The infinitesimal site

This is a follow up to my blogpost Local Systems: The Infinitesimal Perspective. In it, I want to get into some very category-theoretic ways of looking at the ideas in that post. The level is going to be a bit higher here than in the rest of the series. Before, I’ve tried to make sure that people could follow the main picture if they only had an intuitive idea of schemes and sheaves; here I am going to need people to actually be fully comfortable with them.

I won’t refer to this material in the future local systems posts. (Well, hardly ever!) But I hope you’ll read it, because I find it really mind bending.

Before going in, let me explain why you should care about this, even if you don’t enjoy category theory for its own sake. In the previous post, we introduced N^{\infty}(X) and described local systems in terms of vector bundles on N^{\infty}(X). In that post, we gave an explicit description of N^{\infty}(X). In this post, I will explain how working with N^{\infty}(X) is equivalent to working in the category of nilpotent thickenings of X.

When we move to characteristic p, it is the latter notion which will generalize. There are some very strange nilpotent thickenings in characteristic p. For example, it is important to include \mathrm{Spec} \ \mathbb{Z}/p^n as a thickening of \mathrm{Spec} \ \mathbb{Z}/p. If you don’t, you will get the wrong answers! get a cohomology theory with coefficients in a characteristic p ring. So, for example, if you try to compute the number of fixed points of an automorphism using the Lefschetz fixed point theorem, you will only be able to get the number of points modulo p. (Revised due to Matthew Emerton’s comments below.)

When X is defined over a field of characteristic p, the correct analogue of N^{\infty} is a p-adic object. I believe that one can build this object explicitly; it is related to the ring of Witt vectors. The details are very hard though (I haven’t mastered them myself) and so the categorical approach descried below becomes important.

A warning: this is not the only difficulty in characteristic p. There are also problems coming from divided powers; some of which we will discuss in later posts.

By the way, my original plan was just to write a post on this stuff. Wow, would that have been incomprehensible!

Let X be a smooth scheme and let N^{\infty} be the formal neighborhood of the diagonal in X \times X, as discussed in the previous post.

I’ll start off with an idea that Scott Carnahan brought up in the comments. Let A be an integral affine scheme embedded in X. By “embedded”, I mean that A is a closed subscheme of an open subscheme. At first, you’ll want to think of A as being like a point. Later, you’ll want to think of it as like an open chart.

Let S be some nilpotent thickening of A. So A injects into both S and X. By the infinitesimal lifting property (Hartshorne, Exercise II.8.6), we can complete this diagram to A \hookrightarrow S \to X. This completion is in no way unique or canonical, though, and the whole point of this post is to exploit this lack of uniqueness.

To quote Scott:

We say an S-point is a map from a scheme S to X, and two S-points are “close” if the two maps agree on the reduced subscheme S_{red}.

Recall that an “S-point” of X means a map S \to X. So, suppose that we have two different maps p and q from S \to X, each giving the same map A \to X. As Scott says, we should think of these two points as being “near each other”, because their reductions agree, so we should be able to build a natural isomorphism between p^* V and q^* V. Given a B.4 input, we can!

Instead of thinking of two maps p and q from S \to X, think of a single map (p, q): S \to X \times X, such that A lands in the diagonal. Since S is a nilpotent thickening of A, we know that S lands in the formal neighborhood of the identity — which is to say, in N^{\infty}.

\begin{matrix} A & \to & S & & \\ \downarrow & & \downarrow & & \\ X & \to & N^{\infty} & \to & X \times X \end{matrix}

Remember that a B.4 input consists of an isomorphism \alpha : p_1^*(V) \to p_2^*(V) on N^{\infty}(X). If we pull \alpha back along (p,q), we get an isomorphism p^* V \to q^* V.

There should be some way to define our input as being: a vector bundle V on X and, for every (A, S, p, q) as above, an isomorphism p^* V \to q^* V, obeying certain compatibilities. My references limit themselves to the case where A is a Zariski open subset of X, so I’ll do the same, but I don’t think there is any reason for this:

Definition A B.5 input is a vector bundle on V and, for every (A, S, p, q) as above with A a Zariski open affine in X, and an isomorphism \alpha_{pq}: p^* V \to q^* V. We require that \alpha_{pq} \circ \alpha_{qr} = \alpha_{pr}, and we require some sort of compatibility when (p',q') : S' \to X \times X factors through (p,q) : S \to X \times X.

