arXiv subjects

Uriya A. First

Publications and source records attributed to Uriya A. First.

At least 19 recordsLinked to original sources

arXiv2026

Hypergraph Samplers: Typical and Worst Case Behavior

We study the utility and limitations of using $k$-uniform hypergraphs $H = ([n], E)$ ($n \ge \mathrm{poly}(k)$) in the context of error reduction for randomized algorithms for decision problems with one- or two-sided error. Our error reduction idea is sampling a uniformly random hyperedge of $H$, and repeating the algorithm $k$ times using the hyperedge vertices as seeds. This is a general paradigm, which captures every pseudorandom method generating $k$ seeds without repetition. We show two results which imply a gap between the typical and the worst-case behavior of using $H$ for error-reduction. First, in the context of one-sided error reduction, if using a random hyperedge of $H$ decreases the error probability from $p$ to $p^k + \epsilon$, then $H$ cannot have too few edges, i.e., $|E| = \Omega(n k^{-1} \epsilon^{-1})$. Thus, the number of random bits needed for reducing the error from $p$ to $p^k + \epsilon$ cannot be reduced below $\lg n+\lg(\epsilon^{-1})-\lg k+O(1)$. This is also true for hypergraphs of average uniformity $k$. Our result implies new lower bounds for dispersers and vertex-expanders. Second, if the vertex degrees are reasonably distributed, we show that in a $(1-o(1))$-fraction of the cases, choosing $k$ pseudorandom seeds using $H$ will reduce the error probability to at most $o(1)$ above the error probability of using $k$ IID seeds, for both algorithms with one- or two-sided error. Thus, despite our lower bound, for a $(1-o(1))$-fraction of randomized algorithms (and inputs) for decision problems, the advantage of using IID samples over samples obtained from a uniformly random edge of a reasonable hypergraph is negligible. A similar statement holds true for randomized algorithms with two-sided error.

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arXiv2024

Spaces of Generators for Azumaya Algebras with Unitary Involution

Let $A$ be a finite dimensional algebra (possibly with some extra structure) over an infinite field $K$ and let $r\in\mathbb{N}$. The $r$-tuples $(a_1,\dots,a_r)\in A^r$ which fail to generate $A$ are the $K$-points of a closed subvariety $Z_r$ of the affine space underlying $A^r$, the codimension of which may be thought of as quantifying how well a generic $r$-tuple in $A^r$ generates $A$. Taking this intuition one step further, the second author, Reichstein and Williams showed that lower bounds on the codimension of $Z_r$ in $A^r$ (for every $r$) imply upper bounds on the number of generators of \emph{forms} of the $K$-algebra $A$ over finitely generated $K$-rings. That work also demonstrates how finer information on $Z_r$ may be used to construct forms of $A$ which require many elements to generate. The dimension and irreducible components of $Z_r$ are known in a few cases, which in particular lead to upper bounds on the number of generators of Azumaya algebras and Azumaya algebras with involution of the first kind (orthogonal or symplectic). This paper treats the case of Azumaya algebras with a unitary involution by finding the dimension and irreducible components of $Z_r$ when $A$ is the $K$-algebra with involution $(\mathrm{M}_n(K)\times \mathrm{M}_n(K), (a,b)\mapsto (b^{\mathrm{t}},a^{\mathrm{t}}))$. Our analysis implies that every Azumaya algebra with a unitary involution over a finitely generated $K$-ring of Krull dimension $d$ can be generated by $\lfloor \frac{d}{2n-2}+\frac{3}{2} \rfloor$ elements. We also give examples which require at least half that many elements to generate, by building on the work of the second author, Reichstein and Williams. Our method of finding the dimension and irreducible components of $Z_r$ actually applies to all $K$-algebras $A$ satisfying a mild assumption.

math.RA↗

arXiv2024

On Good $2$-Query Locally Testable Codes from Sheaves on High Dimensional Expanders

