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Let $X$ be a set of $K$-rational points in $P^1 \times P^1$ over a field $K$of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, andlet $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper weinvestigate the module of Kaehler differentials $\Omega^1_{R_Y/K}$. We describethis bigraded $R_Y$-module explicitly via a homogeneous short exact sequenceand compute its Hilbert function in a number of special cases, in particularwhen the support $X$ is a complete intersection or an almost completeintersection in $P^1 \times P^1$. Moreover, we introduce a Kaehler differentfor $Y$ and use it to characterize reduced fat point schemes in $P^1 \timesP^1$ having the Cayley-Bacharach property.

Kaehler differentials for fat point schemes in P^1xP^1