1. Spectre premier de <f>Oq(Mn(k))</f> image canonique et se´paration normale
- Author
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Cauchon, Gérard
- Subjects
- *
MATRICES (Mathematics) , *ALGEBRA - Abstract
Given any commutative field
k, denoteR=Oq(Mn(k)) the coordinate ring of quantumn×n matrices overk and assumeq is a nonzero element ink which is not a root of unity. Recall thatR is generated byn2 variablesXi,α ((i,α)∈⟦1,n〉2) subject (only) to the following relations:Ifx y z t 2×2 sub-matrix ofX=(Xi,α), then: (a)yx=q−1xy ,zx=q−1xz ,tz=q−1zt ,ty=q−1yt ,zy=yz; (b)tx=xt−(q−q−1)yz. DenoteR k -algebra generated by the same variablesXi,α subject to the same relations, except relations (b) which are replaced by: (c)tx=xt; so thatR n2 overk. The theory of “derivative elimination” defines a natural embeddingϕ :Spec(R)→Spec( and asserts that:R )- The “canonical image”
ϕ(Spec(R)) is a union of strataSpecw( (in the sense of [Goodearl, Letzter, in: CMS Conf. Proc., Vol. 22 (1998) 39–58]), whereR )w describes some subsetW ofP(⟦1,n〉2) .- The sets
Specw(R):=ϕ−1(Specw( R ))(w∈W) define the Goodearl–LetzterH -stratification ofSpec(R) in the sense of [Goodearl, Letzter, Trans. Amer. Math. Soc. 352 (2000) 1381–1403].W and we compute its cardinality. Using that description and the derivative elimination algorithm, we can verify (Theorems 6.3.1, 6.3.2) thatH -Spec(R) has anH -normal separation (in the sense of [Goodearl, in: Lecture Notes in Pure and Appl. Math. 210 (2000) 205–237]), so thatSpec(R) has normal separation (in the sense of [Brown, Goodearl, Trans. Amer. Math. Soc. 348 (1996) 2465–2502]). This property was conjectured by K. Brown and K. Goodearl. SinceR is Auslander–Regular and Cohen–Macaulay, this implies (by [Goodearl, Lenagan, J. Pure Appl. Algebra 111 (1996) 123–142]) thatR is catenary and satisfies the Tauvel''s height formula. [Copyright &y& Elsevier]- The “canonical image”
- Published
- 2003
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