A differential graded model for derived analytic geometry

Pridham, J. P.

doi:10.48550/arxiv.1805.08538

Cited by 3 publications

(7 citation statements)

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“…rings of overconvergent functions. These give rise to the same theory as that proposed in [Pri16] using rings with entire functional calculus. By Corollary 1.32, this theory is equivalent to the dagger-analytic analogue of the non-Archimedean derived analytic spaces from [PY1,Lur2], and agrees with that theory on spaces without boundary, despite involving much less data; it satisfies similar representability theorems.…”

Section: Introductionmentioning

confidence: 59%

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Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

Pridham¹

2022

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We develop a characterisation of non-Archimedean derived analytic geometry based on dg enhancements of dagger algebras. This allows us to formulate derived analytic moduli functors for many types of pro-étale sheaves, and to construct shifted symplectic structures on them by transgression using arithmetic duality theorems. In order to handle duality functors involving Tate twists, we introduce weighted shifted symplectic structures on formal weighted moduli stacks, with the usual moduli stacks given by taking Gm-invariants.In particular, this establishes the existence of shifted symplectic and Lagrangian structures on derived moduli stacks of ℓ-adic constructible complexes on smooth varieties via Poincaré duality, and on derived moduli stacks of ℓ-adic Galois representations via Tate and Poitou-Tate duality; the latter proves a conjecture of Minhyong Kim.1 for instance BGLr is 2-shifted symplectic J.P.PRIDHAM suitable F -valued sheaves on X with an (n − 2m)-shifted symplectic structure when X is proper (Examples 3.13), or an (n + 1 − 2m)-shifted Lagrangian structure in general (Examples 3.21).The algebraic closure hypothesis is necessary for the examples in §3 because of the Tate twists featuring in duality theorems, so the purpose of Section 4 is to develop a generalisation of the theory of shifted symplectic structures to address cases where the dualising bundle is non-trivial. The considerations introduced here work equally well in algebraic and analytic settings, applying to moduli of maps from spaces which are not Calabi-Yau but have a dualising line bundle, and this weighted theory can be thought of as a special case of the theory of P-shifted symplectic structures from the seminal manuscript [BG]. The idea is to characterise the moduli space as the G m -invariant locus of a natural G m -equivariant formal thickening, with that thickening carrying a shifted symplectic structure of non-zero weight with respect to the G m -action. The resulting constructions have a similar flavour to Iwasawa theory, but with G m -actions rather than Ẑ * -actions. Remark 5.13 describes local forms for weighted shifted symplectic spaces in terms of twisted shifted cotangent bundles, and §4.4 establishes weighted representability results.Section 5 contains the main applications of the paper, constructing weighted shifted symplectic (Examples 5.12) and Lagrangian (Examples 5.17) structures on a wide range of derived analytic moduli stacks of pro-étale sheaves. Poincaré duality for smooth proper schemes and local Tate duality for local fields tends to give rise to symplectic structures for moduli of sheaves, while Poincaré duality for smooth schemes and Poitou-Tate duality for number fields give rise to Lagrangian structures. There are various ways to combine these, including Lagrangian intersections yielding the Selmer-type constructions envisaged by Kim (Example 5.17.( 7)).In Section 6, we set up the theory of shifted Poisson structures in the weighted setting. The main result is Theorem 6.8 and its generalisation in §6.6.2, ext...

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Section: Introductionmentioning

confidence: 59%

“…In §1.4, pro-étale sheaves are associated to derived dagger algebras using their natural topology, a construction which is hard to mimic in other models of analytic geometry. Section 2 then translates the theory of derived analytic symplectic geometry from [Pri16] to this dagger analytic context, where it takes a very natural form.…”

Section: Introductionmentioning

confidence: 99%

Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

Pridham¹

2022

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“…For more details and further references on this approach, see [CR12,Nui18] in the differential setting, and [Pri18b] in the analytic setting. 7 Example 1.11.…”

Section: Dg-algebrasmentioning

confidence: 99%

“…In particular, this is shown in[Pri18b] to be equivalent to the approach via pregeometries in[Lur11a], classical theorems in analysis rendering most of the pregeometric data redundant.8 These dg schemes should not be confused with the DG schemes of[Gai11], which are an alternative characterisation of the derived schemes of Definition 1.21.…”

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confidence: 97%

An introduction to derived (algebraic) geometry

Eugster¹,

Pridham²

2021

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These are notes from an introductory lecture course on derived geometry, given by the second author, mostly aimed at an audience with backgrounds in geometry and homological algebra. The focus is on derived algebraic geometry, mainly in characteristic 0, but we also see the tweaks which extend most of the content to analytic and differential settings.• Footnotes tend to contain details and comments which are tangential to the main thread of the notes; they are excessive in number.• We adhere strictly to the standard convention that the indices in chain complexes and simplicial objects, and related operations and constructions, are denoted with subscripts, while those in cochain complexes and cosimplicial objects are denoted with superscripts; to do otherwise would invite chaos.• We intermittently write chain complexes V as V • to emphasise the structure, and similarly for cochain, simplicial and cosimplicial structures. The presence or absence of bullets in a given expression should not be regarded as significant.• We denote shifts of chain and cochain complexes by [n], and always follow the convention originally developed for cochains, so we have M [n] i := M n+i for cochain complexes, but M [n] i := M n−i for chain complexes.

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“…where A 0 is the ring of holomorphic functions on a some Stein manifold X 0 and ∂ : A 0 → A 1 is an analytic derivation, with A # freely generated over A 0 by a graded projective module (equivalently, functions on a graded vector bundle over X 0 ). Again, introducing a second grading and a chain differential leads to derived structures, which are formulated in detail in [Pri9], using the theory of rings with entire functional calculus. All of the results from the differentiable setting of [Pri10] adapt to the analytic setting, and indeed to any Fermat theory.…”

Section: Tangent Modules Poisson Structures Differential Operators An...mentioning

confidence: 99%

A note on étale atlases for Artin stacks, Poisson structures and quantisation

Pridham

2019

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We explain how any Artin stack X over Q extends to a functor on nonnegatively graded commutative cochain algebras, which we think of as functions on Lie algebroids or stacky affine schemes. There is a notion of étale morphisms for these CDGAs, and Artin stacks admit étale atlases by stacky affines, giving rise to a small étale site of stacky affines over X. This site has the same quasi-coherent sheaves as X and leads to efficient formulations of shifted Poisson structures, differential operators and quantisations for Artin stacks. There are generalisations to higher and derived stacks, and analogues for differentiable and analytic stacks. This note is just a slight elaboration of constructions scattered across several of the author's papers. J.P.PRIDHAM 4.10). There are also analogues for (derived) differentiable stacks, in which setting stacky affines correspond to NQ-manifolds, and for (derived) analytic stacks in both Archimedean and non-Archimedean contexts ( §4.2).

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A differential graded model for derived analytic geometry

Cited by 3 publications

References 13 publications

Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations

An introduction to derived (algebraic) geometry

A note on étale atlases for Artin stacks, Poisson structures and quantisation

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