectrum and n an integer, we sometimes use the notationConvention 1.10. For each integer h ≥ 0, we let ν(h) denote the integerDefinition 1.11. For each nonnegative integer h, let BP h denote the spectrum obtained from BP by quotienting out the Hazewinkel generators v h+1 , v h+2 , . . . ∈ π * (BP).We define W h = Ω ∞ Σ 2ν(h) BP h = BP h 2ν(h) . Steve Wilson proved [Wil75] that any choice of generators v h+1 , v h+2 , . . . yields the same space W h -we choose the Hazewinkel generators solely for concreteness (cf. Remark 2.6).Remark 1.12. Note that W 0 ≃ CP ∞ at all primes p, while W 1 ≃ BU 6 only for p = 2. At odd primes W 1 is a retract of BU 6 .

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Abstract Goerss-Hopkins theory

Cited by 4 publications

References 17 publications

Adams spectral sequences and Franke's algebraicity conjecture

Adams spectral sequences and Franke's algebraicity conjecture

Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

Wilson Spaces, Snaith Constructions, and Elliptic Orientations

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