As sketched above, this is equivalent to a B.4 input.

Now, a really cool idea. (Due, I think, to Grothendieck.)

It is possible to recover a vector bundle from its sections over open sets. Thus, instead of working with vector bundles in the axioms; I’ll shift directly to talking about the sections over open sets.

Definition A B.6 input consists of: (a) for every (A, S), where A is an affine Zariski open of X, an \mathcal{O}(S) module E(A,S) and (b) for every diagram
\begin{matrix}  & A' & \subset & A \\ & \downarrow & & \downarrow \\ p: & S' & \to & S, \end{matrix}
an isomorphism \beta_{p} from E(A, S) \otimes_{\mathcal{O}(S)} \mathcal{O}(S') to E(A', S').

We impose that (1) E(A,S) is a locally free \mathcal{O}(S) (2) that \beta_{p} \circ \beta_{q} = \beta_{q \circ p} and (3) a sheaf-like gluing condition.

Whew, that’s a lot! A few comments:

A map of \mathcal{O}(S')-modules from $E(A, S) \otimes_{\mathcal{O}(S)} \mathcal{O}(S’)$ to E(A', S') is equivalent to a map of $\mathcal{O}(S)$ modules from E(A,S) to E(A',S'). So, if you like, you can think of \beta_p as a map E(A,S) \to E(A', S'). From this perspective, it is more obvious that we are defining something like a sheaf.

The condition that $\beta_p$ in question be an isomorphism should be thought of as a quasi-coherence condition — indeed, when A=S and A'=S', this is exactly the condition that the E(A,A) form a quasi-coherent sheaf.

Above, I defined things only for affine opens. It is easy to extend the definitions to all opens by gluing.

Condition (1) is just to make sure that we are talking about vector bundles, since I wanted to be consistent with what came before. It would be easy, and at this point more natural, to drop this condition and work with all quasi-coherent sheaves.

An input of type B.6 is equivalent to an input of type B.5 (or B.4 or B.3). Here’s how to go from the B.6 data to the B.5 data. First, look at all the cases where S=A. The E(A,A) form a locally free sheaf (in the Zariski topology), so we can use them to build a vector bundle V on X. Next, if we have any (A,S) with A an affine open, we can use the infinitesimal lifting property to get a map p: S \to A and, for every affine open A' in A, we can take the induced map S' \to A' obtained by localizing. This lets us show that the E(S', A') form a sheaf on S, coming from a vector bundle W. Moreover, using the \beta_p, we get an isomorphism p^* V \cong W. Finally, if we have two maps p and q from S \to X, we can compose p^* V \cong W \cong q^* V to get \alpha_{pq}.

There were a lot of details there, so let me emphasize the point I find mindblowing about the B.6 approach: the maps \beta_p are used both to build the vector bundle V and to build the isomorphisms \alpha_{pq}. In the B.6 approach, open inclusions and sections of nilpotent thickenings play the same role. This suggests that there should be some “topology” in which sections of nilpotent thickenings are considered to be open sets, just like we invented the étale topology in which local isomorphisms count as open sets. This can be done, and it is called the infinitesimal site. (If you put in all the necessary gadget to make things work in characteristic p, not all of which I’ve told you yet, you have the crystalline site.)

That is about the limits of my knowledge in this direction. When we return this series, there will be connections, differential equations and \mathcal{D}-modules — possibly the lowest tech perspective yet!

13 thoughts on “The infinitesimal site

  1. I recently discovered the book Models for Smooth Infinitesimal Analysis by Ieke Moerdijk and Gonzalo Reyes. It describes a bunch of topoi that include the category of manifolds but also ‘infinitesimal spaces’ like the space whose algebra of functions is R[x]/. I haven’t looked at this book yet, but I bet some of these topoi exploit ideas related to the infinitesimal site.

    A key idea in some of this work, which distinguishes it from algebraic geometry, is the idea of a “C-infinity ring”. This is an algebraic of the algebraic theory whose n-ary operations are smooth maps from R^n to R. The free C-infinity ring on n generators is the ring of smooth functions on R^n, and the algebra of smooth functions on any paracompact manifold is a finitely presented C-infinity ring.

  2. That funny-looking R[x]/ was my feeble attempt to write R[x] modulo the ideal generated by x^2. The angle brackets I wrote were interpreted in some way that made them and the stuff inside invisible.

