Separable algebras and coflasque resolutions Ballard, Matthew R.; Duncan, Alexander; Lamarche, Alicia ...
Advances in mathematics (New York. 1965),
20/May , Volume:
444
Journal Article
Peer reviewed
Open access
Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between ...Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories.
Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.
We study the behavior of modules of m-integrable derivations of a commutative finitely generated algebra in the sense of Hasse-Schmidt under base change. We focus on the case of separable ring ...extensions over a field of positive characteristic and on the case where the extension is a polynomial ring in an arbitrary number of variables.
We provide a characterization of finite \'etale morphisms in tensor
triangular geometry. They are precisely those functors which have a
conservative right adjoint, satisfy Grothendieck--Neeman ...duality, and for which
the relative dualizing object is trivial (via a canonically-defined map).
In this work, we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a
$\mathbb{Z}$
-graded ideal
$I\subseteq R=\Bbbk x_{1},\ldots ,x_{n}$
. We also ...prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for the Lyubeznik table. For the case of squarefree monomial ideals, we get more insight into the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley–Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field.
We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field
F. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen
...K-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface.
After a classical presentation of quadratic mappings and Clifford algebras over arbitrary rings (commutative, associative, with unit), other topics involve more original methods: interior ...multiplications allow an effective treatment of deformations of Clifford algebras, the relations between automorphisms of quadratic forms and Clifford algebras are based on the concept of the Lipschitz monoid, from which several groups are derived, and the Cartan-Chevalley theory of hyperbolic spaces becomes much more general, precise and effective.
Let
k
be a field and
A
be a separable
k
-algebra. Let
r
≥ 3 be an integer. Generalizing a result of Reichstein (Arch. Math.
88
(
2007
), 12–18), we prove that the essential dimension over
k
of
A
...equals that of the
r
-fold trace form
.
We prove that every separable algebra over an infinite field
F
admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix ...algebras over
F
and for group algebras
FG
, where
G
is a finite group such that the order of
G
is invertible in
F
. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (
A
,
B
) and (
A
′,
B
′) with entries in a commutative ring
R
are automorphically conjugate over the matrix algebra
M
2
(
R
), under an additional assumption that both pairs generate
M
2
(
R
) as an
R
-algebra.
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