Topic
Regular extension
About: Regular extension is a research topic. Over the lifetime, 31 publications have been published within this topic receiving 317 citations.
Papers
TL;DR: In this paper, the authors consider the problem of finding rational functions over a regular extension of a quotient variety, where the objective is to find a rational function that is G-invariant (i.e., constant on orbits) within an isomorphism.
Abstract: We recall that if an algebraic group G operates regularly on a variety V, by a quotient variety is meant a pair (V/G, r), where V/G is a variety and r: V-* V/G is a rational map, everywhere defined and surjective, such that two points of V have the same image under r if and only if they have the same orbit on V, and such that, for any xE V, any rational function on V that is G-invariant (i.e., constant on orbits) and defined at x is actually (under the natural injection of function fields Q(V/G) -*Q(V), Q denoting the universal domain) a rational function on VIG that is defined at rx (cf. [1, expose 8]). Q(V/G) must therefore consist precisely of all G-invariant elements of Q(V), so r is separable. A quotient variety need not exist (obvious necessary condition: all orbits on V must be closed), but when it exists it is clearly unique to within an isomorphism; in this case, for any open subset UC V/G, T-1 U/G exists and equals U. PROPOSITION 1. Let the algebraic group G operate regularly on the variety V, all defined over the field k. Suppose there exists a quotient variety r: V-* V/G. Suppose also that for each point p of V that is algebraic over k there exists an open affine subset of V/G containing the image under r of each of the conjugates of p over k (a vacuous condition if V/G. can. be embedded in a projective space or if V/G and r are known to be defined over a regular extension of k, in particular if k is algebraically closed). Then V/G and r could have been taken so as to be defined over k. The G-invariant elements of Q(V) are generated by those in k(V), in other words there exists a variety W and a generically surjective rational map V-)W, both defined over k, such that for any field K between k and Q, K(W) is the field of G-invariant elements of K(V) [3, Theorem 2]. We have here a field descent problem, and supposing V/G and r to be defined over the extension field K of k, there are two cases to consider: K a regular extension of k, and K algebraic over k. In view of the unicity to within isomorphism of the quotient variety, the criteria of Weil [6] take care of the first case. [Of course this can also be done directly; e.g., supposing k algebraically closed, if
117 citations
TL;DR: The Harbater/Pop theorem for algebraically closed groups was proved in this article, where it was shown that the theorem holds over any large k-variety with a k-point.
Abstract: Let k be a p-adic field. Some time ago, D. Harbater [9] proved that anyfinite group G may be realized as a regular Galois group over the rationalfunction field in one variable k(t), namely there exists a finite field extensionF/k(t), Galois with group G, such that F is a regular extension of k (i.e. kis algebraically closed in F). Moreover, one may arrange that a given k-placeof k(t) be totally split in F. Harbater proved this theorem for k an arbitrarycomplete valued field. Rather formal arguments ([10, §4.5]; §2 hereafter) thenimply that the theorem holds over any ‘large’ field k. This in turn is a specialcase of a result of Pop [15], hence will be referred to as the Harbater/Poptheorem. We refer to [10], [16], [6] for precise references to the literature (workof D`ebes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, andV¨olklein).Most proofs (see [10], [19, 8.4.4, p. 93] and Liu’s contribution to [16]; seehowever [15]) first use direct arguments to establish the theorem when G is acyclic group (here the nature of the ground field is irrelevant), then proceed bypatching, using either formal or rigid geometry, together with GAGA theorems.In the present paper, where I take the case of algebraically closed fieldsfor granted, I show how a technique recently developed by Kolla´r [12] may beused to give a quite different proof of the Harbater/Pop theorem, when the‘large’ field k has characteristic zero. This proof actually gives more than theoriginal result (see comment after statement of Theorem 1).Before I formally state the main result, let us recall what a ‘large’ field is.Let k be a field and let k((y)) be the quotient field of the ring k[[y]] of formalpower series in one variable. Following F. Pop, we shall say that k is ‘large’ ifit satisfies one of the three equivalent properties ([15, Prop. 1.1]):(i) It is existentially closed in k((y)): any k-variety with a k((y))-point hasa k-point.(ii) On a smooth integral k-variety with a k-point, k-points are Zariski dense.(iii) On a smooth integral k-curve with a k-point, k-points are Zariski dense.
