arXiv subjects
Publications and source records attributed to Chirantan Chowdhury.
Lara, Srinivas and Stix showed that the étale fundamental group of a proper scheme over an algebraically closed field is topologically finitely presented. Building on this, we show that the étale fundamental group, as introduced by Noohi, of a proper algebraic stack with finite inertia over an algebraically closed field is topologically finitely presented.
In this article, we study two consequences of abstract six-functor formalisms. Firstly, we show that an abstract six-functor formalism can be extended to specific Ind- and Pro- categories of geometric setups. As an application, we can define the motivic stable homotopy theory for ind-pro-algebraic stacks such as the Hecke stack. Secondly, we also show that Cohomological Purity is functorial using the multisimplicial language developed by Liu-Zheng.
We characterize smooth proper Deligne-Mumford stacks $\mathscr{X}$ that arise as compactifications of ball quotient stacks $[\mathbb{B}^d/Γ]$. Moreover, we show that every ball quotient admits a compactification whose boundary divisor $\mathscr{D}:=\mathscr{X}-[\mathbb{B}^d/Γ]$ is a disjoint union of quotient stacks $[A/G]$, where $A$ is an abelian variety and $G$ is a finite group. This generalizes a result of Deng-Cadorel. Our strategy combines Simpson's non-abelian Hodge correspondence for smooth proper DM-stacks, Mochizuki's generalization of the classical Simpson's correspondence to the log setting, and the uniformization results of Deng-Cadorel.
We construct the Chow weight structure on the derived category of geometric motives with arbitrary coefficients for X a finite type scheme over a field characteristic 0 and G an affine algebraic group. In particular we also show that the heart of this weight structure recovers the category of Chow motives on [X/G].
This article is the last of the series of articles where we reprove the foundational ideas of abstract six-functor formalisms developed by Liu-Zheng. We prove the theorem of partial adjoints, which is a simplicial technique of encoding various functors altogether by taking adjoints along specific directions. Combined with the $\infty$-categorical compactification theorem from the previous article, we can construct abstract six-functor formalisms in reasonable geometric setups of our interest. We also reprove the simplified versions of the DESCENT program due to Liu-Zheng, which allows us to extend such formalisms from smaller to larger geometric setups.
This paper is part of a series of articles in which we reproduce the statements regarding the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem, which is an $\infty$-categorical version for defining the exceptional pushforward functor in an abstract-six functor formalism. The article describes specific combinatorial simplicial sets related to compactifications and pullback squares. This theorem plays a key role in constructing the abstract six-functor formalism, which will be discussed in the forthcoming article.
This article is first in a series of papers where we reprove the statements in constructing the Enhanced Operation Map and the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem regarding constructing functors between simplicial sets using the category of simplices. We shall reprove the statement using the language of marked simplicial sets and studying injective model structure on functor categories. The theorem is a crucial tool and will be used repeatedly in reproving the $\infty$-categorical compactification and constructing the so called Enhanced Operation Map in the forthcoming articles.
In this article, we study the properties of motivic homotopy category $\mathcal{SH}_{\operatorname{ext}}(\mathcal{X})$ developed by Chowdhury and Khan-Ravi for $\mathcal{X}$ a Nis-loc Stack. In particular, we compare the above construction with Voevodsky's original construction using NisLoc topology. Using the techniques developed by Liu-Zheng and Mann's notion of $\infty$-category of correspondences and abstract six-functor formalisms, we also extend the exceptional functors and extend properties like projection formula, base change and purity to the non-representable situation.
The aim of this paper is to extend the definition of motivic homotopy theory from schemes to a large class of algebraic stacks and establish a six functor formalism. The class of algebraic stacks that we consider includes many interesting examples: quasi-separated algebraic spaces, local quotient stacks and moduli stacks of vector bundles. We use the language of $\infty$-categories developed by Lurie. Morever, we use the so-called 'enhanced operation map' due to Liu and Zheng to extend the six functor formalism from schemes to our class of algebraic stacks. We also prove that six functors satisfy properties like homotopy invariance, localization and purity.