Some of the papers can be downloaded as pdf files. I'd also be happy
to send you reprints (where available).
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Polynomial Identity Rings as Rings of Functions, II
Preprint; December 3, 2010 (revised November 12, 2011)
Abstract.
In characteristic zero, Zinovy Reichstein and the
author generalized the usual relationship between irreducible
Zariski closed subsets of the affine space, their defining ideals,
coordinate rings, and function fields, to a non-commutative setting,
where "varieties" carry a PGLn-action, regular and
rational "functions" on them are matrix-valued, "coordinate rings"
are prime polynomial identity algebras, and "function fields" are
central simple algebras of degree n. In the present paper,
much of this is extended to prime characteristic. In addition, a
mistake in the earlier paper is corrected. One of the results is
that the finitely generated prime PI-algebras of degree n are
precisely the rings that arise as "coordinate rings" of
"n-varieties" in this setting. For n = 1 the
definitions and results reduce to those of classical affine
algebraic geometry.
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Central Simple Algebras with Involution: A Geometric
Approach
Manuscripta Mathematica 128 (2009), 453-467
Abstract.
Let k be
an algebraically closed base field of characteristic zero. The
category equivalence between central simple algebras and
irreducible, generically free PGLn-varieties is extended to the context of central
simple algebras with involution. The associated variety of a
central simple algebra with involution comes with an action of the
semidirect product Pn,τ
:= PGLn \rtimes
<τ>,
where τ is the
automorphism of PGLn
given by τ(h)=(h-1)transpose. Basic
properties of an involution are described in terms of the action
of Pn,τ
on the associated variety, and in particular in terms of the
stabilizer in general position for this action.
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Tame Group Actions on Central Simple Algebras
(with Zinovy
Reichstein)
J. Algebra 318 (2007), 1039-1056
Abstract.
We study actions of linear algebraic groups on finite-dimensional
central simple algebras. We describe the fixed algebra for a broad
class of such actions.
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Group Actions and Invariants in Algebras of Generic Matrices
(with Zinovy
Reichstein)
Advances in Applied Mathematics 37, no.4 (2006), 481-500
(in the special issue in honor of Amitai Regev on his 65th birthday)
Abstract.
We show that the fixed elements for the
natural GLm-action on
the universal division algebra UD(m, n)
of m
generic n x
n-matrices form a division subalgebra of
degree n,
assuming n ≥ 3 and 2 ≤ m ≤ n2 - 2. This allows us to describe the
asymptotic behavior of the dimension of the space
of SLm-invariant homogeneous central
polynomials p(X1, ..., Xm) for
n x
n-matrices. Here the base field is assumed to be of
characteristic zero.
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Group Actions on Central Simple Algebras: A Geometric
Approach
(with Zinovy
Reichstein)
J. Algebra 304 (2006), 1160-1192
Abstract.
We study actions of linear algebraic groups on central simple
algebras using algebro-geometric techniques. Suppose an algebraic
group
G acts on a central simple algebra
A of degree
n. We are interested in questions of the
following type:
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Do the G-fixed elements form a central
simple subalgebra of A of degree n?
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Does A have a G-invariant
maximal subfield?
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Does A have a splitting field with
a G-action, extending the G-action on the center
of A?
Somewhat surprisingly, we find that under mild assumptions
on A and the actions, one can answer these questions by using
techniques from birational invariant theory (i.e., the study of
group actions on algebraic varieties, up to equivariant birational
isomorphisms). In fact, group actions on central simple algebras
turn out to be related to some of the central problems in birational
invariant theory, such as the existence of sections, stabilizers in
general position, affine models, etc. In this paper we explain these
connections and explore them to give partial answers to questions
(a) - (c).
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Polynomial Identity Rings as Rings of Functions
(with Zinovy
Reichstein)
J. Algebra 310, no. 2 (2007), 624-647
Abstract.
We generalize the usual relationship between irreducible Zariski
closed subsets of the affine space, their defining ideals,
coordinate rings, and function fields, to a non-commutative setting,
where "varieties" carry a PGLn-action, regular and
rational "functions" on them are matrix-valued, "coordinate rings"
are prime polynomial identity algebras, and "function fields" are
central simple algebras of degree n. In particular, a prime
polynomial identity algebra of degree n is finitely generated
if and only if it arises as the "coordinate ring" of a "variety" in
this setting. For n = 1 our definitions and results reduce
to those of classical affine algebraic geometry.
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Actions of Solvable Algebraic Groups on Central Simple
Algebras
Algebras and Representation Theory 10 (2007), 413-427
Abstract.
