Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ Q= \mathbb{C}_{-1}[x_1,...x_n]$ generated as a graded ring in degree 1 with relations $x_i*x_j=-x_j*x_i$ for $i\neq j$. Set $u_i=x_i^2$ consider central homogeneous functions of the form $f(u_1,...,u_n)$ and the quotient ring R. The question is when, if ever, does Proj(R) have finite homological dimension. By Proj(R) I mean the quotient abelian category Gr(R)/Tor(R), where Gr(R) consists of finitely generated left modules and Tor(R) is the subcategory of torsion modules? Based upon some calculations, I believe the answer should be whenever $C[u_1,...u_n]/(u_idf/du_i)$ is finite dimensional.This is a condition about how the zero locus of f intersecting the coordinate axes $u_i$, which I believe should have a geometric interpretation in terms of the map $Z(Q)=C[u_1,...u_n] \mapsto Q$ when the $u_i$ are not zero, the fibers are matrix algebras which degenerate at the locus when some of the $u_i=0$. By analogy with the ordinary projective space, I'd like to think of this as a smooth hypersurface in the quantum projective space $P_{-1}^{n-1}$ but I can't seem to find this type of stuff analyzed anywhere. Does this ring a bell for anyone? If not, does anyone understand the map on primes $Spec(Q)\mapsto Spec Z(Q)$?
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Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $ k[x_1,...x_n]$ Q= \mathbb{C}_{-1}[x_1,...x_n]$ as a graded ring in degree 1 with relations $x_i*x_j=-x_j*x_i$ for $i\neq j$. Now Set $u_i=x_i^2$ consider central homogeneous functions of the form $x_1^{2m} \pm x_2^{2m}... \pm x_n^{2m}$ f(u_1,...,u_n)$ and the quotient ring R. The question is when, if ever, does Proj(R) have finite homological dimension. By Proj(R) I mean the quotient abelian category Gr(R)/Tor(R), where Gr(R) consists of finitely generated left modules and Tor(R) is the subcategory of torsion modules? Based upon some calculations, I believe the answer should be whenever $C[u_1,...u_n]/(u_idf/du_i)$ is finite dimensional.This is a condition about how the zero locus of f intersecting the coordinate axes $u_i$, which I believe should have a geometric interpretation in terms of the map $Z(Q)=C[u_1,...u_n] \mapsto Q$ when the $u_i$ are not zero, the fibers are matrix algebras which degenerate at the locus when some of the $u_i=0$. By analogy with the ordinary projective space, I'd like to think of this as a smooth hypersurface in the quantum projective space $P_q^{n-1}$ P_{-1}^{n-1}$ but I can't seem to find this type of stuff analyzed anywhere..anywhere. Does this ring a bell for anyone? If not, does anyone understand the map on primes $Spec(Q)\mapsto Spec Z(Q)$? |
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Something I've been thinking about lately has led me to wonder about the following. Consider the quantum polynomial ring $k[x_1,...x_n]$ as a graded ring in degree 1 with relations $x_i*x_j=-x_j*x_i$ for $i\neq j$. Now consider homogeneous functions of the form $x_1^{2m} \pm x_2^{2m}... \pm x_n^{2m}$ and the quotient ring R. The question is when, if ever,is Proj(R)= ever, does Proj(R) have finite homological dimension. By Proj(R) I mean the quotient abelian category Gr(R)/Tor(R), where Gr(R)= Gr(R) consists of finitely generated left modules and Tor(R) is the subcategory of torsion moduleshave finite homological dimension? By analogy with the ordinary problemprojective space, I'd like to think of this as a smooth hypersurface in this the quantum projective space $P_q^{n-1}$ but I can't seem to find this type of stuff analyzed anywhere... |
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