Uriya A. First’s research while affiliated with University of Haifa and other places

Let F be a field. We show that the largest irredundant generating sets for the algebra of $n\times n$ matrices over F have $2n-1$ elements when $n>1$. (A result of Laffey states that the answer is $2n-2$ when $n>2$, but its proof contains an error.) We further give a classification of the largest irredundant generating sets when $n\in\{2,3\}$ and F is algebraically closed. We use this description to compute the dimension of the variety of $(2n-1)$-tuples of $n\times n$ matrices which form an irredundant generating set when $n\in\{2,3\}$, and draw some consequences to Zariski-locally redundant generation of Azumaya algebras. In the course of proving the classification, we also determine the largest sets S of subspaces of $F^3$ with the property that every $V\in S$ admits a matrix stabilizing every subspace in $S-\{V\}$ and not stabilizing V.

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.

Suppose A is an Azumaya algebra over a ring R and $\sigma$ is an involution of A extending an order-2 automorphism $\lambda:R\to R$. We say $\sigma$ is extraordinary if there does not exist a Brauer-trivial Azumaya algebra $\mathrm{End}_R(P)$ over R carrying an involution $\tau$ so that $(A, \sigma)$ and $(\mathrm{End}_R(P), \tau)$ become isomorphic over some faithfully flat extension of the fixed ring of $\lambda:R\to R$. We give, for the first time, an example of such an algebra and involution. We do this by finding suitable cohomological obstructions and showing they do not always vanish. We also give an example of a commutative ring R with involution $\lambda$ so that the scheme-theoretic fixed locus Z of $\lambda:\mathrm{Spec} R\to \mathrm{Spec} R$ is disconnected, but such that every Azumaya algebra over R with involution extending $\lambda$ is either orthogonal at every point of Z, or symplectic at every point of Z. No examples of this kind were previously known.

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

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 $m,n\in\mathbb{N}$ and a fixed semilocal $\mathbb{Z}[\frac{1}{n},e^{2\pi i/n}]$-ring R with infinite residue fields, the symbol length of Azumaya algebras over R having degree m and period n is uniformly bounded. Under mild assumptions on R, e.g., if R is local, the bound depends only on m and n, and not on R.

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.

We study the coboundary expansion of graphs, but instead of using $\mathbb{F}_2$ as the coefficient group when forming the cohomology, we use a sheaf on the graph. We prove that if the graph under discussion is a good expander, then it is also a good coboundary expander relative to any constant augmented sheaf (equivalently, relative to any coefficient group R); this, however, may fail for locally constant sheaves. We moreover show that if we take the quotient of a constant augmented sheaf on an excellent expander graph by a "small" subsheaf, then the quotient sheaf is still a good coboundary expander. Along the way, we prove a new version of the Expander Mixing Lemma applying to r-partite weighted graphs.

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 ⩽2$\leqslant 2$ (or ⩽4$\leqslant 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.