## Other Attempts at Cohomology Theories

I should first point out that I’m basically sketching out Grothendieck’s article on crystals in Dix Expose, so if you want to see more that’s where you should look. Let’s first answer those questions from last time and explain exactly what it is that we are looking for in a cohomology theory.

If ${X/k}$ is of finite type and ${k}$ a perfect field of positive characteristic, then we want to keep all of those properties from our earlier theories. The important ones are that the cohomologie groups are modules over an integral domain that has the property of having characteristic ${0}$ fraction field. We also want to keep the formal properties that we checked for the earlier ones like functoriality, being finite dimensional when ${X}$ is proper, having some sort of duality, having a Kunneth formula, having a flat or smooth base change theorem, and the list continues.

We already have theories that do these things, so remember the key thing we need to be able to add in is that we want information about the ${p}$-torsion of the singular cohomology. For reasons we won’t go into we can’t take our coefficients to be ${\mathbb{Z}_p}$ or ${\mathbb{Q}_p}$. But we’ve already put in the work to see that ${W(k)}$ is a nice choice since it has residue field ${k}$ and fraction field of characteristic ${0}$.

There have been a few failed attempts. Without giving rigorous definitions of the attempts, I’ll just point them out. One might try to build an analogue of the ${\ell}$-adic attmept but using a different site. It was attempted to do this using the fppf site (it was a bit more complicated than just using the fppf site, though). What happens is you get a theory that works great in dimension ${1}$, but then you lose things like Poincare duality for ${\dim(X)\geq 2}$. This theory should still give some nice connections with our original one via Dieudonne modules. If we talk about this later it will be defined more rigorously.

The next attempt by Monsky and Washnitzer was quite beautiful. The idea is that everything works if ${k}$ is of characteristic ${0}$, so let’s just put ourselves in that situation. Let ${S= \mathrm{Spec}(W)}$ then we can consider a lifting of ${X}$ to characteristic ${0}$ by say ${M\rightarrow S}$. Since we are now in characteristic ${0}$, we may as well use de Rham, so the theory should just be ${H^i_{dR}(M/S)}$. Of course, one needs to check several things, the first of which is that ${H^i_{dR}(M/S)}$ is independent of the choice of lift.

With this approach we do get lots of nice things like the correct Betti numbers and finite dimensionality when ${M/S}$ is proper. There is however a very huge problem with the approach. There exists schemes with no lift to characteristic ${0}$. What do we do about this? Well, one can try to get around it by trying to construct a formal lift ${\frak{X}\rightarrow S}$ and then considering the hypercohomology ${\mathbf{H}^*(\frak{X}, \Omega_{\frak{X}/S}^\cdot)}$ using limits, but some problems arise. Again, there isn’t always even a formal lift, and even if there was you can check that this doesn’t give finite dimensional answers.

They actually carefully constructed a way to not need the lift, and when one exists they get ${H^i_{MW}(X)=H^i_{dR}(M/S)}$. Unfortunately this theory still uses differential forms and hence requires some nicenesss hypothesis (maybe even smooth) to make sure everything works. Also, many of the properties we want are unknown to be true like being finite ${W}$-modules when ${X}$ is proper. Admittedly, this theory is the best so far that we’ve looked at, and by the fact that ${H^i_{MW}(X)}$ is defined without the lift, it proves that ${H^i_{dR}(M/S)}$ is independent of lift when one exists.

## Problems with de Rham and l-adic

Let’s begin by reviewing the fact that when we work over ${\mathbb{C}}$ we have a nice comparison theorem. Last time it was briefly mentioned that if ${X}$ is smooth over ${\mathbb{C}}$, then we can consider the analytic topology on the ${\mathbb{C}}$-valued points which we’ll call ${X^{an}}$. Using GAGA and the degeneration of the Hodge-de Rham spectral sequence from last time we immediately get that ${\mathbf{H}^*(X, \Omega_{X/\mathbb{C}}^\cdot)\stackrel{\sim}{\rightarrow} \mathbf{H}^*(X^{an}, \Omega_{X^{an}}^\cdot)\simeq H^*(X, \mathbb{C})}$.

