arXiv subjects

Eliyahu Matzri

Publications and source records attributed to Eliyahu Matzri.

11 recordsLinked to original sources

arXiv2023

Semiassociative algebras over a field

An associative central simple algebra is a form of matrices, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrices, which we are led to define and investigate. Semiassociative algebras modulo skew matrices compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.

math.RA↗

arXiv2017

Triple Massey Products with weights in Galois cohomology

Fix an arbitrary prime $p$. Let $F$ be a field containing a primitive $p$-th root of unity, with absolute Galois group $G_F$, and let $H^n$ denote its mod $p$ cohomology group $H^n(G_F,\mathbb{Z}/p\mathbb{Z})$. The triple Massey product of weight $(n,k,m)\in \mathbb{N}^3$ is a partially defined, multi-valued function $\langle \cdot,\cdot,\cdot \rangle: H^n\times H^k\times H^m\rightarrow H^{n+k+m-1}.$ %(in the mod-$p$ Galois cohomology) In this work we prove that for an arbitrary prime $p$, any defined $3MP$ of weight $(n,1,m)$, where the first and third entries are assumed to be symbols, contains zero; and that for $p=2$ any defined $3MP$ of the weight $(1,k,1)$, where the middle entry is a symbol, contains zero. Finally, we use the description of the kernel of multiplication by a symbol to study general 3MP where the middle slot is a symbol. The main tools we will be using is Lemma 4.1 concerning the the annihilator of cup product with an $H^1$ element, and Theorem 5.2, generalizing a Theorem of Tignol on quaternion algebras with trivial corestriction along a separable quadratic extension.

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arXiv2016

Bimodule Structure of Central Simple Algebras

For a maximal separable subfield $K$ of a central simple algebra $A$, we provide a semiring isomorphism between $K$-$K$-bimodules $A$ and $H$-$H$ bisets of $G = \Gal(L/F)$, where $F = \operatorname{Z}(A)$, $L$ is the Galois closure of $K/F$, and $H = \Gal(L/K)$. This leads to a combinatorial interpretation of the growth of $\dim_K((KaK)^i)$, for fixed $a \in A$, especially in terms of Kummer sets.

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arXiv2015

Triple Massey products and absolute Galois groups

Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Minac and Tan, we prove that the triple Massey product $H^1(G_F)^3\to H^2(G_F)$ contains $0$ whenever it is nonempty. This gives a new restriction on the possible profinite group structure of $G_F$.

math.NT↗

arXiv2015

Kummer Spaces in Cyclic Algebras of Prime Degree

We classify the monomial Kummer subspaces of division cyclic algebras of prime degree $p$, showing that every such space is standard, and in particular the dimension is no greater than $p+1$. It follows that in a generic cyclic algebra, the dimension of any Kummer subspace is at most $p+1$.

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arXiv2015

Vanishing of Massey products and Brauer groups

Let p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Minac, and Tan, to classical results in the theory of central simple algebras. For global fields, we prove a stronger form of the vanishing property.

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arXiv2014

Triple Massey products in Galois cohomology

Fix an arbitrary prime $p$. Let $F$ be a field, containing a primitive $p$-th root of unity, with absolute Galois group $G_F$. The triple Massey product (in the mod-$p$ Galois cohomology) is a partially defined, multi-valued function $\langle \cdot,\cdot,\cdot \rangle: H^1(G_F)^3\rightarrow H^2(G_F).$ In this work we prove a conjecture made in [11] stating that any defined triple Massey product contains zero. As a result the pro-$p$ groups appearing in [11] are excluded from being absolute Galois groups of fields $F$ as above.

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arXiv2014

$\mathbb{Z}_3\times \mathbb{Z}_3$ crossed products

Let $A$ be the generic abelian crossed product with respect to $\mathbb{Z}_3\times \mathbb{Z}_3$, in this note we show that $A$ is similar to the tensor product of 4 symbol algebras (3 of degree 9 and one of degree 3) and if $A$ is of exponent $3$ it is similar to the product of 31 symbol algebras of degree $3$. We then use \cite{RS} to prove that if $A$ is any algebra of degree $9$ then $A$ is similar to the product of $35840$ symbol algebras ($8960$ of degree $3$ and $26880$ of degree $9$) and if $A$ is of exponent $3$ it is similar to the product of $277760$ symbol algebras of degree $3$. We then show that the essential $3$-dimension of the class of $A$ is at most $6$.

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arXiv2014

Symbol length in the Brauer group of a field

We bound the symbol length of elements in the Brauer group of a field $K$ containing a $C_m$ field (for example any field containing an algebraically closed field or a finite field), and solve the local exponent-index problem for a $C_m$ field $F$. In particular, for a $C_m$ field $F$, we show that every $F$ central simple algebra of exponent $p^t$ is similar to the tensor product of at most $len(p^t,F)\leq t(p^{m-1}-1)$ symbol algebras of degree $p^t$. We then use this bound on the symbol length to show that the index of such algebras is bounded by $(p^t)^{(p^{m-1}-1)}$, which in turn gives a bound for any algebra of exponent $n$ via the primary decomposition. Finally for a field $K$ containing a $C_m$ field $F$, we show that every $F$ central simple algebra of exponent $p^t$ and degree $p^s$ is similar to the tensor product of at most $len(p^t,p^s,K)\leq len(p^t,L)$ symbol algebras of degree $p^t$, where $L$ is a $C_{m+ed_L(A)+p^{s-t}-1}$ field.

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arXiv2014

All dihedral division algebras of degree five are cyclic

Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree $2n$ of the center, $n$ odd, is in fact cyclic. The proof requires roots of unity of order $n$ in the center. We show that for $n=5$, this assumption can be removed. It then follows that ${}_{5\!\!\!\:}Br(F)$, the $5$-torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev on the $2$ and $3$ torsion parts.

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arXiv2013

Diophantine and cohomological dimensions

We give explicit linear bounds on the p-cohomological dimension of a field in terms of its Diophantine dimension. In particular, we show that for a field of Diophantine dimension at most 4, the 3-cohomological dimension is less than or equal to the Diophantine dimension.

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