Quantum polynomial rings and singularities - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:07:06Z http://mathoverflow.net/feeds/question/45903 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45903/quantum-polynomial-rings-and-singularities Quantum polynomial rings and singularities Daniel Pomerleano 2010-11-13T08:48:08Z 2010-11-21T19:25:48Z <p>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)\$?</p>