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David Rydh

Publications and source records attributed to David Rydh.

At least 19 recordsLinked to original sources
arXiv2025

Deformation to the normal bundle and blow-ups via derived Weil restrictions

We develop an analogue of the deformation to the normal cone in the context of derived algebraic geometry. This provides any given morphism of derived stacks with a degeneration to the zero section of its normal bundle (i.e., its 1-shifted relative tangent bundle). The construction is realized via the derived Weil restriction along the zero section of the affine line. We prove a general algebraicity theorem for derived Weil restrictions along finite but possibly non-flat morphisms. As an application of the theory, we study derived blow-ups along arbitrary closed centres, generalizing previous works of the authors in the quasi-smooth case.

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arXiv2025

Functorial flatification of proper morphisms

For proper morphisms, we give a functorial flatification algorithm by blow-ups in the spirit of Hironaka's flatification algorithm. In characteristic zero, this gives functorial flatification by blow-ups in smooth centers. We also give a functorial \'etalification algorithm by Kummer blow-ups in characteristic zero.

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arXiv2023

Absolute noetherian approximation of algebraic stacks

We show that every quasi-compact and quasi-separated algebraic stack can be approximated by a noetherian algebraic stack. We give several applications such as eliminating noetherian hypotheses in the theory of good moduli spaces.

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arXiv2022

Stabilizer Reduction for Derived Stacks and Applications to Sheaf-Theoretic Invariants

We construct a canonical stabilizer reduction $\widetilde{X}$ for any derived $1$-algebraic stack $X$ over $\mathbb{C}$ as a sequence of derived Kirwan blow-ups, under mild natural conditions that include the existence of a good moduli space for the classical truncation $X_{\mathrm{cl}}$. Our construction has several desired features: it naturally generalizes Kirwan's classical partial desingularization algorithm to the context of derived algebraic geometry, preserves quasi-smoothness, and is a derived enhancement of the intrinsic stabilizer reduction constructed by Kiem, Li and the third author. Moreover, if $X$ is $(-1)$-shifted symplectic, we show that the semi-perfect and almost perfect obstruction theory of $\widetilde{X}_{\mathrm{cl}}$ and the associated virtual fundamental cycle and virtual structure sheaf, constructed by the same authors, are naturally induced by $\widetilde{X}$ and its derived tangent complex. As corollaries, we define virtual classes for moduli stacks of semistable sheaves on surfaces, give a fully derived perspective on generalized Donaldson-Thomas invariants of Calabi-Yau threefolds and define new generalized Vafa-Witten invariants for surfaces via Kirwan blow-ups.

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arXiv2022

Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts

We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of \'etale, smooth, or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin.

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arXiv2021

A Luna étale slice theorem for algebraic stacks

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

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arXiv2020

Canonical reduction of stabilizers for Artin stacks with good moduli spaces

We present a complete generalization of Kirwan's partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if $\mathcal{X}$ is an irreducible Artin stack with stable good moduli space $\mathcal{X} \to X$, then there is a canonical sequence of birational morphisms of Artin stacks $\mathcal{X}_n \to \mathcal{X}_{n-1} \to \ldots \to \mathcal{X}_0 = \mathcal{X}$ with the following properties: (1) the maximum dimension of a stabilizer of a point of $\mathcal{X}_{k+1}$ is strictly smaller than the maximum dimension of a stabilizer of $\mathcal{X}_k$ and the final stack $\mathcal{X}_n$ has constant stabilizer dimension; (2) the morphisms $\mathcal{X}_{k+1} \to \mathcal{X}_k$ induce projective and birational morphisms of good moduli spaces $X_{k+1} \to X_{k}$. If in addition the stack $\mathcal{X}$ is smooth, then each of the intermediate stacks $\mathcal{X}_k$ is smooth and the final stack $\mathcal{X}_n$ is a gerbe over a tame stack. In this case the algebraic space $X_n$ has tame quotient singularities and is a partial desingularization of the good moduli space $X$. When $\mathcal{X}$ is smooth our result can be combined with D. Bergh's recent destackification theorem for tame stacks to obtain a full desingularization of the algebraic space $X$.