Thus we have a comparison theorem that allows us to use purely algebraic methods to recover the topological information of the singular cohomology. But what happens if we work over a field that is not ${\mathbb{C}}$? Well let’s assume that ${X}$ is smooth and of finite type over ${k}$. If ${k}$ has characteristic ${0}$, then the standard idea of the Lefschetz principle gives us that everything works out again and we can recover the singular cohomology.

Obviously lots can go wrong in positive characteristic. Say ${\mathrm{char}(k)=p}$, then we’ll throw out the case of affine things because for instance if ${X}$ is an affine curve the de Rham cohomology is not even finitely generated. But if we assume ${X/k}$ is proper and have finite generatedness, we still have problems with de Rham. It doesn’t recover the correct topological information. For instance, we’d like a cohomology that gives us “correct” Betti numbers.

If we define the Hodge Betti numbers to be ${\displaystyle b^i_H=\sum_{p+q=i}\dim_k H^q(X, \Omega^p)}$ and the de Rham Betti numbers ${b^i_{dR}=\dim_k H^i_{dR}(X/k)}$ we can just look at the HdR SS and see that ${b^i_H\leq b^i_{dR}}$ and we have equality if and only if the HdR SS degenerates at ${E_1}$. Degeneration of this spectral sequence is related to having some sort of analogue of ${X}$ in characteristic ${0}$, also known as a “lift” of ${X}$. Even if this spectral sequence degenerates, it merely relates these two Betti numbers, but doesn’t fully relate back to the topological Betti numbers.

Also, despite the fact that there is still a Lefschetz fixed point formula for de Rham cohomology in positive characteristic, it only gives the right answer mod ${p}$. We’ve seen that de Rham cohomology is a nice attempt at coming up with an algebraic way of defining singular cohomology (and is quite useful for lots and lots of things), but it seems very tied to smoothness and characteristic ${0}$ issues.

Anyway, it was good to point that out, but many of you might be frustrated because we already knew that de Rham was going to be a failure in characteristic ${p}$. If you’ve been introduced to ${\ell}$-adic cohomology, then you probably already realized that this was basically invented to solve some of these problems. Now we’ll impose the extra condition that ${k}$ be algebraically closed. Recall that we can just pick a prime ${\ell\neq p}$ and define ${H^i_\ell (X)=H^i(X_{et}, \mathbb{Z}_\ell)=\lim H^i(X_{et}, \mathbb{Z}/\ell^n)}$. This is a ${\mathbb{Z}_\ell}$-module.

Using the comparison theorem we see that if ${k=\mathbb{C}}$, then ${H^i_\ell(X)\simeq H^i(X^{an}, \mathbb{Z})\otimes \mathbb{Z}_\ell}$. This is quite nice. ${H^i_\ell(X)}$ tells us exactly the topological information we sought, since we recover the rank of ${H^i(X, \mathbb{Z})}$ and the ${\ell}$-primary torsion. Thus if we run through all primes ${\ell}$ we can completely recover ${H^i(X, \mathbb{Z})}$ up to isomorphism. Also, we see that the rank of ${H^i_\ell(X)}$ is finite and independent of ${\ell}$.

Here is the first of many problems. Although for a smooth projective scheme in positive characteristic we do get that the rank of ${H^i_\ell(X)}$ is finite, it is unknown if this is a number independent of ${\ell}$. The second much more severe problem is that we only have the reasonable properties listed in the previous paragraph for ${\ell\neq p}$. So even though we can get a bunch of topological information about ${H^i(X, \mathbb{Z})}$, we are unfortunately left clueless about the ${p}$-torsion. We also lose a bunch of classical theorems (even when ${\ell\neq p}$) such as the Lefschetz hyperplane theorem. Or if we consider the induced map from ${X\rightarrow X}$ on ${H^i_\ell(X)\rightarrow H^i_\ell(X)}$ there may be powers of ${p}$ in the denominator of the characteristic polynomial that we won’t ever be able to get information about.