45 citations
Posted Content•
TL;DR: The Harbater/Pop theorem for algebraically closed fields was shown to hold over any large field k as mentioned in this paper, where k is a p-adic field with characteristic zero.
Abstract: Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$, Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10, §4.5]; \S2 hereafter) then imply that the theorem holds over any `large' field k. This in turn is a special case of a result of Pop [15], hence will be referred to as the Harbater/Pop theorem. We refer to [10], [16], [6] for precise references to the literature (work of Debes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, and Volklein).
Most proofs (see [10], [19, 8.4.4, p.~93] and Liu's contribution to [16]; see however [15]) first use direct arguments to establish the theorem when G is a cyclic group (here the nature of the ground field is irrelevant), then proceed by patching, using either formal or rigid geometry, together with GAGA theorems.
In the present paper, where I take the case of algebraically closed fields for granted, I show how a technique recently developed by Kollar [12] may be used to give a quite different proof of the Harbater/Pop theorem, when the `large' field k has characteristic zero. This proof actually gives more than the original result (see comment after statement of Theorem 1).
14 citations
Posted Content•
2 Mar 2007TL;DR: In this paper, it was shown that the rank of this group can be computed in terms of certain ranks of the Neron-Severi groups. But this is not a proof without heights of the Lang-Neron theorem: if $K/k$ is a regular extension of finite type and $A$ is an abelian $K$-variety, the group $A(K)/\Tr_{K/K} A(k)$ is finitely generated, where $\Tr_{k/k} A$ denotes the $K
Abstract: We give a proof without heights of the Lang-Neron theorem: if $K/k$ is a regular extension of finite type and $A$ is an abelian $K$-variety, the group $A(K)/\Tr_{K/k} A(k)$ is finitely generated, where $\Tr_{K/k} A$ denotes the $K/k$-trace of $A$ in the sense of Chow. Our method computes the rank of this group in terms of certain ranks of Neron-Severi groups.
10 citations
TL;DR: In this paper, the authors studied simple representations of U'(L) which may be topologized via the Ziegler topology on the set of injective indecomposable representations and via the Jacobson topology of primitive ideals.
Abstract: Let k be an algebraically closed field of characteristic zero and L = sl(2,k) the Lie algebra of 2 × 2 traceless matrices over k. It is shown that there exists a von Neumann regular extension \( U(L) \subseteq U'(L) \) of the universal enveloping algebra, which is an epimorphism in the category of rings. The article is devoted to the study of the simple representations of U'(L), which may be topologized via the Ziegler topology on the set of injective indecomposable representations of U'(L) or via the Jacobson topology on the set of primitive ideals. These two topologies coincide and the finite dimensional simple representations of L form a dense, discrete and open subset. The field of fractions K(L) of the universal enveloping algebra is another simple representation of U'(L). If the point K(L) is removed from the Ziegler spectrum of U'(L), one obtains a compact totally disconnected topological space, which has the cardinality of the continuum. It is also shown that the lattice of ideals of U'(L) is isomorphic to the lattice of open subsets. The epimorphic ring extension \( U(L) \subseteq U'(L) \) is used to find an axiomatization of the finite dimensional representations of L in the language of left U(L)-modules. A representation V of L is called pseudo-finite dimensional if it satisfies these axioms. It is shown that a representation V of L is pseudo-finite dimensional if and only if for every central idempotent \( e \in U'(L) \) for which \( eK(L)
eq 0 \), whenever the subrepresentation eV is nonzero, then it has a nonzero highest weight space.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 2 |
| 2019 | 2 |
| 2018 | 2 |
| 2016 | 1 |
| 2010 | 1 |