Lim one obstruction to completeness

Let $M_0\stackrel{u_0}{\leftarrow }{M}_1\stackrel{u_1}{\leftarrow }{M}_2\leftarrow \cdots$ be a countable tower of abelian groups and homomorphisms. The product $P={\prod }_{m\ge 0}{M}_m$ has coordinatewise addition, zero and negatives; these operations satisfy the group laws coordinatewise. It contains the all-zero tuple without any choice assumption. Define $\mathrm{\Delta }:P⟶P,\mathrm{\Delta }(x{)}_m=x_m-u_m(x_{m+1})\mathrm{.}$ Additivity of each $u_m$ gives $\mathrm{\Delta }(x+y{)}_m=\mathrm{\Delta }(x{)}_m+\mathrm{\Delta }(y{)}_m$, so this is a homomorphism. Using the subgroup kernels and coset cokernels of Abelian-group model for spectral-sequence computations, set The first consists precisely of tuples satisfying $x_m=u_m(x_{m+1})$ for every $m$. A cone of homomorphisms $f_m:T\to M_m$ factors uniquely by $t\mapsto (f_m(t){)}_m$, which belongs to that subgroup exactly by cone compatibility. Thus it is the categorical limit of Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties. The notation here names this particular cokernel; no unproved identification with general derived functors is included.

For an increasing filtration on an abelian group $A$ as in Exhaustive separated bounded and finite filtration, put $G_m=F_{-m}A$ with inclusion transitions. The term will measure the failure of surjectivity of through the following completion exact sequence. This is a claim about this subgroup tower, not an assertion that every unrelated tower measures completeness of $A$. The present definitions and coordinate formulas require no AC. They also apply to modules over a fixed ring with coordinate scalar multiplication; $\mathrm{\Delta }$ is then linear. Zero groups and zero or identity transitions are allowed.