Galois Descent and Severi-brauer Varieties
نویسنده
چکیده
We say an algebraic object or property over a field k is arithmetic if it becomes trivial or vanishes after finite separable base extension. Since such objects or properties owe their existence to the presence of “arithmetic gaps” in k, i.e., the failure of k to be algebraically closed, we view them as responses to specific arithmetic properties of k, and we study them in order to gain insight into the arithmetic complexity of k, which consists of the features of k responsible for the existence and relative abundance of arithmetic objects and properties.
منابع مشابه
Incompressibility of Products of Weil Transfers of Generalized Severi-brauer Varieties
We generalize the result of [11] on incompressibility of Galois Weil transfer of generalized Severi-Brauer varieties, to direct products of varieties of such type; as shown in [11], this is needed to compute essential dimension of representations of finite groups. We also provide a generalization to non-Galois (separable) Weil transfer.
متن کاملTHE BRAUER-SEVERI VARIETY ASSOCIATED WITH A CENTRAL SIMPLE ALGEBRA: A SURVEY by
— The article describes the one-to-one correspondence between central simple algebras and Brauer-Severi varieties over a field. Non-abelian group cohomology is recalled and Galois descent is worked out in detail. The classifications of central simple algebras as well as of Brauer-Severi varieties by one and the same Galois cohomology set are explained. A whole section is devoted to the discussi...
متن کاملThe Zeta Function of a Class of Rational Varieties
Let k be a field and V be an algebraic fc-scheme of dimension w—1. V is called a Severi-Brauer /;-scheme if there exists a separable algebraic extension L/k such that V xkL and Pw_1(L)=Proj L[XX, • • • , Xn] are isomorphic as /.-schemes [7]. V is said to be split by L/k. V is called a trivial Severi-Brauer ^-scheme if V and -Pw_i(fc) are isomorphic as fcschemes. Let K/k be a finite Galois exten...
متن کاملMorphisms to Brauer–severi Varieties, with Applications to Del Pezzo Surfaces
We classify morphisms from proper varieties to Brauer– Severi varieties, which generalizes the classical correspondence between morphisms to projective space and globally generated invertible sheaves. As an application, we study del Pezzo surfaces of large degree with a view towards Brauer–Severi varieties, and recover classical results on rational points, the Hasse principle, and weak approxim...
متن کاملQuaternionic Algebraic Cycles and Reality
In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic K-theory, and the other one classifies and equivariant cohomology theory Z∗(−) which is a natural recip...
متن کامل