Valuations on algebras with involution
Identifieur interne : 000364 ( Main/Exploration ); précédent : 000363; suivant : 000365Valuations on algebras with involution
Auteurs : J.-P. Tignol [Belgique] ; A. R. Wadsworth [États-Unis]Source :
- Mathematische Annalen [ 0025-5831 ] ; 2011-09-01.
Abstract
Abstract: Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ(x)x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.
Url:
DOI: 10.1007/s00208-010-0580-9
Affiliations:
- Belgique, États-Unis
- Californie, Province du Brabant wallon, Région wallonne
- Louvain-la-Neuve
- Université catholique de Louvain
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Tignol</name><affiliation wicri:level="4"><country xml:lang="fr">Belgique</country><wicri:regionArea>Institute for Information and Communication Technologies, Electronics and Applied Mathematics, Université catholique de Louvain, 1348, Louvain-la-Neuve</wicri:regionArea><orgName type="university">Université catholique de Louvain</orgName><placeName><settlement type="city">Louvain-la-Neuve</settlement><region type="region" nuts="1">Région wallonne</region><region type="province" nuts="1">Province du Brabant wallon</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Belgique</country></affiliation></author><author><name sortKey="Wadsworth, A R" sort="Wadsworth, A R" uniqKey="Wadsworth A" first="A. R." last="Wadsworth">A. R. Wadsworth</name><affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Mathematics, University of California, San Diego, 92093-0112, La Jolla, CA</wicri:regionArea><placeName><region type="state">Californie</region></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">États-Unis</country></affiliation></author></analytic><monogr></monogr><series><title level="j">Mathematische Annalen</title><title level="j" type="abbrev">Math. Ann.</title><idno type="ISSN">0025-5831</idno><idno type="eISSN">1432-1807</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>Berlin/Heidelberg</pubPlace><date type="published" when="2011-09-01">2011-09-01</date><biblScope unit="volume">351</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="109">109</biblScope><biblScope unit="page" to="148">148</biblScope></imprint><idno type="ISSN">0025-5831</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0025-5831</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ(x)x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.</div></front></TEI><affiliations><list><country><li>Belgique</li><li>États-Unis</li></country><region><li>Californie</li><li>Province du Brabant wallon</li><li>Région wallonne</li></region><settlement><li>Louvain-la-Neuve</li></settlement><orgName><li>Université catholique de Louvain</li></orgName></list><tree><country name="Belgique"><region name="Région wallonne"><name sortKey="Tignol, J P" sort="Tignol, J P" uniqKey="Tignol J" first="J.-P." last="Tignol">J.-P. Tignol</name></region><name sortKey="Tignol, J P" sort="Tignol, J P" uniqKey="Tignol J" first="J.-P." last="Tignol">J.-P. Tignol</name></country><country name="États-Unis"><region name="Californie"><name sortKey="Wadsworth, A R" sort="Wadsworth, A R" uniqKey="Wadsworth A" first="A. R." last="Wadsworth">A. R. Wadsworth</name></region><name sortKey="Wadsworth, A R" sort="Wadsworth, A R" uniqKey="Wadsworth A" first="A. R." last="Wadsworth">A. R. Wadsworth</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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