The Stacks project

Example 52.28.3. Let $k$ be a field and let $X$ be a proper variety over $k$. Let $Y \subset X$ be an effective Cartier divisor such that $\mathcal{O}_ X(Y)$ is ample and denote $Y_ n$ its $n$th infinitesimal neighbourhood. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Here are some special cases of Proposition 52.28.1.

  1. If $X$ is a curve, we don't learn anything.

  2. If $X$ is a Cohen-Macaulay (for example normal) surface, then

    \[ H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n}) \]

    is an isomorphism.

  3. If $X$ is a Cohen-Macaulay threefold, then

    \[ H^0(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^0(Y_ n, \mathcal{E}|_{Y_ n}) \quad \text{and}\quad H^1(X, \mathcal{E}) \to \mathop{\mathrm{lim}}\nolimits H^1(Y_ n, \mathcal{E}|_{Y_ n}) \]

    are isomorphisms.

Presumably the pattern is clear. If $X$ is a normal threefold, then we can conclude the result for $H^0$ but not for $H^1$.

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