Derived products and completion defects of inverse limits

Abstract

We compute derived products and inverse limits in Grothendieck categories by exact coresolutions. For the product of $r$ doubled affine lines over $B[s]$, where $B$ is any nonzero commutative ring,the exact dimensions of products, epimorphic countable limits, and arbitrary countable limits are $r$, $r$, and $r+1$. We identify the top derived functors for every $B$-module of coefficients. Both obstruction modules contain the countable reduced product as a natural $B$-module direct summand, and a formal-series map relates them by an explicit cokernel sequence. A relative cofree coresolution gives degree shifts over an arbitrary Grothendieck category and computes the localization defect for arbitrary locally nilpotent diagrams. We also construct transfinite scalar extensions preserving a prescribed kernel and its derived limit;these give nonzero completion defects beyond any fixed ordinal.The general epimorphic top-degree vanishing question is not settled by these constructions.