nLab L-complete module

Idea

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.

Motivation and definition

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

$$\hat M = \lim_{\leftarrow} M/I^n M \,.$$

Let $i:M\rightarrow \hat M$ be the canonical map.

Definition

An $R$-module $M$ is said to be

1. quasi-complete if $i$ is surjective

2. separated if $i$ is injective

3. 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.

Definition

An $R$-module $M$ is called $L$-complete if the canonical map

$$M\rightarrow L(M)$$

is an isomorphism.

Proposition

Let $M$ be an $R$-module.

• If $M$ is finitely generated, then $\hat M=L(M)$.
• If $M$ is complete, then it is $L$-complete.
• The canonical map $M\rightarrow \hat M$ factor through the canonical map $M\rightarrow L(M)$, and the map $L(M)\rightarrow \hat M$ is surjective. In particular, an $L$-complete module is always quasi-complete.

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.

Proposition

(Salch 20)

The category of $L$-complete modules is the smallest abelian full subcategory of RMod which contains the $I$-adically complete modules.

References

• Andrew Salch, Approximation of subcategories by abelian subcategories (arXiv:1006.0048)

• Charles Rezk, Analytic completion (pdf)