Algebra A with Spec(A) reduced and Rep_n(A) non-reduced - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T11:56:03Z http://mathoverflow.net/feeds/question/21380 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21380/algebra-a-with-speca-reduced-and-rep-na-non-reduced Algebra A with Spec(A) reduced and Rep_n(A) non-reduced Peter Samuelson 2010-04-14T19:46:47Z 2010-04-14T19:46:47Z <p>As always, corrections to my misconceptions/misstatements are appreciated. This question is related to the following one, but in this question the algebras considered are commutative: <a href="http://mathoverflow.net/questions/9738/non-smooth-algebra-with-smooth-representation-variety" rel="nofollow">http://mathoverflow.net/questions/9738/non-smooth-algebra-with-smooth-representation-variety</a></p> <p>The "commuting scheme" X is defined intuitively as "pairs of nxn commuting matrices." More precisely, it's the subscheme of the affine space $M_n(\mathbb{C})\times M_n(\mathbb{C})$ defined by the equations in the entries of the matrices corresponding to the matrix equations "XY = YX," which are $n^2$ homogenous equations of degree n. To the best of my knowledge, the question of whether X is reduced is an old problem that is still open. </p> <p>The commuting scheme is naturally isomorphic to $Rep^n_\mathbb{C}(\mathbb{C}[x,y])$. This leads to my question: Is there an algebra A such that $Spec(A)$ is reduced but $Rep_\mathbb{C}^n(A)$ is not reduced? Even stronger, is there an A with $Spec(A)$ smooth and $Rep_\mathbb{C}^n(A)$ non-reduced? My guess would be yes for both, but I'm not sure how to find one for either one.</p>