Clearly, ${\ell}$-adic cohomology isn’t going to quite do the trick. All I seem to be doing is complaining about how these cohomology theories are failing what I want, but I haven’t ever told you exactly what it is that I do want from a cohomology theory. So next time we’ll start working on stating those properties and seeing if anything we’ve come across satisfies them and if not how we’re going to construct something that does.

## Hodge and de Rham Cohomology Revisited

I was going to talk about how the moduli of K3 surfaces is stratified by height in positive characteristic and some of the cool properties of this (for instance, “most” K3 surfaces have height 1). Instead I’m going to shift gears a little. We’ve talked about ${\ell}$-adic étale cohomology, Witt cohomology, cohomology on any site you want to put on ${X}$, de Rham cohomology, and we’ve implicitly used Hodge theory in places. Secretly we’ve been heading straight towards cyrstalline cohomology. I think it might be neat to start a series of posts on how each of these relate to eachother and then really motivate the need for crystalline stuff.

Away from this blog I’ve been thinking about degeneration of the Hodge-de Rham Spectral Sequence a lot. Suppose for a minute we’re in the nicest situation possible. We have a smooth variety ${X}$ over ${\mathbb{C}}$. This means we can look at the ${\mathbb{C}}$-points and get an actual complex manifold. We defined the algebraic de Rham cohomology awhile ago to be ${H^i_{dR}(X/\mathbb{C}):=\mathbf{H}^i(\Omega_{X/\mathbb{C}}^\cdot)}$ the hypercohomology of the complex ${0\rightarrow \mathcal{O}_X\rightarrow \Omega^1\rightarrow \Omega^2\rightarrow \cdots}$. Since we’re in this nice case, this actually agrees perfectly with the standard singular cohomology on the manifold with coefficients in ${\mathbb{C}}$ (and by the de Rham theorem, the standard de Rham cohomology).

On a complex manifold we also have a nice working notion of Hodge theory. The Hodge numbers are $h^{ij}=\dim_{\mathbb{C}}H^j(X, \Omega^i)$ which we would normally derive through the Dolbeault resolution. We also have a Hodge decomposition ${H_{dR}^j(X/\mathbb{C})=\bigoplus_{p+q=j} H^q(X, \Omega^p)}$.

How do we see this using fancy language? Well, merely from the fact that de Rham cohomology is defined as the hypercohomology of a complex, we get the spectral sequence arising from hypercohomology. Without doing any work we can just check what this spectral sequence is and we find ${E_1^{ij}=H^j(X, \Omega^i)\Rightarrow H_{dR}^{i+j}(X/\mathbb{C})}$. This is because the ${E_1^{ij}}$ terms come from resolving each individual part of the complex which by definition just gives sheaf cohomology of the ${\Omega^i}$.

Of course, there was nothing special about ${X}$ being over ${\mathbb{C}}$, we could just as easily be over an arbitrary field and all of this still works. There is a great theorem that says that this spectral sequence degenerates at ${E_1}$ if ${X}$ is smooth over a characteristic ${0}$ field. There are several known proofs, some more analytic and some more algebraic. The coolest one is certainly by Deligne and Illusie.

They prove this preliminary result that if ${X}$ is smooth over a field ${k}$ of characteristic ${p}$ where ${p>\dim(X)}$ and ${X}$ has a lift to ${W_2(k)}$, then the Hodge-de Rham spectral sequence degenerates at ${E_1}$. Maybe we’ll talk about how this is done some other day, but if you know about the Cartier isomorphism then it is related to that. Using this side result that seems to be about as unrelated to the characteristic ${0}$ case as possible they then amazingly prove the characteristic ${0}$ case by reducing to positive characteristic and using a Lefschetz principle type argument.

