Descent for semiorthogonal decompositions

Antieau, Benjamin; Elmanto, Elden

doi:10.1016/j.aim.2021.107600

Cited by 3 publications

(31 citation statements)

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“…This result is based on the linear reductivity of finite groups, so our proof works under the assumption that U is defined in characteristic 0. Independently of the current work, in a more general setting and using derived algebro-geometric methods, an fppf descent result for semiorthogonal decompositions for arbitrary families of enhanced triangulated categories was obtained in [1,Theorem 1.4]. This implies point (1) without the assumption on characteristic.…”

mentioning

confidence: 82%

“…Independently of the current work, in a more general setting and using derived algebro-geometric methods, an fppf descent result for semiorthogonal decompositions for arbitrary families of enhanced triangulated categories was obtained in [1,Theorem 1.4]. This implies point (1) without the assumption on characteristic. We come back to the relationship between this result and ours in Remarks 4.5 and 8.9.…”

mentioning

confidence: 82%

“…By arguments similar to those for the constructions of s and t , one obtains a morphism u : a → b [1] such that u| U i = u i . Thus we have obtained a sequence…”

Section: Proof Let Us Define the Subcategorymentioning

confidence: 99%

“…In this paper we give a proof for point (1) as an application of the descent results in [23]. This result is based on the linear reductivity of finite groups, so our proof works under the assumption that U is defined in characteristic 0.…”

mentioning

confidence: 99%

“…The case where D = Perf X (with the appropriate enhancement) is Theorem A. Proving it will likely require some techniques from derived algebraic geometry, as in [1], and is related to the construction of moduli of objects in dg categories from [65].…”

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confidence: 99%

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Moduli spaces of semiorthogonal decompositions in families

Belmans¹,

Okawa²,

Ricolfi³

2020

Preprint

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For a smooth projective family of schemes we prove that a semiorthogonal decomposition of the bounded derived category of coherent sheaves of the fiber of a point uniquely deforms over an étale neighborhood of the point. We do this using a comparison theorem between semiorthogonal decompositions and decomposition triangles of the structure sheaf of the diagonal. We then apply to the latter a deformation theory for morphisms with a fixed lift of the target, which is developed in the appendix.Using this as a key ingredient we introduce a moduli space which classifies semiorthogonal decompositions of the category of perfect complexes of a smooth projective family.We show that this is an étale algebraic space over the base scheme of the family, which can be non-quasicompact and non-separated. This is done using Artin's criterion for a functor to be an étale algebraic space over the base.We generalize this to families of geometric noncommutative schemes in the sense of Orlov. We also define the open subspace classifying non-trivial semiorthogonal decompositions only, which is used in a companion paper to study indecomposability of derived categories in various examples.

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confidence: 82%

mentioning

confidence: 82%

“…By arguments similar to those for the constructions of s and t , one obtains a morphism u : a → b [1] such that u| U i = u i . Thus we have obtained a sequence…”

Section: Proof Let Us Define the Subcategorymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Moduli spaces of semiorthogonal decompositions in families

Belmans¹,

Okawa²,

Ricolfi³

2020

Preprint

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Stability conditions in families

et al. 2021

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We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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Picard sheaves, local Brauer groups, and topological modular forms

Antieau,

Meier,

Stojanoska

2024

Journal of Topology

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We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group.

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Descent for semiorthogonal decompositions

Cited by 3 publications

References 20 publications

Moduli spaces of semiorthogonal decompositions in families

Moduli spaces of semiorthogonal decompositions in families

Stability conditions in families

Picard sheaves, local Brauer groups, and topological modular forms

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