SOME REMARKS ON CATEGORIES OF MODULES MODULO MORPHISMS WITH ESSENTIAL KERNEL OR SUPERFLUOUS IMAGE

Abstract

For an ideal <TEX>$\mathcal{I}$</TEX> of a preadditive category <TEX>$\mathcal{A}$</TEX>, we study when the canonical functor <TEX>$\mathcal{C}:\mathcal{A}{\rightarrow}\mathcal{A}/\mathcal{I}$</TEX> is local. We prove that there exists a largest full subcategory <TEX>$\mathcal{C}$</TEX> of <TEX>$\mathcal{A}$</TEX>, for which the canonical functor <TEX>$\mathcal{C}:\mathcal{C}{\rightarrow}\mathcal{C}/\mathcal{I}$</TEX> is local. Under this condition, the functor <TEX>$\mathcal{C}$</TEX>, turns out to be a weak equivalence between <TEX>$\mathcal{C}$</TEX>, and <TEX>$\mathcal{C}/\mathcal{I}$</TEX>. If <TEX>$\mathcal{A}$</TEX> is additive (with splitting idempotents), then <TEX>$\mathcal{C}$</TEX> is additive (with splitting idempotents). The category <TEX>$\mathcal{C}$</TEX> is ample in several cases, such as the case when <TEX>$\mathcal{A}$</TEX>=Mod-R and <TEX>$\mathcal{I}$</TEX> is the ideal <TEX>${\Delta}$</TEX> of all morphisms with essential kernel. In this case, the category <TEX>$\mathcal{C}$</TEX> contains, for instance, the full subcategory <TEX>$\mathcal{F}$</TEX> of Mod-R whose objects are all the continuous modules. The advantage in passing from the category <TEX>$\mathcal{F}$</TEX> to the category <TEX>$\mathcal{F}/\mathcal{I}$</TEX> lies in the fact that, although the two categories <TEX>$\mathcal{F}$</TEX> and <TEX>$\mathcal{F}/\mathcal{I}$</TEX> are weakly equivalent, every endomorphism has a kernel and a cokernel in <TEX>$\mathcal{F}/{\Delta}$</TEX>, which is not true in <TEX>$\mathcal{F}$</TEX>. In the final section, we extend our theory from the case of one ideal<TEX>$\mathcal{I}$</TEX> to the case of <TEX>$n$</TEX> ideals <TEX>$\mathcal{I}_$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$\mathca{l}_n$</TEX>.