Let k be an algebraically closed base field of arbitrary
characteristic. In this paper, we study actions of a connected
solvable linear algebraic group G on a central simple
algebra Q. The main result is the following: Q can be
split G-equivariantly by a finite-dimensional splitting
field, provided that G acts "algebraically'', i.e., provided
that Q contains a G-stable order on which the action
is rational. As an application, it is shown that rational torus
actions on prime PI-algebras are induced by actions on commutative
domains.
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Stable Affine Models for Algebraic Group Actions
(with Zinovy
Reichstein)
J. Lie Theory 14 (2004), 563-568
This paper is freely available from the online version of
the Journal
of Lie Theory (one of the journals available through
the Electronic
Library of Mathematics).
Abstract.
Let G be a reductive linear algebraic group defined over an
algebraically closed base field k of characteristic
zero. A G-variety is an algebraic variety with a regular
action of G, defined over k . An
affine G-variety is called stable if its points in general
position have closed G-orbits. We give a simple necessary and
sufficient condition for a G-variety to have a stable affine
birational model.
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Actions of Algebraic Groups on the Spectrum of Rational
Ideals, II
J. Algebra 208 (1998), 216-261
Abstract.
We study rational actions of a linear algebraic group G on an
algebra V, and the induced actions on Rat(V), the
spectrum of rational ideals of V (a subset of Spec(V)
which often includes all primitive ideals). This work extends
results of Moeglin and Rentschler to prime characteristic, often
also relaxing their finiteness assumptions on V. In
particular, we relate properties of a rational ideal J and
its orb (J:G) =
∩γ∈G γ(G).
The rational ideals of V containing the orb of J are
precisely those in the Zariski-closure X of the orbit
of J in Rat(V). The
G-stratum of J consists of all rational ideals in
X whose orbit is dense in X (i.e., whose orb is equal
to the orb of J). We show that the G-stratum of a
rational ideal consists of exactly one G-orbit, and that
rational ideals are maximal in their strata in a strong sense. These
results are useful for studying prime and primitive spectra of
certain algebras, cf. recent work by Goodearl and Letzter. We
further show that the orbit of J is open in its closure in
Rat(V), provided that J is locally closed. Among other
results, we show that the semiprime ideal (J:G) is Goldie,
and we relate the uniform and Gelfand-Kirillov dimensions
of V/J and V/(J:G).
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Actions of Algebraic Groups on the Spectrum of Rational
Ideals
J. Algebra 182 (1996), 383-400
Abstract.
Let k be an algebraically closed field and G a linear
algebraic group over k acting rationally on
a k-algebra
V. Generalizing work of Moeglin and Rentschler in
characteristic zero, we study the action of G on the spectrum
of rational ideals of V. The main result is the
following. Suppose that V is semiprime left Goldie.
Let L be a G-stable commutative semisimple subalgebra
of the total ring of fractions Q(V) of V such
that LG = k ⋅ 1L. This occurs, for
example, if the zero ideal of V is G-rational
and L is the center of Q(V). Then there is, for some
closed subgroup H of G, a G-equivariant
embedding ν of L into Q(G/H) (the algebra of
rational functions on G/H) such that Q(G/H) is purely
inseparable over ν(L). This has applications to the
closure of the orbit of a rational ideal.
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Free Subgroups of Division Algebras
(with Zinovy
Reichstein)
Comm. Algebra 23 (1995), 2181-2185
Abstract.
A. I. Lichtman asked whether or not the multiplicative group of a
non-commutative division ring D always contains a copy of the
free group on two generators. This is known to be true in several
cases, in particular if D is finite-dimensional over its
center. We give an affirmative answer in the case that the
center K of D is uncountable and D contains a
non-central element which is algebraic over K.
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Rational Central Simple Algebras
(with Zinovy
Reichstein)
Israel J. Math. 95 (1996), 253-280
Abstract.
We introduce a notion of rationality (called toroidal or
t-rationality) for central simple algebras which extends Demazure's
characterization of rational algebraic varieties via torus
actions. We prove a structure theorem for t-rational central
simple algebras and study the interplay among t-rationality, crossed
products and rationality of the center in the setting of universal
division algebras.
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An Embedding Property of Universal Division Algebras
(with Zinovy
Reichstein)
J. Algebra 177 (1995), 451-462
Abstract.
Let A be a central simple algebra of degree n and let
k be a subfield of its center. We show that A contains a copy
of the universal division algebra Dm,n(k)
generated by m generic n x n matrices if and
only if trdegk A ≥
trdegk Dm,n(k)
= (m-1) n2 + 1.
Moreover, if in addition the center of A is finitely and
separably generated over k then "almost all" division
subalgebras of A generated by m elements are
isomorphic to Dm,n(k). In the last section
we give an application of our main result to the question of
embedding free groups in division algebras.
