A note on the Matlis category equivalence

Abstract

We provide additional information concerning the important Matlis category equivalence between the categories of h-divisible torsion and R-complete torsion-free R-modules over any domain R. We identify the h-divisible modules that correspond under this category equivalence to the Enochs cotorsion torsion-free modules as the weak-injective modules. As a consequence, we can characterize weak-injective modules in terms of their flat covers. We also determine the range of the injective dimensions of pure-injective modules.

h-Divisible modules over integral domains

“…Suppose that in the above diagram we start with the bottom exact sequence with (W , γ ) as the weak-injective envelope of M; the existence of such an exact sequence is guaranteed by [9] or [7]. Next we take the flat cover (E, β) of W ; in view of [10], flat covers of weak-injectives are torsion-free injectives, so E is torsion-free injective. Let H = Ker β, and define α : F → M as the restriction of β.…”

On weak-injective modules over integral domains

“…Lee [10] proved that the statement that all weak-injective R-modules are pure-injective is equivalent to saying that all torsion-free Enochs-cotorsion modules are pure-injective by showing that over a domain R, in the Matlis category equivalence, the weak-injective (resp. the h-divisible pure-injective) torsion modules D correspond to the Enochs-cotorsion (resp.…”

“…Hence E 2 = 0 = W 2 , and W is the weak-injective envelope of A. Conversely, suppose that in the above diagram we start with the bottom exact sequence with (W , γ ) as the weak-injective envelope of A; the existence of such an exact sequence is guaranteed by Lee [9] or Göbel and Trlifaj [7]. Next we take the flat cover (E, β) of W ; in view of Lee [10], flat covers of weak-injectives are torsion-free injectives, so E is torsion-free injective. Let H = Ker β, and define α : F → A as the restriction of β.…”

“…By the Matlis category equivalence, the h-divisible torsion T corresponds to the R-complete torsion-free module H . As H is Enochs-cotorsion, by Lee[10] T , and hence also W , is weak-injective. That E is a flat pre-cover of W follows from the exactness of the sequence…”

Weak-injectivity and almost perfect domains

“…In general, the class of weak-injectives lies strictly between the classes of h-divisible and injective R-modules. For their main properties we refer to papers [8,9].…”