arXiv subjects
Publications and source records attributed to Philippe Gille.
We prove Springer's Odd Degree Theorem for quadratic forms over LG rings, and Scharlau's and Knebusch's norm principles for quadratic forms over semilocal rings. We present applications to the flat cohomology of spin groups and {\'e}tale norm groups.
Let $k$ be a field, and let $G$ be a simply connected semisimple k-group which is isotropic and contains a strictly proper parabolic $k$-subgroup $P$. Let $D$ be a discrete valuation ring which is a local ring of a smooth algebraic curve over $k$. Let $K$ be the fraction field of $D$. We show that the corresponding non-stable $K_1$-functor (for $G$ and $P$, also called the Whitehead group of $G$) coincide over $D$ and $K$. As a consequence, $K^G_1 (D)$ coincides with the (generalized) Manin's $R$-equivalence class group of $G(D)$.
Let $K$ be a complete discretely valued field with residue field $\bar K$ of dimension $1$ (not necessarily perfect). This occurs if and only if $K$ has dimension $2$. We prove the following statements on the arithmetic of such fields: - The "period equals index" property holds for central simple $K$-algebras. - For every prime $p$, every class in the Milnor $\mathrm{K}$-theory modulo $p$ is represented by a symbol. - Serre's Conjecture II holds for the field $K$. That is, for every semisimple and simply connected $K$-group $G$, the set $H^1(K,G)$ is trivial.
Loop torsors over Laurent polynomial rings in characteristic 0 were originally introduced in relation to infinite dimensional Lie theory. Applications to other areas require a theory that can yields results in positive characteristic, and for group schemes that are not of finite type. The relation between loop and so-called toral torsors, is one of the central questions in the area. The present paper addresses this question in full generality.
We prove several results on reductive group schemes over LG-rings, e.g., existence of maximal tori and conjugacy of parabolic subgroups. These were proven in SGA3 for the special case of semilocal rings. We apply these results to establish cancellation theorems for hermitian and quadratic forms over LG-rings and show that the Brauer classes of Azumaya algebras over connected LG-rings have a unique representative and allow Brauer decomposition.
We compare different local-global principles for torsors under a reductive group G defined over a semiglobal field F. In particular if the F-group G s a retract rational F-variety, we prove that the local global principle holds for the completions with respect to divisorial valuations of F.
We define the notion of loop torsors under certain group schemes defined over the localization of a regular henselian ring A at a strict normal crossing divisor D. We provide a Galois cohomological criterion for classifying those torsors. We revisit also the related theory of loop torsors on Laurent polynomial rings.
For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this power operation has many good properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $\xi^{\Diamond n}$ then coincides with the $n$-th power of $\xi$ in this group. On the other hand, we show that a power operation on $H^1(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using this power operation, for a cohomology class $\xi$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(\xi)$ to be the least integer $n\ge 1$ such that $\xi^{\Diamond n}=1$. We define the index ${\rm ind}(\xi)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi$. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ defined over a local or global field $K$, we show that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$ and that ${\rm ind}(\xi)$ may be strictly greater than ${\rm per}(\xi)$, but they always have the same prime factors.
We construct a globalization of Ferrand's norm functor over rings which generalizes it to the setting of a finite locally free morphism of schemes $T\to S$ of constant rank. It sends quasi-coherent modules over $T$ to quasi-coherent modules over $S$. These functors restrict to the category of quasi-coherent algebras. We also assemble these functors into a norm morphism from the stack of quasi-coherent modules over a finite locally free of constant rank extension of the base scheme into the stack of quasi-coherent modules. This morphism also restricts to the analogous stacks of algebras. Restricting our attention to finite \'etale covers, we give a cohomological description of the norm morphism in terms of the Segre embedding. Using this cohomological description, we show that the norm gives an equivalence of stacks of algebras $A_1^2 \equiv D_2$, akin to the result shown in The Book of Involutions.
We provide a criterion for certain algebraic objects over Jacobson schemes to be forms of each other based on their behaviour at closed fibres. This criterion permits to answer a question that I. Burban had asked the authors.
We define the notion of reductive group schemes defined over the localization of a regular henselian ring A at a strict normal crossing divisor $D$. We provide a criterion for the existence for parabolic subgroups of a given type.
We provide a characterization of homogeneous spaces under a reductive group scheme such that the geometric stabilizers are maximal tori. The quasi-split case over a semilocal base is of special interest and permits to answer a question raised by Marc Levine on homogeneous SL$_n$-spaces. At the end, we provide an application to the local-global principles for embeddings of \'etale algebras with involution into central simple algebras with involution.
We prove a local-global principle for twisted flag varieties over a semiglobal field.
We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme S as defined by Calm{\`e}s and Fasel, generalizing the case of quadratic pairs on central simple algebras over a field (Knus, Merkurjev, Rost, Tignol). We describe a cohomological obstruction for an Azumaya algebra over S with orthogonal involution to admit a quadratic pair. When S is affine this obstruction vanishes, however it is non-trivial in general. In particular, we construct explicit examples with non-trivial obstructions.
We define R-equivalence for group schemes over a semilocal ring and relate this with rational properties. Two main cases are investigated: tori and isotropic semisimple simply connected group schemes where we show in certain cases that R-equivalence coincide with Karoubi-Villamayor equivalence and is also related to the Kneser-Tits problem in this setting. Finally we construct specialization maps for R-equivalence in the case of regular algebras containing a field.
A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is {é}tale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.
We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite {é}tale cover.
We investigate topological properties of torsors in algebraic geometry over adelic rings.