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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? |
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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. |
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