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arXiv2019

The \'etale local structure of algebraic stacks

We prove that an algebraic stack with affine stabilizers over an arbitrary base is \'etale-locally a quotient stack around any point with a linearly reductive stabilizer. This generalizes earlier work by the authors of this article (stacks over algebraically closed fields) and by Abramovich, Olsson and Vistoli (stacks with finite inertia). In addition, we prove a number of foundational results, which are new even over a field. These include various coherent completeness and effectivity results for adic sequences of algebraic stacks. Finally, we give several applications of our results and methods, such as structure theorems for linearly reductive group schemes and generalizations to the relative setting of Sumihiro's theorem on torus actions and Luna's \'etale slice theorem.

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arXiv2019

Functorial destackification and weak factorization of orbifolds

Let X be a smooth and tame stack with finite inertia. We prove that there is a functorial sequence of blow-ups with smooth centers after which the stabilizers of X become abelian. Using this result, we can extend the destackification results of the first author to any smooth tame stack. We give applications to resolution of tame quotient singularities, prime-to-l alterations of singularities and weak factorization of Deligne-Mumford stacks. We also extend the abelianization result to infinite stabilizers in characteristic zero, generalizing earlier work of Reichstein-Youssin.

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arXiv2018

The complexity of a flat groupoid

Grothendieck proved that any finite epimorphism of noetherian schemes factors into a finite sequence of effective epimorphisms. We define the complexity of a flat groupoid $R\rightrightarrows X$ with finite stabilizer to be the length of the canonical sequence of the finite map $R\to X\times\_{X/R} X$, where $X/R$ is the Keel-Mori geometric quotient. For groupoids of complexity at most 1, we prove a theorem of descent along the quotient $X\to X/R$ and a theorem of quotient of a groupoid by a normal subgroupoid. We expect that the complexity could play an important role in the finer study of quotients by groupoids.

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arXiv2018

Virtual Cartier divisors and blow-ups

We prove a universal property for blow-ups in regularly immersed subschemes, based on a notion we call "virtual effective Cartier divisor". We also construct blow-ups of quasi-smooth closed immersions in derived algebraic geometry.

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arXiv2017

Addendum: Étale dévissage, descent and pushouts of stacks

Using Nisnevich coverings and a Hilbert stack of stacky points, we prove étale dévissage results for non-representable étale and quasi-finite flat coverings. We give applications to absolute noetherian approximation of algebraic stacks and compact generation of derived categories.

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arXiv2017

Perfect complexes on algebraic stacks

We develop a theory of unbounded derived categories of quasi-coherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau--Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived Deligne--Mumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

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arXiv2017

The telescope conjecture for algebraic stacks

Using Balmer--Favi's generalized idempotents, we establish the telescope conjecture for many algebraic stacks. Along the way, we classify the thick tensor ideals of perfect complexes of stacks.

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arXiv2016

Mayer-Vietoris squares in algebraic geometry

We consider various notions of Mayer--Vietoris squares in algebraic geometry. We use these to generalize a number of gluing and pushout results of Moret-Bailly, Ferrand--Raynaud, Joyet and Bhatt. An important intermediate step is Gabber's rigidity theorem for henselian pairs, which our methods give a new proof of.

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arXiv2015

One positive and two negative results for derived categories of algebraic stacks

Let $X$ be a quasi-compact and quasi-separated scheme. There are two fundamental and pervasive facts about the unbounded derived category of $X$: (1) $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated by perfect complexes and (2) if $X$ is noetherian or has affine diagonal, then the functor $Ψ_X \colon \mathsf{D}(\mathsf{QCoh}(X)) \to \mathsf{D}_{\mathrm{qc}}(X)$ is an equivalence. Our main results are that for algebraic stacks in positive characteristic, the assertions (1) and (2) are typically false.

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