arXiv subjects

Uriya A. First

Publications and source records attributed to Uriya A. First.

29 records · Page 2Linked to original sources

arXiv2017

Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups

Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck--Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat--Tits theory.

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arXiv2016

On the number of generators of an algebra

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by "algebra". Here we allow arbitrary finite algebras, not necessarily unital, commutative or associative. Forster's theorem can be recovered as a special case by viewing a module as an algebra where the product of any two elements is $0$.

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arXiv2016

The Ramanujan Property for Simplicial Complexes

Let $G$ be a topological group acting on a simplicial complex $\mathcal{X}$ satisfying some mild assumptions. For example, consider a $k$-regular tree and its automorphism group, or more generally, a regular affine Bruhat-Tits building and its automorphism group. We define and study various types of high-dimensional spectra of quotients of $\mathcal{X}$ by subgroups of $G$. These spectra include the spectrum of many natural operators associated with the quotients, e.g. the high-dimensional Laplacians. We prove a theorem in the spirit of the Alon-Boppana Theorem, leading to a notion of Ramanujan quotients of $\mathcal{X}$. Ramanujan $k$-regular graphs and Ramanuajn complexes in the sense of Lubotzky, Samuels and Vishne are Ramanujan in dimension $0$ according to our definition (for $\mathcal{X}$, $G$ suitably chosen). We give a criterion for a quotient of $\mathcal{X}$ to be Ramanujan which is phrased in terms of representations of $G$, and use it, together with deep results about automorphic representations, to show that affine buildings of inner forms of $\mathbf{GL}_n$ over local fields of positive characteristic admit infinitely many quotients which are Ramanujan in all dimensions. The Ramanujan (in dimension $0$) complexes constructed by Lubotzky, Samuels and Vishne arise as a special case of our construction. Our construction also gives rise to Ramanujan graphs which are apparently new. Other applications are also discussed. For example, we show that there are non-isomorphic simiplicial complexes which are isospectral in all dimensions.

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arXiv2016

Highlights from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]

This paper brings the main definitions and results from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]. No proofs are given. Given a simplicial complex $\mathcal{X}$ and a group $G$ acting on $\mathcal{X}$, we define Ramanujan quotients of $\mathcal{X}$. For $G$ and $\mathcal{X}$ suitably chosen this recovers Ramanujan $k$-regular graphs and Ramanujan complexes in the sense of Lubotzky, Samuels and Vishne. Deep results in automorphic representations are used to give new examples of Ramanujan quotients when $\mathcal{X}$ is the affine building of an inner form of $\mathbf{GL}_n$ over a local field of positive characteristic.

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arXiv2016

Patching and Weak Approximation in Isometry Groups

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of $R$ and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (non necessarily commutative) $R$-orders is always a finite power of $2$, and under further assumptions, e.g. that the order is hereditary, this number is $1$. The same result is also shown for related objects, e.g. systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings. The appendix proves that the isometry group of a quadratic space over an $R$-order with involution can be regarded as a smooth affine group scheme under mild assumptions.

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arXiv2015

Witt's Extension Theorem for Quadratic Spaces over Semiperfect Rings

We prove that every isometry of between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the broader class of semilocal rings, but with certain restrictions on the base modules, which cannot be removed in general. Our result implies that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. This improves a cancellation result of Quebbemann, Scharlau and Schulte, which applies to quadratic spaces over hermitian categories. Combining this with other known results yields further cancellation theorems. For instance, we prove cancellation of (1) systems of sesquilinear forms over henselian local rings, and (2) non-unimodular hermitian forms over (arbitrary) valuation rings. Finally, we determine the group generated by the reflections of a unimodular quadratic space over a semiperfect ring.

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arXiv2015

Rings That Are Morita Equivalent to Their Opposites

We consider the following problem: Under what assumptions do one or more of the following are equivalent for a ring $R$: (A) $R$ is Morita equivalent to a ring with involution, (B) $R$ is Morita equivalent to a ring with an anti-automorphism, (C) $R$ is Morita equivalent to its opposite ring. The problem is motivated by a theorem of Saltman which roughly states that all conditions are equivalent for Azumaya algebras. Basing on the recent "general bilinear forms", we present a general machinery to attack the problem, and use it to show that (C)$\iff$(B) when $R$ is semilocal or $\mathbb{Q}$-finite. Further results of similar flavor are also obtained, for example: If $R$ is a semilocal ring such that $\mathrm{M}_{n}(R)$ has an involution, then $\mathrm{M}_{2}(R)$ has an involution, and under further mild assumptions, $R$ itself has an involution. In contrast to that, we demonstrate that (B) does not imply (A). Our methods also give a new perspective on the Knus-Parimala-Srinivas proof of Saltman's Theorem. Finally, we give a method to test Azumaya algebras of exponent $2$ for the existence of involutions, and use it to construct explicit examples of such algebras.

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arXiv2014

An Elementary Proof That Rationally Isometric Quadratic Forms Are Isometric

Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We give an elementary proof of the following known result: Two unimodular quadratic forms over $R$ are isometric over $K$ if and only if they are isometric over $R$. Our proof does not use Witt's Cancelation Theorem and yields an explicit algorithm to construct an isometry over $R$ from a given isometry over $K$. The statement actually holds for hermitian forms over valuated involutary division rings, provided mild assumptions. A python implementation of the algorithm derived from the proof can be found on the author's home page.

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arXiv2013

Hermitian Categories, Extension of Scalars and Systems of Sesquilinear Forms

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e. the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends a result obtained by E. Bayer-Fluckiger and D. Moldovan. We use the equivalence to define a Witt ring of sesquilinear forms over a hermitian category, and also to generalize various results (e.g.: Witt's Cancelation Theorem, Springer's Theorem, the weak Hasse principle, finiteness of genus) to systems of sesquilinear forms over hermitian categories.

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arXiv2013

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers and bilinear pairings as special cases. We prove a Krull-Schmidt Theorem in this general context, which results in a Krull-Schmidt Theorem for the special cases just mentioned. We also show that cancellation holds under milder assumptions. Using similar ideas we prove a version of Fitting's Lemma for natural transformations between functors.

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arXiv2013

Semi-Invariant Subrings

We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring $S\supseteq R$ and a set $X\subseteq S$ such that $R_0=\Cent_R(X):={r\in R \suchthat xr=rx \forall x\in X}$; in particular, centralizers of subsets of $R$ are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring is semiprimary (resp. right perfect). (2) If $M$ is a finitely presented module over a "good" semiperfect ring (e.g. an inverse limit of semiprimary rings), then $(M)$ is semiperfect, hence $M$ has a Krull-Schmidt decomposition. (This generalizes results of Bjork and Rowen). (3) If $ρ$ is a representation of a monoid or a ring over a module with a "good" semiperfect endomorphism ring (in the sense of (2)), then $ρ$ has a Krull-Schmidt decomposition. (4) If $S$ is a "good" commutative semiperfect ring and $R$ is an $S$-algebra that is f.p.\ as an $S$-module, then $R$ is semiperfect. (5) Let $R\subseteq S$ be rings and let $M$ be a right $S$-module. If $(M_R)$ is semiprimary (resp. right perfect), then $(M_S)$ is semiprimary (resp. right perfect).

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