We expose a strong connection between good $2$-query locally testable codes (LTCs) and high dimensional expanders. Here, an LTC is called good if it has constant rate and linear distance. Our emphasis in this work is on LTCs testable with only $2$ queries, which are of particular interest to theoretical computer science. This is done by introducing a new object called a sheaf that is put on top of a high dimensional expander. Sheaves are vastly studied in topology. Here, we introduce sheaves on simplicial complexes. Moreover, we define a notion of an expanding sheaf that has not been studied before. We present a framework to get good infinite families of $2$-query LTCs from expanding sheaves on high dimensional expanders, utilizing towers of coverings of these high dimensional expanders. Starting with a high dimensional expander and an expanding sheaf, our framework produces an infinite family of codes admitting a $2$-query tester. We show that if the initial sheaved high dimensional expander satisfies some conditions, which can be checked in constant time, then these codes form a family of good $2$-query LTCs. We give candidates for sheaved high dimensional expanders which can be fed into our framework, in the form of an iterative process which conjecturally produces such candidates given a high dimensional expander and a special auxiliary sheaf. (We could not verify the prerequisites of our framework for these candidates directly because of computational limitations.) We analyse this process experimentally and heuristically, and identify some properties of the fundamental group of the high dimensional expander at hand which are sufficient (but not necessary) to get the desired sheaf, and consequently an infinite family of good $2$-query LTCs.

math.CO↗

arXiv2024

Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes

We study sheaves on posets, showing that cosystolic expansion of such sheaves can be derived from local expansion conditions of the sheaf and the poset (typically a high dimensional expander). When the poset at hand is a cell complex, a sheaf on it may be thought of as generalizing coefficient groups used for defining homology and cohomology, by letting the coefficient group vary along the cell complex. Previous works established local criteria for cosystolic expansion only for simplicial complexes and with respect to constant coefficients. Cosystolic expansion of sheaves is related to property testing. We use this relation and our local criterion for cosystolic expansion to give two applications to locally testable codes (LTCs). First, we show the existence of good $2$-query LTCs. These codes are related to the recent good $q$-query LTCs of Dinur et. al and Panteleev-Kalachev, being the formers' so-called line codes, but we get them from a new, more illuminating perspective, namely, by realizing them as cocycle codes of sheaves over posets. We then derive their good properties directly from our criterion for cosystolic expansion. Second, we give a local criterion for a a lifted code (with some auxiliary structure) to be locally testable. This improves on a previous work of Dikstein et. al, where it was shown that one can obtain local testability of lifted codes from a mixture of local and global conditions.

math.CO↗

arXiv2024

Algebraic Groups with Torsors That Are Versal for All Affine Varieties

Let $k$ be a field and let $G$ be an affine algebraic group over $k$. Call a $G$-torsor weakly versal for a class of $k$-schemes $\cal C$ if it specializes to every $G$-torsor over a scheme in $\cal C$. A recent result of the first author, Reichstein and Williams says that for any $d\geq 0$, there exists a $G$-torsor over a finite type $k$-scheme that is weakly versal for finite type affine $k$-schemes of dimension at most $d$. The first author also observed that if $G$ is unipotent, then $G$ admits a torsor over a finite type $k$-scheme that is weakly versal for all affine $k$-schemes, and that the converse holds if $\operatorname{char} k=0$. In this work, we extend this to all fields, showing that $G$ is unipotent if and only if it admits a $G$-torsor over a quasi-compact base that is weakly versal for all finite type regular affine $k$-schemes. Our proof is characteristic-free and it also gives rise to a quantitative statement: If $G$ is a non-unipotent subgroup of $\mathbf{GL}_n$, then a $G$-torsor over a quasi-projective $k$-scheme of dimension $d$ is not weakly versal for finite type regular affine $k$-schemes of dimension $n(d+1)+2$. This means in particular that every such $G$ admits a nontrivial torsor over a regular affine $(n+2)$-dimensional variety. When $G$ contains a nontrivial torus, we show that nontrivial torsors already exist over $3$-dimensional smooth affine varieties (even when $G$ is special), and this is optimal in general. In the course of the proof, we show that for every $m,\ell\in\mathbb{N}\cup\{0\}$ with $\ell\neq 1$, there exists a smooth affine $k$-scheme $X$ carrying an $\ell$-torsion line bundle that cannot be generated by $m$ global sections. We moreover study the minimal possible dimension of such an $X$ and show that it is $m$, $m+1$ or $m+2$.

