Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular affine algebraic subgroup $H \subseteq G$ is a complete intersection in $G$? Or are such subgroups always complete intersections? (I would imagine not, but I don't know any counterexamples.) If it helps I'm happy to assume that $G$ is semisimple.

EDIT: I should probably say "smooth k-group scheme" here instead of "nonsingular."

  • $\begingroup$ Can you explain the use of "nonsingular" in the context of algebraic $k$-groups? What would a "singular" example look like? $\endgroup$ – Jim Humphreys Oct 13 '13 at 23:34
  • $\begingroup$ I want to rule out things like infinitesimal groups (eg Frobenius kernels). Is "smooth" a better term? $\endgroup$ – Chuck Hague Oct 14 '13 at 1:18
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    $\begingroup$ I think the minimal parabolic subgroup of $PGL_2 = SO_3$ is an example of a non-complete intersection. Any function which cut it out would pull back to a function that cuts out the subgroup of upper triangular matrices in $GL_2$, which would have to be the coordinate of that matric entry times a power of the determinant. But none of those functions are invariant to the action of the diagonal. $\endgroup$ – Will Sawin Oct 14 '13 at 3:06
  • $\begingroup$ @Chuck: The usual terminology for group schemes uses the distinction of being "reduced" (no nilpotents in structure sheaf), in which case the group of points over an algebraically closed field usually carries the same information. Frobenius kernels are non-reduced. But "smooth" or "nonsingular" comes up in the more classical geometric theory of varieties. $\endgroup$ – Jim Humphreys Oct 14 '13 at 12:47

This answer is just to record that Will Sawin's example works: The Borel subgroup of $PGL_2$ is not a complete intersection in $PGL_2$. Recall that the coordinate ring of $GL_2$ is $k[w,x,y,z,\Delta^{-1}]$ where $\Delta=wz-xy$. We are thinking of $w$, $x$, $y$, $z$ as entries of the matrix $\left( \begin{smallmatrix} w & x \\ y & z \end{smallmatrix} \right)$. The coordinate ring of $PGL_2$ is the subring of homogenous elements, where $w$, $x$, $y$ and $z$ are graded in degree $1$ and hence $\Delta^{-1}$ has degree $-2$.

The Borel is given by the equations $wy/\Delta=xy/\Delta=y^2/\Delta=yz/\Delta=0$. Conceptually, the defining equation is $y=0$, but the element $y$ isn't in the coordinate ring of $PGL_2$. Since the Borel is a hypersurface, if it were a complete intersection, it would be defined by the vanishing of a single equation, say $g(w,x,y,z)/\Delta^k$, where $g$ is a homogeneous polynomial of degree $2k$. We may also assume that $\Delta$ does not divide $g$. Since $g$ has even degree, it must either vanish to even order along $y=0$, or else vanish along some other hypersurface besides $y=0$. Either way, $g/\Delta^k$ does not define the ideal of the Borel.

  • $\begingroup$ Great, this is a helpful example - thanks! $\endgroup$ – Chuck Hague Oct 14 '13 at 15:37

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