Degree 3 relative invariant for unitary involutions

IF 0.5 Q3 MATHEMATICS
Demba Barry, Alexandre Masquelein, Anne Qu'eguiner-Mathieu
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引用次数: 0

Abstract

. Using the Rost invariant for non split simply connected groups, we define a relative degree 3 cohomological invariant for pairs of orthogonal or unitary involutions having isomorphic Clifford or discriminant algebras. The main purpose of this paper is to study general properties of this invariant in the unitary case, that is for torsors under groups of outer type A . If the underlying algebra is split, it can be reinterpreted in terms of the Arason invariant of quadratic forms, using the trace form of a hermitian form. When the algebra with unitary involution has a symplectic or orthogonal descent, or a symplectic or orthogonal quadratic extension, we provide comparison theorems between the corresponding invariants of unitary and orthogonal or symplectic types. We also prove the relative invariant is classifying in degree 4, at least up to conjugation by the non-trivial automorphism of the underlying quadratic extension. In general, choosing a particular base point, the relative invariant also produces absolute Arason invariants, under some additional condition on the underlying algebra. Notably, if the algebra has even co-index, so that it admits a hyperbolic involution, which is unique up to isomorphism, we get a so-called hyperbolic Arason invariant. Assuming in addition the algebra has degree 8, we may also define a decomposable Arason invariant. It generally does not coincide with the hyperbolic Arason invariant, as the hyperbolic involution need not be totally decomposable.
单位对合的3次相对不变量
.使用非分裂单连通群的Rost不变量,我们定义了具有同构Cliff ord或判别代数的正交或酉对合对的相对度3上同调不变量。本文的主要目的是研究这种不变量在酉情形下的一般性质,即在外型A群下的扭子。如果底层代数是分裂的,它可以用二次形式的Arason不变量重新解释,使用hermitian形式的迹形式。当具有酉对合的代数具有辛或正交下降,或辛或正交二次扩张时,我们给出了酉型、正交型或辛型对应不变量之间的比较定理。我们还证明了相对不变量的分类为4次,至少直到通过下面的二次扩张的非平凡自同构共轭。一般来说,在底层代数上的一些附加条件下,选择一个特定的基点,相对不变量也会产生绝对Arason不变量。值得注意的是,如果代数具有偶数共索引,从而允许双曲对合,这在同构之前是唯一的,我们得到了所谓的双曲Arason不变量。此外,假设代数的阶数为8,我们还可以定义一个可分解的Arason不变量。它通常与双曲Arason不变量不一致,因为双曲对合不需要是完全可分解的。
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来源期刊
Annals of K-Theory
Annals of K-Theory MATHEMATICS-
CiteScore
1.10
自引率
0.00%
发文量
12
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