Constructing derived moduli stacks

Pridham, J. P.

doi:10.2140/gt.2013.17.1417

Cited by 9 publications

(10 citation statements)

References 31 publications

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“…This example will be adapted further in [Pri2], constructing geometric derived n-stacks from DG Lie algebras similar to those used in [CFK2] and [CFK1].…”

Section: Deriving Functorsmentioning

confidence: 99%

Representability of derived stacks

Pridham¹

2012

J K-Theor

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Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify for many applications. Provided a derived analogue of Schlessinger's condition holds, the theorem reduces to verifying conditions on the underived part and on cohomology groups. Another simplification is that functors need only be defined on nilpotent extensions of discrete rings. Finally, there is a pre-representability theorem, which can be applied to associate explicit geometric stacks to dg-manifolds and related objects. ContentsProof.[Lur] Theorem 7.5.1.Readers unfamiliar with the conditions of this theorem should not despair, since the conditions will be explained and considerably simplified over the course of this paper.Remark 1.2. Note that there are slight differences in terminology between [TV] and [Lur]. In the former, only disjoint unions of affine schemes are 0-representable, so arbitrary schemes are 2-geometric stacks, and Artin stacks are 1-geometric stacks if and only if they

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“…This example will be adapted further in [Pri2], constructing geometric derived n-stacks from DG Lie algebras similar to those used in [CFK2] and [CFK1].…”

Section: Deriving Functorsmentioning

confidence: 99%

Representability of derived stacks

Pridham¹

2012

J K-Theor

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“…In contrast to the indirect approach of satisfying a representability theorem, [CFK2] and [CFK1] construct explicit derived Hilbert and Quot schemes as dg-schemes with the necessary properties, but there no universal family is given, so the derived moduli spaces lack functorial interpretations. In [Pri2], we use the results of this paper to compare these approaches, thereby giving explicit presentations for the derived moduli spaces constructed here.…”

Section: Introductionmentioning

confidence: 99%

Derived moduli of schemes and sheaves

Pridham¹

2011

J K-Theor

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“…The problem, however, is that the 'injective resolution' RBG is 'too big' and complicated, and it is hard to understand RLoc G (X) (in particular, to compute π i [RLoc G (X)] for i > 0) even in simplest examples. Kapranov's definition of the derived moduli space of G-local systems was refined and generalized within the framework of derived algebraic geometry by Toën, Vezzosi, Pridham and others (see, e.g., [50,52,53,36,38,39,40] and [51] for a general overview) 4 ; however, explicit computations seem still to be missing.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The referee has suggested to us that the main results of this Appendix (namely, Proposition A.1 and the equivalence (A.2)) can be also deduced from the work of J. P. Pridham [40]. Specifically, assuming that the constructions of [40] extend to higher Deligne-Mumford stacks, for any (finite) reduced simplicial set X, one can consider the derived moduli stack of G-torsors on the Deligne-Mumford hypergroupoid Spec(k) × X ∈ sAff defined by (Spec(k) × X) n := Spec(k ×Xn ). In this case, the functor C…”

Section: Appendix a Relation To Derived Algebraic Geometrymentioning

confidence: 92%

Vanishing theorems for representation homology and the derived cotangent complex

Berest

Ramadoss

Yeung

2019

Algebr. Geom. Topol.

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Let G be a reductive affine algebraic group defined over a field k of characteristic zero. In this paper, we study the cotangent complex of the derived G-representation scheme DRep G (X) of a pointed connected topological space X. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G (X) to the representation homology HR * (X, G) := π * O[DRep G (X)] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in R 3 and generalized lens spaces. In particular, for any f.g. virtually free group Γ, we show that HR i (BΓ, G) = 0 for all i > 0. For a closed Riemann surface Σg of genus g ≥ 1, we have HR i (Σg, G) = 0 for all i > dim G. The sharp vanishing bounds for Σg depend actually on the genus: we conjecture that if g = 1, then HR i (Σg , G) = 0 for i > rank G, and if g ≥ 2, then HR i (Σg, G) = 0 for i > dim Z(G) , where Z(G) is the center of G. We prove these bounds locally on the smooth locus of the representation scheme Rep G [π 1 (Σg )] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined K-theoretic virtual fundamental class for DRep G (X) in the sense of Ciocan-Fontanine and Kapranov [12]. We give a new "Tor formula" for this class in terms of functor homology.

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Constructing derived moduli stacks

Cited by 9 publications

References 31 publications

Representability of derived stacks

Representability of derived stacks

Derived moduli of schemes and sheaves

Vanishing theorems for representation homology and the derived cotangent complex

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