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Amit Hogadi

Publications and source records attributed to Amit Hogadi.

At least 19 recordsLinked to original sources
arXiv2026

Zero cycles on Severi--Brauer flag varieties

Let \(A\) be a central simple algebra over a field \(F\) with index \(n\) and let \(\mathrm{SB}_r(A)\) denote the \(r\)-th generalized Severi--Brauer variety associated with \(A\). We prove that the Chow group of zero cycles of degree zero \(\mathrm{A_0}(\mathrm{SB}_r(A))\) is \((d, n/d)\)-torsion where \(d = (r,n)\). Our approach reduces the general case to division algebras of prime power index and yields several new instances in which \(\mathrm{A_0}\) is trivial, together with sharper torsion bounds in general.\\ We also show that if \(F\) is a local or global field, then \(\mathrm{A_0}(\mathrm{SB}_r(A))=0\). Since Severi--Brauer flag varieties are stably birational to generalized Severi--Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for \(\mathrm{A_0}(X)\), where \(X\) is stably birational to \(\mathrm{SB}_r(A)\).

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arXiv2026

Corrigendum: Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649.)

The proof of Lemma 5.1 in the paper Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649) is incomplete as it relies on some results of Choudhury-Hagadi, the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of this lemma.

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arXiv2024

A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks

We develop a motivic cohomology theory, representable in the Voevodsky's triangulated category of motives, for smooth separated Deligne-Mumford stacks and show that the resulting higher Chow groups are canonically isomorphic to the higher $K$-theory of such stacks. This generalises the Grothendieck-Riemann-Roch theorem to the category of smooth Deligne-Mumford stacks.

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arXiv2024

Non-finite type \'etale sites over fields

We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the \'etale site of a field. For a given field $k$, we conjecture that the \'etale site of $Sm/k$ is of finite type if and only if the field $k$ admits a finite extension of finite cohomological dimension. We prove this conjecture in some cases, e.g. in the case when $k$ is countable, or in the case when the $p$-cohomological dimension $cd_p(k)$ is infinite for infinitely many primes $p$.

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arXiv2023

Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups

We show that the sheaf of $\mathbb A^1$-connected components of a reductive algebraic group over a perfect field is strongly $\mathbb A^1$-invariant. As a consequence, torsors under such groups give rise to $\mathbb A^1$-fiber sequences. We also show that sections of $\mathbb A^1$-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $R$-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

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arXiv2022

$\mathbb{A}^1$-connectivity of moduli of vector bundles on a curve

In this note we prove that the moduli stack of vector bundles on a curve, with a fixed determinant is $\mathbb{A}^1$-connected. We obtain this result by classifying vector bundles on a curve upto $\mathbb{A}^1$-concordance. Consequently we classify$\mathbb{P}^n$- bundles on a curve upto $\mathbb{A}^1$-weak equivalence, extending a result of Asok-Morel. We also give an explicit example of a variety which is $\mathbb{A}^1$-h-cobordant to a projective bundle over $\mathbb{P}^2$ but does not have the structure of a projective bundle over $\mathbb{P}^2$, thus answering a question of Asok-Kebekus-Wendt

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arXiv2022

Nisnevich local Good compactifications

For a local complete intersection morphism, we establish fiberwise denseness in the $n$-dimensional irreducible components of the compactification Nisnevich locally.

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arXiv2022

Milnor-Witt cycle modules over an excellent DVR

The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R5 is vacuously satisfied when the base is a perfect field and it is also satisfied by $K^{MW}$ over any base. As a corollary, we obtain the strict $\mathbb{A}^1$-invariance and the exactness of Gersten complex for $K^{MW}$ over an excellent DVR.

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arXiv2022

The Nisnevich Motive of an Algebraic Stack

We construct the motive of an algebraic stack in the Nisnevich topology. For stacks which are Nisnevich locally quotient stacks, we give a presentation of the motive in terms of simplicial schemes. We also show that for quotient stacks the motivic cohomology agrees with the Edidin-Graham-Totaro Chow groups with integer coefficients.

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arXiv2021

Grothendieck-Serre Conjecture for Quasi-split Reductive Groups

We prove the Grothendieck-Serre conjecture for quasi-split reductive groups schemes. Our method involves reducing to the Borel subgroup in order to conclude the result from purity for tori and the structure theorem for unipotent radicals of parabolic subgroups in a reductive group.

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arXiv2021

Geometric criteria for $\mathbb A^1$-connectedness and applications to norm varieties

We show that $\mathbb A^1$-connectedness of a large class of varieties over a field $k$ can be characterized as the condition that their generic point can be connected to a $k$-rational point using (not necessarily naive) $\mathbb A^1$-homotopies. We also show that symmetric powers of $\mathbb A^1$-connected varieties (over an arbitrary field), as well as smooth proper models of them (over an algebraically closed field of characteristic $0$), are $\mathbb A^1$-connected. As an application of these results, we show that the standard norm varieties over a field $k$ of characteristic 0 become $\mathbb A^1$-connected (and consequently, universally $R$-trivial) after base change to an algebraic closure of $k$.

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arXiv2021

The Hurewicz map in motivic homotopy theory

For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$, we prove that the Hurewicz map $π_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ is surjective. We also observe that the Hurewicz map for $¶^1_k$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $ϕ$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $ϕ$ are also strongly $\A^1$-invariant.

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arXiv2020

Quasi-affineness and the 1-Resolution Property

We prove that, under mild hypothesis, every normal algebraic space which satisfies the $1$-resolution property is quasi-affine. More generally, we show that for algebraic stacks satisfying similar hypotheses, the 1-resolution property guarantees the existence of a finite flat cover by a quasi-affine scheme.

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arXiv2018

Gabber's presentation lemma for finite fields

We give a proof of Gabber's presentation lemma for finite fields. We use ideas from Poonen's proof of Bertini's theorem to prove this lemma in the special case of open subsets of the affine plane. We then reduce the case of general smooth varieties to this special case.

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arXiv2017

On the comparison of two constructions of Witt vectors of non-commutative rings

Let $A$ be any associative ring , possibly non-commutative, and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be that constructed by Hesselholt. The goal of this paper is to answer the following question by Hesselholt: Is $HH_0(E(A)) $ isomorphic to $W(A)$? We show that in the case $p=2$, there is no such isomorphism possible if one insists it to be compatible with the Verscheibung operator and the Teichmüller map.

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arXiv2015

A^1-connected components of schemes

A conjecture of Morel asserts that the sheaf of A^1-connected components of a simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that A^1-connected components of smooth k-schemes coincide with their A^1-chain-connected components and are birational invariants of smooth proper schemes. In this article, we exhibit examples of schemes for which Asok-Morel's conjectures fail to hold and whose Sing_* is not A^1-local. We also give equivalent conditions for Morel's conjecture to hold. A method suggested by these results is then used to prove Morel's conjecture for non-uniruled surfaces over a field k.

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