math.AG↗

arXiv2023

Highly Versal Torsors

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-variety is said to be versal if it specializes to every $G$-torsor over any $k$-field. The existence of versal torsors is well-known. We show that there exist $G$-torsors that admit even stronger versality properties. For example, for every $d\in\mathbb{N}$, there exists a $G$-torsor over a smooth quasi-projective $k$-scheme that specializes to every torsor over a quasi-projective $k$-scheme after removing some codimension-$d$ closed subset from the latter. Moreover, such specializations are abundant in a well-defined sense. Similar results hold if we replace $k$ with an arbitrary base-scheme. In the course of the proof we show that every globally generated rank-$n$ vector bundle over a $d$-dimensional $k$-scheme of finite type can be generated by $n+d$ global sections. When $G$ can be embedded in a group scheme of unipotent upper-triangular matrices, we further show that there exist $G$-torsors specializing to every $G$-torsor over any affine $k$-scheme. We show that the converse holds when $\operatorname{char} k=0$. We apply our highly versal torsors to show that, for fixed $m,n\in\mathbb{N}$, the symbol length of any degree-$m$ period-$n$ Azumaya algebra over any local $\mathbb{Z}[\frac{1}{n},e^{2πi/n}]$-ring is uniformly bounded. A similar statement holds in the semilocal case, but under mild restrictions on the base ring.

math.AG↗

arXiv2022

The Cheeger Inequality and Coboundary Expansion: Beyond Constant Coefficients

The Cheeger constant of a graph, or equivalently its coboundary expansion, quantifies the expansion of the graph. This notion assumes an implicit choice of a coefficient group, namely, $\mathbb{F}_2$. In this paper, we study Cheeger-type inequalities for graphs endowed with a generalized coefficient group, called a sheaf; this is motivated by applications to cosystolic expansion and locally testable codes. We prove that a graph is a good spectral expander if and only if it has good coboundary expansion relative to any (resp. some) constant sheaf, or equivalently, relative to any `ordinary' coefficient group. We moreover show that sheaves that are close to being constant in a well-defined sense are also good coboundary expanders, provided that their underlying graph is an expander, thus giving the first example of good coboundary expansion in non-cosntant sheaves on sparse graphs. By contrast, we observe that for general sheaves on graphs, it is impossible to relate the expansion of the graph and the coboundary expansion of the sheaf. We specialize our results to sheaves on (finite) spherical buildings. Specifically, we show that the normalized second eigenvalue of the (weighted) graph underlying a $q$-thick $d$-dimensional spherical building is $O(\frac{1}{\sqrt{q}-3d})$ if $q>9d^2$. Plugging this into our results about coboundary expansion gives explicit lower bounds on the coboundary expansion of some constant and non-constant sheaves on spherical buildings; for a fixed dimension $d$, the bounds approach a constant as the thickness $q$ grows. Along the way, we prove a new version of the Expander Mixing Lemma for $r$-partite weighted graphs.

math.CO↗

arXiv2022

On the Grothendieck-Serre Conjecture for Classical Groups

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.

math.AG↗

arXiv2022

On The Gersten-Witt Complex of an Azumaya Algebra with Involution

Let $(A,σ)$ be an Azumaya algebra with involution over a regular ring $R$. We prove that the Gersten-Witt complex of $(A,σ)$ defined by Gille is isomorphic to the Gersten-Witt complex of $(A,σ)$ defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten-Witt complex is exact when $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $σ$ is orthogonal or symplectic. This means that the Grothendieck-Serre conjecture holds for the group $R$-scheme of $σ$-unitary elements in $A$ under the same hypotheses; $R$ is not required to contain a field.

math.AG↗

arXiv2022

Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Let $(A,\sigma)$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^\times$. We show that if $(A,\sigma)$ admits an improper isometry, i.e., an element $a\in A$ with $\sigma(a)a=1$ and $\mathrm{Nrd}_{A/R}(a)=-1$, then the Brauer class of $A$ is trivial. An analogue of this statement also holds for Azumaya algebras with quadratic pair when $2\notin R^\times$. We also show that at this level of generality, the hypotheses do not guarantee that $A$ is a matrix algebra over $R$.