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Torus Actions on Rings
(with Zinovy
Reichstein)
J. Algebra 170 (1994), 781-804
Abstract.
We study torus actions on non-commutative rings, focusing on upper
bounds on the dimensions of tori for which faithful actions are
possible. We give sharp bounds for actions on algebras of generic
matrices and their trace rings.
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Rings with Polynomial Identities: An Elementary
Introduction
Proceedings of the XII Brazilian Algebra Meeting, Matemática
Contemporânea 7 (1994), 199-231
Abstract.
These notes form an elementary introduction to the theory of rings
satisfying a polynomial identity. No background in non-commutative
ring theory is assumed; the only prerequisite is familiarity with
the fundamental concepts of commutative algebra.
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Mapping Galois Extensions into Division Algebras
Proc. Amer. Math. Soc. 119 (1993), 1061-1068
Abstract.
Let A be a ring with a finite group of
automorphisms G, and let f1
and f2 be homomorphisms from A into some
division algebra D such that f1
and f2 agree on the fixed
ring AG. Assuming certain additional
assumptions, it is shown that f1 and
f2 differ only by an automorphism in G and
an inner automorphism of D.
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Integrality for PI-Rings
(with Amiram
Braun)
J. Algebra 151 (1992), 39-79
Abstract.
In this paper, we study (Schelter) integral extensions of
PI-rings. We prove in particular lying over, going up and
incomparability for prime ideals. A major result is
transitivity of integrality: If R ⊆ S
⊆ B are PI-rings such that B is integral
over S and S is integral over R, then B
is integral over R. Next, we obtain a powerful criterion for
integrality: If S is a prime PI-ring such that its center is
integral over a Noetherian subring R of S,
then S is integral over R. This allows interesting
applications to the theory of finite group actions. Further topics
concern Eakin-Nagata type results and embeddings of quotient rings
for integral extensions. Finally, we analyze the relationship
between module-finite extensions and finitely generated integral
extensions, obtaining positive results for affine Noetherian
PI-algebras and algebras satisfying certain restrictions on
PI-degrees (e.g., algebras of low PI-degree).
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Actions of Linearly Reductive Groups on PI-Algebras
Trans. Amer. Math. Soc. 335 (1993), 425-442
Abstract.
Let G be a linearly reductive group acting rationally on a
PI-algebra R. We study the relationship between R and
the fixed ring RG, generalizing earlier results
obtained under the additional hypothesis that R is affine.
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Prime Ideals in Intermediate Integral Centralizing
Extensions of PI-Rings
Comm. Algebra 19 (1991), 795-802
Abstract.
In this short note, we study the behavior of prime ideals in
intermediate integral centralizing extensions of PI-rings. Using
this, we give a PI-theoretic proof of E. Letzter's lying over
theorem for arbitrary finite extensions of right Noetherian
PI-rings. Similar techniques yield also an Artin-Tate lemma for
finite extensions of right Noetherian PI-algebras.
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Actions of Linearly Reductive Groups on Affine
PI-Algebras
Mem. Amer. Math. Soc. 414 (1989), 106 pages
Abstract.
Let
k be an algebraically closed field, and let
R be
an affine (i.e., finitely generated)
k-algebra satisfying a
polynomial identity. Let
G be a linearly reductive group
acting rationally on
R. In this paper, the relationship
between
R and the fixed ring
RG is
studied. Best results have been obtained if
R is left
Noetherian, or even an Azumaya algebra, or if
G acts by inner
automorphisms.
Among the results for left Noetherian algebras are the
following. (a) The fixed ring RG is affine - this
is an extension of Hilbert's famous theorem for commutative
algebras. (b) "Lying over" holds. That is, given a prime
ideal p of RG, there is a prime
ideal P of R such that p is a minimal prime
over P ∩ RG. (c) Further results concern
localization. E.g., if R is prime, then RG
has a total ring of fractions which is Artinian and which is
contained in the total ring of fractions of R. This means in
particular that the regular elements of RG are
also regular in R.
These and other results actually characterize linearly reductive
groups: If G is a linear algebraic group which is not
linearly reductive, then a rational action of G on an affine
prime Noetherian PI-algebra R is constructed such
that RG is neither affine nor Noetherian, and
lying over does not hold. This is an important difference to
commutative invariant theory where in prime characteristic most
results can be proven for reductive groups. If one, however, assumes
that R is a finite module over its center, then the above
properties hold in prime characteristic also for actions of
reductive groups.
Finally, the question is studied whether and when one can define a
"map" from the prime spectrum of R to the spectrum
of RG, and what the obstacles are.
This is the published version of:
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Actions of Linearly Reductive Groups on Affine
PI-Algebras
Ph.D.-thesis, Massachusetts Institute of Technology, May 1988.
