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For a commutative ring $R$ and a finitely generated ideal $I$, the category of $I$-adically complete modules is not abelian. The category of $L$-complete modules is, in some precise sense, the smallest abelian subcategory of RMod which contains the $I$-adically complete modules. Roughly speaking, it is obtained by formally adjoining cokernels of morphisms between $I$-adically complete modules.
Let $R$ be a commutative ring and $I \subset R$ a finitely generated ideal. The $I$-adic completion of an $R$-module $M$ is the inverse limit
Let $i:M\rightarrow \hat M$ be the canonical map.
An $R$-module $M$ is said to be
quasi-complete if $i$ is surjective
separated if $i$ is injective
complete if $i$ is bijective.
The completion functor is idempotent (warning: this is not true in general if $I$ is not finitely generated), hence induces an idempotent monad on the category RMod, and is left adjoint to the inclusion of the category $\widehat{R Mod}$ of complete modules into all modules. Hence, $\widehat{R Mod}$ is a reflective subcategory of $R Mod$. It is not, however, an abelian subcategory: the quotient of two complete modules is not, in general, complete.
A key observation is that the completion functor is not right exact when seen as an endofunctor of $R Mod$. Hence, let $L$ be the 0th left derived functor of the completion functor, which is right exact by definition.
An $R$-module $M$ is called $L$-complete if the canonical map
is an isomorphism.
Let $M$ be an $R$-module.
Let $(m_i)_{i\in \mathbb{N}}$ be a sequence of elements in $M$ such that for all $n$, all but finitely many of the $m_i$‘s belong to $I^n$. Then $M$ is complete if and only if for every such sequence, the partial sums $\sum_{i\leq k} m_i$ have a well-defined limit in $M$. The above proposition shows an $L$-complete needs not to be separated, hence, this limit might not exists. However, informally speaking, $L$-complete modules are those for which we can still make sense of some sort of limit.
(Salch 20)
The category of $L$-complete modules is the smallest abelian full subcategory of RMod which contains the $I$-adically complete modules.
Andrew Salch, Approximation of subcategories by abelian subcategories (arXiv:2011.01827)
Charles Rezk, Analytic completion (pdf)
Last revised on June 26, 2021 at 11:07:23. See the history of this page for a list of all contributions to it.