A simple problem on commutative algebra related to G.I.T - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:10:14Z http://mathoverflow.net/feeds/question/73352 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73352/a-simple-problem-on-commutative-algebra-related-to-g-i-t A simple problem on commutative algebra related to G.I.T Xin Nie 2011-08-21T19:28:06Z 2011-08-21T19:28:06Z <p>Let <code>\$G\$</code> be a geometrically reductive algebraic group over an algebraically closed field <code>\$k\$</code>. Let <code>\$X\$</code> be an affine variety over <code>\$k\$</code> on which <code>\$G\$</code> acts regularly. Then <code>\$G\$</code> acts on the coordinate ring <code>\$A\$</code> of <code>\$X\$</code> by automorphisms, and we denote by <code>\$A^G\$</code> the subring consisting of invariant elements.</p> <p><strong>Problem:</strong> Prove that the inclusion \$A^G\subset A\$ induces a surjective map of the ring spectra.</p> <p><strong>Background:</strong> A reffined statement is that for a reducitve group action on an affine variety, the categorical quotient map is surjective.</p> <p>In the book "Lectures on Invariant Theory" by Igor Dolgachev, it is proved that such a categorical quotient is "good" in the sense of G.I.T., but it seems that his proof is incomplete and the probleme above is the missing part.</p>