Let $X$ be an algebraic variety. Let $\mathcal I_{\Delta}\subset\mathcal O_{X\times_kX}$ be the ideal sheaf defining the diagonal $\Delta\subset X\times_kX$. Regard $\mathcal O_{X\times_kX}/\mathcal I^{n+1}$ and $\mathcal I/\mathcal I^{n+1}$ as $\mathcal O_X$-modules through the first projection.

The question is: Are these $\mathcal O_X$-modules locally free of constant rank over some open dense set of $X$? When $n=1$ is well known, what about for n>1?


1 Answer 1


They are locally free of constant rank over the smooth locus $U$ of $X$, which is dense and open if $X$ is reduced and irreducible. (I have taken the phrase "variety over a field $k$" to mean: of finite type over $k$, reduced and absolutely irreducible.) The reason, starting from $n=1$, is that $\mathcal O_{X\times X}/\mathcal I^{n+1}$ has a filtration whose graded pieces are $\mathcal I^r/\mathcal I^{r+1}$, which is isomorphic over $U$ to the symmetric product $Symm^r(\mathcal I/\mathcal I^2)$ and, as you say, $\mathcal I/\mathcal I^2$ is locally free there.

  • $\begingroup$ I think he wanted to look at the direct image of these sheaves with respect to the first projection, isn't it ? $\endgroup$
    – diverietti
    Jan 13, 2012 at 9:36
  • 2
    $\begingroup$ @diverietti -- What inkspot says is correct. The direct image under the first projection is locally free over any open subset of the target copy of $X$ which is smooth over $k$. $\endgroup$ Jan 13, 2012 at 14:13
  • $\begingroup$ Indeed, when one says that $\mathcal I/\mathcal I^{n+1}$ is a $\mathcal O_X$-module, in fact one means that $p_∗\mathcal I/\mathcal I^{n+1}$ is a $\mathcal O_X$-module, where $p$ is the first projection. Anyway, thank you for the answer and the comments! $\endgroup$
    – user20544
    Jan 16, 2012 at 8:31

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