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L-complete module

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For a commutative ring RR and a finitely generated ideal II, the category of II-adically complete modules is not abelian. The category of LL-complete modules is, in some precise sense, the smallest abelian subcategory of RMod which contains the II-adically complete modules. Roughly speaking, it is obtained by formally adjoining cokernels of morphisms between II-adically complete modules.

Motivation and definition

Let RR be a commutative ring and IRI \subset R a finitely generated ideal. The II-adic completion of an RR-module MM is the inverse limit

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

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

Definition

An RR-module MM is said to be

  1. quasi-complete if ii is surjective

  2. separated if ii is injective

  3. complete if ii is bijective.

The completion functor is idempotent (warning: this is not true in general if II is not finitely generated), hence induces an idempotent monad on the category RMod, and is left adjoint to the inclusion of the category RMod^\widehat{R Mod} of complete modules into all modules. Hence, RMod^\widehat{R Mod} is a reflective subcategory of RModR 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 RModR Mod. Hence, let LL be the 0th left derived functor of the completion functor, which is right exact by definition.

Definition

An RR-module MM is called LL-complete if the canonical map

ML(M)M\rightarrow L(M)

is an isomorphism.

Proposition

Let MM be an RR-module.

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

Let (m i) i(m_i)_{i\in \mathbb{N}} be a sequence of elements in MM such that for all nn, all but finitely many of the m im_i‘s belong to I nI^n. Then MM is complete if and only if for every such sequence, the partial sums ikm i\sum_{i\leq k} m_i have a well-defined limit in MM. The above proposition shows an LL-complete needs not to be separated, hence, this limit might not exists. However, informally speaking, LL-complete modules are those for which we can still make sense of some sort of limit.

Proposition

(Salch 20)

The category of LL-complete modules is the smallest abelian full subcategory of RMod which contains the II-adically complete modules.

References

Last revised on June 26, 2021 at 07:07:23. See the history of this page for a list of all contributions to it.