Decomposition of the Kähler differentials

Abstract

This thesis is concerned with the module of Kähler differentials of an affine scheme and its primary decomposition. For a smooth scheme, these are the local building blocks of the cotangent bundle. Each inclusion of a closed subscheme induces a reversed map of Kähler differentials. We aim to study the kernel of this map, as well as its primary decomposition. First we review Kähler differentials, then we review primary decompositions. Next we relate the primary decomposition of the closed subscheme ideal to the kernel of the map of differentials. Lastly, we look at the geometric aspect of the theory of differentials. We will be working in the case of curves in the affine plane, but we expect that many of our findings remain true for more general schemes and their closed subschemes.