Rlim

Let N be the natural numbers. Think of N as a category with a unique morphisms n —> m whenever m ≥ n and endow it with the chaotic topology to get a site. Then a sheaf of abelian groups on N is an inverse system (M_e) and H^0 corresponds to the limit lim M_e of the system. The higher cohomology groups H^i correspond to the right dervided functors R^i lim. A good exercise everybody should do is prove directly that R^i lim is zero when i > 1.

Consider D(Ab N). This is the derived category of the abelian category of inverse systems. An object of D(Ab N) is a complex of inverse systems (and not an inverse system of complexes — we will get back to this). The functor RΓ(-) corresponds to a functor Rlim. Given a bounded below complex of inverse systems where all the transition maps are surjective, then Rlim is computed by simply taking the lim in each degree.

What if we have an inverse system with values in D(Ab)? In other words, we have objects K_e in D(Ab) and transition maps K_{e + 1} —> K_e in D(Ab). By choosing suitable complexes K^*_e representing each K_e we can assume that there are actual maps of complexes K^*_{e + 1} —> K^*_e representing the transition maps in D(Ab). Thus we obtain an object K of D(Ab N) lifting the inverse system (K_e). Having made this choice, we can compute Rlim K.

During the lecture on crystalline cohomology yesterday morning I asked the following question: Is Rlim K independent of choices? The reason for this question is that there are a priori many isomorphism classes of objects K in D(Ab N) which give rise to the inverse system (K_e) in D(Ab). It turns out that Rlim K is somewhat independent of choices, as Bhargav explained to me after the lecture. Namely, you can identify Rlim K with the homotopy limit, i.e., Rlim K sits in a distinguished triangle

Rlim K —–> Π K_e —–> Π K_e ——> Rlim K[1]

in D(Ab) where the second map is given by 1 – transition maps. And this homotopy limit depends, up to non-unique isomorphism, only on the inverse system in D(Ab).

One of the things I enjoy about derived categories is how things don’t work, but how in the end it sort of works anyway. The above is a nice illustration of this phenomenon.

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