math.RA↗

arXiv2021

On the number of generators of an algebra over a commutative ring

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull dimension $d$, then any projective $R$-module of rank $n$ can be generated by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than $d+n$ elements. We view projective $R$-modules as $R$-forms of the non-unital $R$-algebra where the product of any two elements is $0$. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for étale algebras. In this paper, we prove new upper and lower bound on the number of generators of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ has finite transcendence degree $d$ over $k$. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack $BG$, where $G$ is the automorphism group of the algebra in question, by algebraic spaces of a certain form.

math.RA↗

arXiv2021

An $8$-Periodic Exact Sequence of Witt Groups of Azumaya Algebras with Involution

Given an Azumaya algebra with involution $(A,σ)$ over a commutative ring $R$ and some auxiliary data, we construct an $8$-periodic chain complex involving the Witt groups of $(A,σ)$ and other algebras with involution, and prove it is exact when $R$ is semilocal. When $R$ is a field, this recovers an $8$-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala--Sridharan--Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck--Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $\mathbf{GL}_n$ and $\mathbf{Sp}_{2n}$, provided some assumptions on $R$. We show that a $1$-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring $R$ is isotropic if and only if its trace (a quadratic form over $R$) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $W(R)\to W(S)$ when $R$ is a (non-semilocal) $2$-dimensional regular domain and $S$ is a quadratic étale $R$-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.

math.AG↗

arXiv2020

On the non-neutral component of outer forms of the orthogonal group

Let $(A,σ)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,σ)$ contains elements of reduced norm $-1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary.

math.AG↗

arXiv2020

Involutions of Azumaya algebras

We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, {\mathcal{O}}_{\mathbf{ X}})$ where the latter carries a (possibly trivial) involution, denoted $λ$. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending $λ$ are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra $A$ is Brauer equivalent to an algebra carrying an involution extending $λ$, by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, nontrivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then $A$ is Brauer equivalent to an Azumaya algebra of degree $2n$ carrying an involution. By comparison with the case of topological spaces, we show that the integer $2n$ is minimal, even in the case of a nonsingular affine variety $X$ with a fixed-point free involution. As an incidental step, we show that if $R$ is a commutative ring with involution for which the fixed ring $S$ is local, then either $R$ is local or $R/S$ is a quadratic étale extension of rings.

math.AG↗

arXiv2019

Pfister's Local--Global Principle and Systems of Quadratic Forms

Let $q$ be a unimodular quadratic form over a field $K$. Pfister's famous local--global principle asserts that $q$ represents a torsion class in the Witt group of $K$ if and only if it has signature $0$, and that in this case, the order of Witt class of $q$ is a power of $2$. We give two analogues of this result to systems of quadratic forms, the second of which applying only to nonsingular pairs. We also prove a counterpart of Pfister's theorem for finite-dimensional $K$-algebras with involution, generalizing a result of Lewis and Unger.

math.NT↗

arXiv2019

Azumaya Algebras Without Involution

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a $2$-torsion class in the Brauer group if and only if there is an algebra $A'$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A'$ such that $\mathrm{deg}\, A'=2\mathrm{deg}\, A$. We show that $2\mathrm{deg}\, A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra $A$ of degree $4$ and period $2$ such that the degree of any algebra $A'$ in the Brauer class of $A$ admitting an involution is divisible by $8$. Separately, we provide examples of split and non-split Azumaya algebras of degree $2$ admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.

math.AG↗

arXiv2018

Orders that are Étale-Locally Isomorphic

Let $R$ be a semilocal Dedekind domain with fraction field $F$. We show that two hereditary $R$-orders in central simple $F$-algebras which become isomorphic after tensoring with $F$ and with some faithfully flat étale $R$-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for hermitian forms over hereditary $R$-orders with involution. The results can be restated by means of étale cohomology and can be seen as variations of the Grothendieck--Serre conjecture on principal homogeneous bundles of reductive group schemes. Connections with Bruhat--Tits theory are also discussed.

math.AG↗

arXiv2018

On Uniform Admissibility of Unitary and Smooth Representations

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are uniformly admissible. We also show that the latter property is inherited by finite-index subgroups and overgroups of $G$.

math.RT↗