In an earlier paper [KRS] we defined and investigated the properties of the na¨ıve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one na¨ıvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noeth...

Let R be a commutative Noetherian ring of prime characteristic p. Assume R (or, more generally, a finitely generated R-module N with SuppR(N) = Spec(R)) has finite F-representation type (abbreviated FFRT) by finitely generated R-modules. Then, for every c ∈ R◦, there is a uniform test exponent Q = p for c and for all R-modules. As a consequence, we show the existence of uniform test exponents o...

It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised. The flat...

Residue complexes were introduced by Grothendieck in algebraic geometry. These are canonical complexes of injective modules that enjoy remarkable functorial properties (traces). In this paper we study residue complexes over noncommutative rings. These objects are even more complicated than in the commutative case, since they are complexes of bimodules. We develop methods to prove uniqueness, ex...