Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:10:44Z http://mathoverflow.net/feeds/question/43581 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43581/are-schematic-fixed-points-of-a-cohen-macaulay-scheme-cohen-macaulay Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay? Ben Webster 2010-10-25T21:30:31Z 2010-10-27T05:55:40Z <p>I'm not sure how long this <a href="http://mathoverflow.net/questions/43438/" rel="nofollow">iterative</a> <a href="http://mathoverflow.net/questions/43255/" rel="nofollow">questions</a> can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the coordinate ring). Are the schematic fixed points $X^{\mathbb{G}_m}$ of $X$ Cohen-Macaulay? </p> http://mathoverflow.net/questions/43581/are-schematic-fixed-points-of-a-cohen-macaulay-scheme-cohen-macaulay/43588#43588 Answer by Hailong Dao for Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay? Hailong Dao 2010-10-25T22:32:45Z 2010-10-26T05:48:51Z <p>Edit: the following does not answer Ben's question. It gives an example of the <em>subring</em> fixed by $G_m$ being not CM, while the question asked about the <em>subscheme</em> of fixed points, see the comments for more details. </p> <p>Let $R$ be the (homogenous) cone of a curve $C$ of genus $g>0$, for example $R=\mathbb C[x,y,z]/(x^3+y^3+z^3)$. Let $S=R[u,v]$, $X=\text{Spec}(S)$ and $G_m$ acts by </p> <p>$a.(x,y,z,u,v) = (ax,ay,az,a^{-1}u, a^{-1}v)$. </p> <p>Then $A= S^{G_m}$ would be a homogenous coordinate ring for $Y= C\times \mathbb P^1$, so it is not Cohen-Macaulay (if $A$ is CM, it would mean that $H^1(Y,\mathcal O_Y)=0$, impossible, see <a href="http://mathoverflow.net/questions/1652/simple-example-of-a-ring-which-is-normal-but-not-cm/2194#2194" rel="nofollow">here</a> for an explanation).</p> <p>(I learned this idea from Hochster, let me try to find a reference)</p> http://mathoverflow.net/questions/43581/are-schematic-fixed-points-of-a-cohen-macaulay-scheme-cohen-macaulay/43766#43766 Answer by Angelo for Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay? Angelo 2010-10-27T05:55:40Z 2010-10-27T05:55:40Z <p>Here is a counterexample. Consider the action of $\mathbb G_{\rm m}$ on $\mathbb A^4$ defined by $t \cdot(x,y,z,w) = (x, y, tz, t^{-1}w)$, and let $X$ be the invariant closed subscheme with ideal $(xy, y^2 + zw)$; this is a complete intersection, hence it is Cohen-Macaulay. The fixed point subscheme is obtained by intersecting with the fixed point subscheme in $\mathbb A^4$, which is given by $z = w = 0$; hence it is the subscheme of $\mathbb A^2$ given by $xy = y^2 = 0$, which is of course the canonical example of a non Cohen-Macaulay scheme.</p> <p>Developing this idea a little, one can show that any kind of horrible singularity can appear in the fixed point subscheme of a $\mathbb G_{\rm m}$-action on a complete intersection variety.</p>