Notes on prismatic cohomology

Suppose one is trying to write down a functor $F$ from ${\mathbf{Mod}}_A$ to some abelian category which is right exact and commutes with filtered colimits. Then it is enough to specify the values of $F$ on arbitrary finite free $A$-modules: every module is a cokernel of a morphism between two free modules, each of which is itself a filtered colimit of finite free modules. Furthermore, using projective resolutions by free modules, we can compute the left derived functors of $F$ from this.

The construction of nonabelian derived functors allows us to do something similar starting from the category ${\mathbf{Ring}}_A\text{.}$ The free objects in this case (i.e., the essential image of the left adjoint of the forgetful functor to $\mathbf{Set}$) are polynomial rings. In order to replace modules to rings, we need to reconceptualize some familiar constructions without reference to the additive structure of the category ${\mathbf{Mod}}_A\text{;}$ for example, in ${\mathbf{Mod}}_A$ we can form the equalizer of two maps $f_1,f_2:M\to N$ as the kernel of the difference $f_1-f_2\text{,}$ but now we need to forgo this shortcut.

The resulting process amounts to the transition from homological algebra to homotopical algebra in the sense of Quillen [101]. Nowadays this is usually done in the framework of $\mathrm{\infty }$-categories, as in [93]; we will keep ourselves in a very limited part of the picture so as to keep the prerequities for the discussion under control.

By way of motivation, you should imagine that $\mathrm{\Delta }[n]$ represents an $n$-dimensional simplex and the product $U\times \mathrm{\Delta }[n]$ represents taking the product of some geometric object corresponding to $U$ with this simplex. This motivates the definition of homotopies between maps of simplicial objects, as in Definition 16.2.5.

Just as in homological algebra, one would like to work in the derived category to enforce that any object is “interchangeable” with a sufficiently nice resolution, in the simplicial realm one wants to to replace objects with simplicial objects that are more flexible (in the sense of being fibrant or cofibrant). The general story is out of scope for these notes (in part due to the need to develop robust combinatorial formalism, as in the language of $\mathrm{\infty }$-categories, to keep track of homotopy coherence); here we limit ourselves to a few critical examples, such as Example 16.2.9.

It should be stressed that while the standard resolution is a “natural” (and functorial) way to construct simplicial resolutions, the resulting resolutions are not preferred in any mathematical sense. In particular, if one starts performing operations one quickly ends up with simplicial resolutions that are not the standard ones but are homotopy equivalent, and the distinction will carry no value (if anything it is more of a hindrance).

Note that in Definition 16.4.1, both the commutativity of the second diagram in Figure 16.4.2 and the uniqueness of $\sigma$ are well-posed because a natural transformation is specified by a collection of morphisms between prescribed sources and targets, so the comparison of these is set-theoretic and not category-theoretic.

As usual with a definition via a universal property, the use of the definite article is justified by the observation that any two objects satisfying the definition are uniquely isomorphic. However, $\alpha$ is not itself guaranteed to be an isomorphism of functors; that is, $G$ is not necessarily isomorphic to the restriction of $L$ along $F\text{.}$

In practice, we will be considering cases in which $F$ can be lifted to a functor $\tilde{F}:{\mathbf{Poly}}_A\to \mathbf{Comp}(\mathbf{Ab})\text{,}$ in which case the colimit in part (2) of Proposition 16.4.6 can be interpreted as the totalization of a double complex made out of the terms $L\tilde{F}(P^{\bullet })\text{.}$ Otherwise, one should replace the derived category $D(\mathbf{Ab})$ with its $\mathrm{\infty }$-categorical analogue and take the colimit there (where it can be reinterpreted as the geometric realization).

To give a concrete example of the effect of the colimit, note that if $B$ is the coequalizer of two maps $f_0,f_1:P_1\to P_0\text{,}$ then $LF(B)=\mathrm{Cone}(f_0-f_1)\text{.}$

An important basic example will be given by the exterior power ${\wedge }^i:{\mathbf{Mod}}_A\to {\mathbf{Mod}}_A\text{,}$ which will give us the derived exterior power $L{\wedge }^i:{\mathbf{Mod}}_A\to D(A)\text{.}$ This in turn extends to a functor $L{\wedge }^i:D^{\le 0}(A)\to D(A)\text{.}$ (As indicated in [18], Lecture VII, Remark 1.4, this is a point at which we are forced to be a bit sloppy by not working in the language of $\mathrm{\infty }$-categories, but never mind.)

In what follows we will frequently use the following construction without explicit comment. Let $G^{\prime }:{\mathcal{C}}_1\to {\mathcal{C}}_3$ be another functor admitting a left Kan extension $(L^{\prime }:{\mathcal{C}}_2\to {\mathcal{C}}_3,\alpha :G^{\prime }\to L^{\prime }\circ F)\text{,}$ and suppose that $\gamma :G\to G^{\prime }$ is a natural transformation. Then we obtain a natural transformation $\alpha \circ \gamma :G\to L^{\prime }\circ F$ to which we may apply the universal property of the left Kan extension of $L\text{,}$ so as to obtain a natural transformation $\sigma :L\to L^{\prime }\text{.}$ That is, a natural transformation between two functors from ${\mathcal{C}}_2$ to ${\mathcal{C}}_3$ can be uniquely specified by giving its restriction (along $F$) to ${\mathcal{C}}_1\text{.}$

It was mentioned in passing earlier that the derived category of $A$-modules, for some $A\in \mathbf{Ring}\text{,}$ is more robust to work with in the language of (stable) $\mathrm{\infty }$-categories. This allows us to be more careful about making identifications “up to homotopy”; rather than simply declaring two morphisms of complexes to be equal if there is a homotopy between them, in the homotopical approach one records the data of the homotopy witness and keeps track of it as one performs further operations.

One reason this is advantageous is that the formation of mapping cones is not functorial in the derived category as we have described it, but it becomes functorial in the stable $\mathrm{\infty }$-category (because of the retention of the homotopy data). A minimal example is given by the map from $A\to 0$ to $0\to A\text{.}$

Another reason is that one cannot perform any reasonable descent on the functor $A\mapsto D(A)$ without the homotopical data: for instance, for a Zariski covering of three or more opens, it is not generally possible to lift descent data from objects in derived categories to chain complexes. Again, recording the homotopy data makes it possible to perform this lifting.