Adjunctions and inverse limits of derived categories

Question

Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its nerve), sending $t$ to the derived category $D(A_{t})$ and $t+1\to t$ to $A_{t}\otimes_{A_{t+1}}^{\text{L}}-$, etc.

Now let $A:=\varprojlim_{t}A_{t}$. Then there is a functor $\text{L}\colon D(A) \to \varprojlim_{t}D(A_{t})$ which, roughly speaking, sends a complex $M$ of $A$-modules to the data of the complexes $A_{t}\otimes_{A}^{\text{L}}M$ for all $t$. If I am not mistaken, Proposition 5.5.3.13 in Lurie's Higher Topos Theory implies that $\text{L}$ is a left adjoint. Fix a right adjoint $\text{R}$.

Here is my question: How do we describe $\text{R}$ explicitely? I understand that intuitively, we should take a limit of a given system $M_{\bullet}=\left( M_{t}\right)_{t}$ of complexes of $A_{t}$-modules; but one always takes the limit of a diagram. How precisely do we produce a suitable diagram $\mathbb{N}^{\text{op}}\to D(A)$ from a given object $M_{\bullet}\in\varprojlim_{t}D(A_{t})$?

Answer 1

A reference for exactly this type of problem in general is a paper by Horev and Yanovski called "On conjugates and adjoint descent".

Given a diagram $C \to D_i$ of left adjoints $f_i$ with right adjoints $g_i$, they construct an $I$-indexed diagram of functors $\lim_I D_i \to C$ whose value at $i\in I$ is the $i$th projection followed by $g_i$ and whose transition maps are "what you expect" (except of course that they actually construct the whole diagram) and they prove that the limit of this diagram is a right adjoint for the induced functor $C\to \lim_I D_i$.

Now, this is the general case. In your specific situation, $\mathbb N$ is a much simpler shape than an arbitrary $I$ and it turns out that $\mathbb N$ is free as an $\infty$-category on the simplicial set $0 \to 1 \to ... \to n\to ...$, by which I mean only arrows $i \to i+1$ and no composites. Thus to specify a functor out of $\mathbb N$ it suffices to specify the objects, and the transition maps.

So given your object $M_\bullet$, you can simply view each $M_t$ as an $A$-module via restriction of scalars alonf $A\to A_t$, and the data of an object in a limit provides maps $M_{t+1}\otimes_{A_{t+1}}A_t \to M_t$, which provides maps $M_{t+1}\to M_t$ that are $A_{t+1}$-linear and hence $A$-linear.

Answer 2

Not an answer, but too long for a comment: there are many details to adjust. I am not familiar with derived categories of rings, but I guess you are looking for an explicit description of the limit of $\infty$-categories along towers.

Firstly, recall that limits in an $\infty$-category arising from a model structure (as $\textrm{Cat}_{\infty}$ itself is) are computed as homotopy limits. Your diagram is not strict, but I think a rectification should always exist for diagrams of simplicial sets (as it does for topological spaces - see Vogt Theorem). Let's call $D(A_t)'$ the rectified version of your diagram.

Secondly, homotopy limits of towers can be computed nicely by taking a 'fibrant resolution'. Let us consider the Joyal model structure on simplicial sets. Since derived categories are $\infty$-categories, your functor is made of fibrant objects. If you look at proposition 3.4 in the nLab page above (or, equivalently, cor 2.4.6.5 in HTT), to prove that the diagram is fibrant we should also prove that $D(A_{t+1}) \to D(A_t)$ is an inner fibration and an isofibration (the same nlab page contains relevant definitions), and then transfer this information to $D(A_t)'$. An alternative approach to the homotopy limit computation, if the maps turn out not to be fibrations, is to use the so-called "Bousfield-Kan formula". It is an analog of the telescopic formula for homotopy colimits of cotowers of spaces: morally, it multiplies each term $D(A_t)'$ for a contractible simplicial set and then take an appropriate limit.

If we assume that the above is true, we get that the $D(A)$ is weakly equivalent to the limit of $D(A_t)'$. Since the Joyal model structure is Quillen equivalent to the Quillen model structure (via 'groupoidification'), weak equivalences should be also equivalences in the Joyal model structure. This means that when looking for a right adjoint departing from $\textrm{Cat}_{\infty}$ we can substitute $D(A)$ with $\lim_t D(A_t)'$, where the latter is meant to be the classical limit of simplicial sets.

At this point, the right-adjoint functor is probably given by the limit you thought, since an element of $D(A)$ is a sequence of compatible objects in $D(A_t)'$... I hope it works.