Now despite the fact that H-dR degenerating being the norm in characteristic ${0}$, it turns out to be not so much the case in positive characteristic, so it is really shocking that to prove the characteristic ${0}$ case they moved themselves to this situation where it was likely not to degenerate. But we’ll get a better intuition later for why this wasn’t as risky as it sounds. Namely that since it came from characteristic ${0}$, there wasn’t going to be a problem lifting it back so the lifting to ${W_2(k)}$ was not a problem. It seems that the obstruction to being able to do this is almost exactly the failure of degeneracy. Recall that every K3 surface lifts to characteristic ${0}$, so (if you don’t know the proof of this) you’d expect the H-dR SS to degenerate at ${E_1}$. It might be a fun exercise for you to try to figure out why this is (very important hint: there are no global vector fields on a K3 so ${h^{1,0}=0}$).

Before ending this post it should be pointed out that all of this can be done in the relative setting as well. We actually originally defined de Rham cohomology purely in the relative setting without thinking about it over a field like we did today. Suppose ${\pi: X\rightarrow S}$ is a smooth scheme. The relative H-dR SS is given by ${E_1^{ij}=H^j(X, \Omega^i_{X/S})\Rightarrow \mathbf{R}^{i+j}\pi_*(\Omega_{X/S}^\cdot)=H_{dR}^{i+j}(X/S)}$.

We’ll continue with this next time, but I’ll just leave you with the thought that you can basically formulate for any class of schemes you want a large open problem by asking yourself whether or not the HdR SS degenerates at ${E_1}$ or at all.

## Gauss-Manin Connection 2

It’s probably been awhile since you read the first post in this series, so I’ll quickly remind you of the key point. ${S}$ is a smooth scheme over a field ${k}$. We fixed connection ${\rho}$ on ${\mathcal{E}}$. Then given a derivation ${\delta}$ corresponding to ${D: \Omega\rightarrow \mathcal{O}_S}$, then for any element of ${\mathcal{D}er_k(\mathcal{O}_S)}$, the sheaf of germs of ${k}$-derivations we can compose the maps we have and we get ${\overline{D}\in \mathcal{E}nd_k(\mathcal{E})}$.
So every connection gives a map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$, ${\delta\mapsto \overline{D}}$ and we had a relation between ${D}$ and ${\overline{D}}$, and any such map satisfying the relation comes from a connection.

Now we’ll go back to the construction of the Gauss-Manin connection from last time. We haven’t actually checked that it is a connection or that it is flat. Recall that it is just one of the maps we get from the spectral sequence associated to the filtration of the complex ${\Omega_{X/k}^\cdot}$. Now the filtration is compatible with taking wedge products (${F^i\wedge F^j\subset F^{i+j}}$) and the functors ${\mathbf{R}^q\pi_*}$ are multiplicative, so we have a product structure on the terms of the spectral sequence as follows.

If we take sections of the sheaves over an open, then ${E^{p,q}_r\times E^{p',q'}_r\rightarrow E^{p+p', q+q'}_r}$ by ${(e,e')\mapsto e\cdot e'}$. If you want the actual construction see Godement. The product satisfies a few important properties. We have a type of anti-commutativity ${e\cdot e'=(-1)^{(p+q)(p'+q')}e'\cdot e}$. Also we know how it behaves under the differential map: ${d_r(e\cdot e')=d_r(e)\cdot e'+ (-1)^{p+q}e\cdot d_r(e')}$.

In particular, let’s look at what this product rule for the differential is for the Gauss-Manin map. For ${\nabla=d_1^{0,q}:E_1^{0, q}\rightarrow E_1^{1,q}}$ which is really mapping ${\mathcal{H}^q_{dR}(X/S)\rightarrow \Omega_{S/k}\otimes \mathcal{H}^q_{dR}(X/S)}$, the differential is really just ${d_{S/k}\otimes Id}$. Thus that rule says that ${\nabla(\omega\cdot e)=d\omega\cdot e+(-1)^0\omega\cdot \nabla(e)}$. So it is a connection!

