Spectre premier de <f>Oq(Mn(k))</f> image canonique et se´paration normale

Given any commutative field <f>k,</f> denote <f>R=Oq(Mn(k))</f> the coordinate ring of quantum <f>n×n</f> matrices over <f>k</f> and assume <f>q</f> is a nonzero element in <f>k</f> which is not a root of unity. Recall that <f>R</f> is generated by <f>n2</f> variables <f>Xi,α</f> <f>((i,α)∈⟦1,n〉2)</f> subject (only) to the following relations:If <f><fen><cp type="lpar" STYLE="S"><ar><r><c ca="c" CSPAN="1" RSPAN="1" RA="T">x</c><c ca="c" CSPAN="1" RSPAN="1" RA="T">y</c></r><r><c ca="c" CSPAN="1" RSPAN="1" RA="T">z</c><c ca="c" CSPAN="1" RSPAN="1" RA="T">t</c></r></ar><cp type="rpar" STYLE="S"></fen></f> is any <f>2×2</f> sub-matrix of <f>X=(Xi,α),</f> then: (a) <f>yx=q−1xy</f>, <f>zx=q−1xz</f>, <f>tz=q−1zt</f>, <f>ty=q−1yt</f>, <f>zy=yz;</f> (b) <f>tx=xt−(q−q−1)yz.</f>Denote <f><ovl type="bar" STYLE="S">R</ovl></f> the <f>k</f>-algebra generated by the same variables <f>Xi,α</f> subject to the same relations, except relations (b) which are replaced by: (c) <f>tx=xt;</f> so that <f><ovl type="bar" STYLE="S">R</ovl></f> is just the algebra of regular functions on some quantum affine space of dimension <f>n2</f> over <f>k.</f>The theory of “derivative elimination” defines a natural embedding <f>ϕ :Spec(R)→Spec(<ovl type="bar" STYLE="S">R</ovl>)</f> and asserts that: <l type="unord"><li>The “canonical image” <f>ϕ(Spec(R))</f> is a union of strata <f>Specw(<ovl type="bar" STYLE="S">R</ovl>)</f> (in the sense of [Goodearl, Letzter, in: CMS Conf. Proc., Vol. 22 (1998) 39–58]), where <f>w</f> describes some subset <f>W</f> of <f>P(⟦1,n〉2)</f>.</li><li>The sets <f>Specw(R):=ϕ−1(Specw(<ovl type="bar" STYLE="S">R</ovl>))</f> <f>(w∈W)</f> define the Goodearl–Letzter <f>H</f>-stratification of <f>Spec(R)</f> in the sense of [Goodearl, Letzter, Trans. Amer. Math. Soc. 352 (2000) 1381–1403].</li></l>In this paper, we give the precise description of the set <f>W</f> and we compute its cardinality. Using that description and the derivative elimination algorithm, we can verify (Theorems 6.3.1, 6.3.2) that <f>H</f>-<f>Spec(R)</f> has an <f>H</f>-normal separation (in the sense of [Goodearl, in: Lecture Notes in Pure and Appl. Math. 210 (2000) 205–237]), so that <f>Spec(R)</f> 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. Since <f>R</f> is Auslander–Regular and Cohen–Macaulay, this implies (by [Goodearl, Lenagan, J. Pure Appl. Algebra 111 (1996) 123–142]) that <f>R</f> is catenary and satisfies the Tauvel''s height formula. [Copyright &y& Elsevier]