    It would be really great to have a tiny box somewhere on the side of this blog that said how to write stuff in TeX. Lacking the knowledge of how to do, I’ll try something now and see if it works:

    $R[x]/\langle x^2 \rangle$

  3. To use LaTeX, simply write $ latex R[x]/\langle x^2 \rangle$ without the space between the dollar sign and LaTeX. This is the system used on all wordpress blogs.

    But, you’re right, we should put a post up in the side bar explaining this.

    The C^{\infty} ring sounds fascinating. How much do I need to know to read this book?

  4. In the mod p situation, it is not that one get the wrong answer by not including Z/p^n as a thickening of Z/p; it all depends on what one is trying to compute.

    In general, suppose that k is a perfect field of char. p, and let W be the ring of Witt vectors of k. (So if k is Z/p, then W is Z_p.)

    If X is a smooth k-scheme, then X is also a W-scheme. So when we form the infinitesimal site of X, we have (at least) two choices: we can look at thickenings of Zariski open subsets which are themselves k-schemes (so we keep everything in char. p), or we look at more general thickenings which are just W-schemes (so we allow thickening in the “p direction”, as well as in the geometric directions).

    If we compute cohomology for the site over k, we will end up with de Rham cohomology of X, which will be k-vector spaces. If we compute cohomology for the site over W, we will end up with the crystalline cohomology of X, which will be W-modules. Both are interesting, although crystalline cohomology has the advantage of being over a ring of char. 0 (so is better from the point of view of studying things like zeta functions of varieties, as in the Weil conjectures).

    Here I have ignored the fact that one actually has to replace the infinitesimal site by the crystalline site (i.e. consider divided power thickenings rather than arbitrary thickenings). This is technical, but crucial; in char. p the infinitesimal site (as defined in this post, for example) is very rigid, and doesn’t compute the full de Rham/crystalline cohomology. (Also, there is another technicality: I think that one should assume that X is proper to get the cohomology computations to work out well.)

  5. Thanks for the first correction, I was being to glib. See if you like the revised statement.

    As for the distinction between crystalline and infinitesimal sites, I’ve been trying to warn people where the difficulty is without actually introducing divided powers. Of course, if you want to write up a guide to divided power algebras, I’d gladly link to it!

  6. There is another, rather bizarre version of sheaves on the infinitesimal site. The two projections from N^\infty X to X together with the diagonal embedding gives us the structure of a formal groupoid, which Beilinson and Drinfeld call the universal formal groupoid of X. Taking a quotient of X by the action yields a structure called a c-stack (c for crystalline) which is in general non-algebraic, and if X is smooth, it has dimension zero. The category of O-modules on this c-stack is equivalent to the category of O-modules on the infinitesimal site, and I guess this might be tautological if you choose the right definitions. There is also a PD version of c-stacks which as far as I know doesn’t appear in the literature.

  7. For those who are completely baffled by the notion of a “formal groupoid”, it will help to realize that this is more like a generalization of an equivalence relation than it is like a generalization of a group. Recall that a subset of X \times X is called an equivalence relation if it is reflexive, symmetric and transitive. Apparently, we can generalize from working with actual subsets of X \times X to working with formal subschemes.

    Although I am not completely baffled, I am still confused. The problem, most likely, is that I don’t know how to write down a quotient stack in this level of generality. Does someone out there get this?

  8. David,

    Think about the functor of points: a map from a scheme into this quotient of X (which I’ve learned to call “the deRham stack”) is a map of the reduction into X.

  9. There is a way to look at this in terms of groups. The formal completion of any point has a formal group structure (I think I should assume smoothness here), that in characteristic zero is noncanonically isomorphic to a product of formal additive groups. N^\infty(X) is the equivalence relation that describes its action by infinitesimal translations, and makes any two nearby points equivalent. Modulo details, this yields the functor of points that Ben described.

  10. Now I’ve managed to confuse myself – it looks like in dimension greater than one, there might be an obstruction to getting a formal group structure on the completion of a point, arising from some kind of curvature.

  11. Please disregard both sentences in the previous two comments containing the word “completion”. I have limited introspective powers, but I think I was on some foolhardy quest to describe the equivalence relation by completing the image of a map.

    (I still think it’s pretty neat that the tangent space of any point in this gadget is the zero vector space.)

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