Vanishing of $\varprojlim^1$ on Mittag-Leffler sequences story.

Question

I'm trying to clarify myself on some points about the story of the first derived functor $\varprojlim^1$ of the projective limit functor vanishing on some kind of filtered inverse systems in arbitrary (complete) abelian categories (with enough injectives). I am interested in the case of towers (countable inverse systems), so the set of indices will be the set of nonnegative integers. I will say that an inverse system of objects $\{A_i,f_{i}\}$ (where $f_{i}:A_{i+1} \rightarrow A_i$, for $i \geq 0$) in an abelian category

• is M(ittag)L(effler) if for any $i$ there exists $j \geq i$ such that $Im(A_k \rightarrow A_i)=Im(A_j \rightarrow A_i)$ for all $k \geq j$ (like in Weibel);

• is epi if all the maps $f_{i}$ are surjective (which is "Mittag-Leffler sequence" in Roos and Neeman);

• is pro-zero if for any $i$ there exists $j \geq i$ such that the map $A_j \rightarrow A_i$ is the zero map.

The last two are particular cases of the first one. Moreover, an ML inverse system fits as the middle term in a short exact sequence where the first term is an epi system (the inverse system $\{f_{n}(A_{n+1})\}$) and the last one is pro-zero (the inverse system of quotients).

Now, I know that Roos proved, 40 years after his first wrong statement, that $\varprojlim^1$ vanishes on epi towers when the category is abelian, satisfies (AB3) and (AB4*) and has a generator. My question is, does $\varprojlim^1$ vanishes on ML towers in this setting as well?

One should have the vanishing on pro-zero systems, too, in order to conclude that, but in which setting does that occur?

I know this all works in module categories, by the way. I also heard somewhere, but cannot figure it out, that in the particular case of towers, ML condition is equivalent to epi condition in some sense. Can anybody help me clarify everything?

Answer 1

I assume that you have a basic knowledge about pro-categories. If $\mathbf{C}$ is a category, then $\mathbf{pro\text{-}C}$ denotes its pro-category which is the category of all inverse systems in $\mathbf{C}$. You only consider the full subcategory $\mathbf{tow\text{-}C}$ of towers, i.e. inverse systems indexed by the integers. Let us consider the case that $\mathbf{C}$ is the category of abelian groups. I think the case of a general abelian category can be treated similarly.

Let $A = (A_i,f_i)$ be a tower. Then it easy to show that $A$ satisfies (ML) iff it is isomorphic in $\mathbf{tow\text{-}C}$ to a system $A'$ satisfying (epi). Moreover, $A$ satisfies (pro-zero) iff it is isomorphic in $\mathbf{tow\text{-}C}$ to the trivial system $0$ consisting only of zero objects. The first derived limit is in fact a functor on $\mathbf{pro\text{-}C}$ which explains why it vanishes if $A$ is ML.

Here are some references where you can find material on $\mathbf{pro\text{-}C}$ :

Edwards, David A., and Harold M. Hastings. Cech and Steenrod homotopy theories with applications to geometric topology. Vol. 542. Springer, 2006.

Geoghegan, Ross. Topological methods in group theory. Vol. 243. Springer Science & Business Media, 2007.

Mardesic, Sibe. Strong shape and homology. Springer Science & Business Media, 2013.

And of course Grothendieck's work.

Added: Proof that $A$ being (ML) implies $A$ isomorphic to $A'$ satisfying (epi).

By definition the images of $A_j \to A_i$ "stabilize" for each $i$. This implies we can construct a strictly increasing function $u : \mathbb{N} \to \mathbb{N}$ such that $im(f_i^k) = im(f_i^{u(i)})$ for all $k \ge u(i)$. Here $f_i^k = f_i \circ ... \circ f_{k-1} : A_k \to A_i$. Set $A'_i = im(f_i^{u(i)}) \subset A_i$. Clearly $f_i(A'_{i+1}) = f_i(f_{i+1}^{u(i+1)}(A_{u(i+1)})) = f_i^{u(i+1)}(A_{u(i+1)}) = f_i^{u(i)}(A_{u(i)}) = A'_i$ which shows that the $f_i$ restrict to epimorphisms $f'_i : A'_{i+1} \to A'_i$. Therefore $A' = (A'_i,f'_i)$ is an inverse system such that the inclusions $\alpha_i : A'_i \to A_i$ form a level-morphism $\alpha$ between inverse systems. Its inverse in the pro-category is given by $\beta = (u,\beta_i)$ with $\beta_i= f_i^{u(i)} : A_{u(i)} \to A'_i$. The function $u$ is the "index function"; we have $\beta_i \circ f_{u(i)}^{u(i+1)} = f'_i \circ \beta_{i+1}$. (The proof that $\alpha$ and $\beta$ are inverse to each other is very easy.)

Added: As we have seen, it suffices to show that $\varprojlim^1$ is a functor on $\mathbf{\text{pro-C}}$ (it would even be sufficient to have it on $\mathbf{\text{tow-C}}$). Once this has been done, we can conclude that $\varprojlim^1$ vanishes on (ML) sequences iff it vanishes on (epi) sequences. The latter was proved by Roos (under certain assumptions on $\mathbf{C}$).

In section 6 of the paper

Duskin, J. "Pro-objects (after Verdier)." Dualité de Poincaré (Seminaire Heidelberg-Strasbourg 1966/67), IRMA Strasbourg 3 (1969)

the author considers the derived functors of $\varprojlim : \mathbf{\text{pro-C}} \to \mathbf{C}$. If $\mathbf{C}$ satisfies (AB4$^\ast$) and has enough injectives, he proves the existence of derived limit functors living on $\mathbf{\text{pro-C}}$. See the arguments after Theorem 6.1.

Concerning $\mathbf{\text{pro-C}}$ as it occurs in this paper it may be useful to consult

Porter, Timothy. "Essential properties of pro-objects in Grothendieck categories." Cahiers de topologie et géométrie différentielle catégoriques 20.1 (1979): 3-57

It gives an alternative interpretation of $\mathbf{\text{pro-C}}$ which in my opinion is "more accessible''. See page 9 of Porter's paper.