arXiv subjects

Marcin Lara

Publications and source records attributed to Marcin Lara.

11 recordsLinked to original sources

arXiv2026

Tame fundamental groups of rigid spaces

We introduce the tame \'etale fundamental group $\pi_1^t(X/K)$ of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then $\pi_1^t(X/K)$ is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then $\pi_1^t(X/K)$ is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log \'etale fundamental group, and on the 'vertical compactification' of a map of adic spaces.

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arXiv2025

The condensed homotopy type of a scheme

We study a condensed version of the \'etale homotopy type of a scheme, which refines both the usual \'etale homotopy type of Friedlander-Artin-Mazur and the pro\'etale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line $\mathbf{A}^1_{\mathbf{C}}$ over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the pro\'etale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

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arXiv2024

Logarithmic geometry beyond fs

We develop the foundations of logarithmic structures beyond the standard finiteness conditions. The motivation is the study of semistable models over general valuation rings. The key new notion is that of a morphism of finite presentation up to saturation (sfp), which is one that is qcqs and which is locally isomorphic to the saturated base change of a finitely presented morphism between fs log schemes. As in the case of schemes, sfp maps can (locally on the base) be approximated by maps between fs log schemes of finite type over $\mathbb{Z}$. Based on sfp maps, we define smooth, \'etale, and Kummer \'etale maps. Importantly, the maps of schemes underlying such maps are no longer of finite type in general, though surprisingly they are if the base is the spectrum of a valuation ring with algebraically closed field of fractions. These foundations allow us to extend beyond the fs case the theory of the Kummer \'etale site and of the Kummer \'etale fundamental group.

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arXiv2023

Fundamental groups of proper varieties are finitely presented

It was recently proven by Esnault, Shusterman and the second named author, that the étale fundamental group of a connected smooth projective variety over an algebraically closed field $k$ is finitely presented. In this note, we extend this result to all connected proper schemes over $k$.

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arXiv2022

Fundamental Exact Sequence for the Pro-Étale Fundamental Group

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers. We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e. that the sequence $$1 \rightarrow π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map $π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

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arXiv2022

Geometric arcs and fundamental groups of rigid spaces

We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are étale local on X. If X is connected, its geometric coverings form a tame infinite Galois category, and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs," making it similar to geometric coverings of schemes studied by Bhatt and Scholze as well as semicoverings of topological spaces introduced by Brazas.

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arXiv2022

Variants of the de Jong fundamental group

For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite etale coverings: we show that this class is different from the one used by de Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.

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arXiv2022

A theorem on meromorphic descent and the specialization of the pro-étale fundamental group

Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-étale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the pro-étale fundamental group of a connected scheme $X$ of finite type over a field as coproducts and quotients of the free group and the étale fundamental groups of the normalizations of the irreducible components of $X$ and those of its singular loci.

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arXiv2021

Specialization for the pro-étale fundamental group

For a formal scheme $\mathfrak{X}$ of finite type over a complete rank one valuation ring, we construct a specialization morphism \[ π^{\rm dJ}_1(\mathfrak{X}_η) \to π^{\rm proet}_1(\mathfrak{X}_k) \] from the de Jong fundamental group of the rigid generic fiber to the Bhatt-Scholze pro-étale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak{X}$ and normalizations of the irreducible components of $\mathfrak{X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.

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arXiv2019

Homotopy Exact Sequence for the Pro-Étale Fundamental Group II

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite étale covers. We prove exactness of the general homotopy sequence for the pro-étale fundamental group, i.e. that for a geometric point $\bar{s}$ on $S$ and a flat proper morphism $X \rightarrow S$ of finite presentation whose geometric fibres are connected and reduced, the sequence $$ π_1^{\mathrm{proet}}(X_{\bar{s}}) \rightarrow π_1^{\mathrm{proet}}(X) \rightarrow π_1^{\mathrm{proet}}(S) \rightarrow 1 $$ is "nearly exact". This generalizes a theorem of Grothendieck from finite étale covers to geometric coverings. We achieve the proof by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization in this setting.

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arXiv2019

On the inverse problem for deformations of finite group representations

Let $s$ be even and $q=p^s$. We show that the ring $W(\mathbb{F}_{q})[\![X]\!]/(X^2-pX)$ is a quotient of the universal deformation ring of a representation of a finite group. This amounts to giving an example of a finite group and its $\mathbb{F}_q$-representation that lifts to $W(\mathbb{F}_q)$ in two different ways and satisfies certain subtle extra conditions. We achieve this by studying representations of $\mathrm{SL}(2,\mathbb{F}_{p^2})$.

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