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  1. For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of $P^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t = V (I^t)$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for $X$, formulated in terms of the cotangent complex. 
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  2. Let $R = k[x_1,...,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p > 0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H_I^i (R)$ where $I$ is an ideal of $R$. This algorithm has not been implemented because its complexity grows very rapidly with the growth of p which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory. 
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