Extension of Coefficients and Descent

Abstract

The main theme is to discuss the process of coefficient extension for modules, and its reverse, descent. For example, consider a module M over a ring R and let R⟶R′ be an extension of rings. Then extending coefficients from R to R′ on M means that one passes from M to the tensor product M⊗ _R R′, viewed as an R′-module. The chapter starts with the construction of tensor products of general type for modules and their morphisms. Special attention is payed to so-called flat and even faithfully flat extensions. These are particularly well adapted to the formation of kernels, cokernels, and images of module morphisms under the process of coefficient extension. The next step is to look at module properties that are maintained under coefficient extension, or descent. For this to work well, the extension R⟶R′ is assumed to be flat, and even faithfully flat in the descent case. Finally, Grothendieck’s descent theory is studied for modules and their morphisms.