The curvature is easily seen to be ${d_1^{1,q}\circ d_1^{0,q}}$ and since the ${d_1}$‘s are maps of a complex we get that it is ${0}$, and hence ${\nabla}$ is flat and hence the Gauss-Manin connection is integrable. We’ve now proved the theorem that any smooth ${k}$-morphism of smooth ${k}$-schemes gives rise to a canonical integrable connection on the relative de Rham cohomology sheaves that is compatible with the cup product.

If you want a more explicit way to see what the map is see the paper, but it is kind of tedious since writing out how it appears in the spectral sequence you will quickly find that it is the connecting homomorphism when taking the long exact sequence after applying the functor ${\mathbf{R}^q\pi_*}$ to the exact sequence ${0\rightarrow \mathrm{gr}^{p+1}\rightarrow F^p/F^{p+2}\rightarrow \mathrm{gr}^p\rightarrow 0}$.

This is the third post on this topic, and I haven’t given you a reason to care yet. Here’s why we should care. One would hope (via a conjecture of Grothendieck) that there is some sort of relative de Rham Leray Spectral Sequence: ${E_2^{p,q}=\mathbf{H}^p(S, \Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S))\Rightarrow H_{dR}^{p+q}(X/k)}$. For the ${E_2}$-term to make any sort of sense we needed ${\Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S)}$ to be a complex, and since the Gauss-Manin connection is integrable it is a complex. Also, ${H_{dR}^{p+q}(X/k)}$ is defined to be ${\mathbf{H}^{p+q}(X, \Omega_{X/k}^\cdot)}$.

It turns out that when ${S}$ is affine such a spectral sequence exists. In case you’re wondering, affineness is needed for a nice proof of this because it makes certain cohomologies vanish. Deligne has proved it in a more complicated way when ${S}$ is not affine (but still with our standing assumptions). This is of great importance in proving every K3 surface lifts from characteristic ${p}$ to characteristic ${0}$.

## Connections and Curvature … on Schemes! … in Characteristic p?!

I feel bad about my absence. I lasted posted during winter break, and now winter quarter is completely over. I kept meaning to do a series on “well-known” algebraic geometry results and constructions that don’t appear with any amount of thoroughness in the references. I thought it would be good to get that information out there. Unfortunately, I had already written these things down into a notebook and just couldn’t motivate myself to type something up that I already had. Anyway, one thing led to another and I didn’t do any posts. I’m not sure why I’m trying to justify my absence with an excuse.

Recently I’ve been typing up a translation of Deligne’s argument (written down by Illusie) that every K3 surface in characteristic ${p>0}$ lifts to characteristic ${0}$. I’m not to the point of trying to understand it, but I wanted a typed version, so that when I get the background material (namely crystalline cohomology!) and go to understand it, I can just fill in the details into my typed notes quickly and easily. I also was curioius as to the overall format of the argument.

This led me to the 1968 paper by Katz and Oda called On the Differentiation of de Rham Cohomology Classes with Respect to Parameters. The next few posts will be about the main result from this paper. It is really quite amazing.

First, some definitions. We’ll always be working with a smooth scheme ${S}$ over a field ${k}$ (no assumptions here!). Let ${\mathcal{E}}$ be a quasi-coherent sheaf of ${\mathcal{O}_S}$-modules. We’ll write ${\Omega}$ for ${\Omega^1_{S/k}}$ and unless otherwise noted, all tensor products will be over ${\mathcal{O}_S}$. We say that ${\nabla}$ is a connection on ${\mathcal{E}}$ if it is a homomorphism ${\nabla: \mathcal{E}\rightarrow \Omega\otimes \mathcal{E}}$ that satisfies the “Leibniz rule”.

In other words, ${\nabla(fg)=f\nabla(g)+df\otimes g}$. This is the standard shorthand meaning ${\nabla(U): \mathcal{E}(U)\rightarrow \Omega(U)\otimes \mathcal{E}(U)}$ satisfies the rule where ${f\in \mathcal{O}_S(U)}$, ${g\in \mathcal{E}(U)}$ and ${df}$ is the image of ${f}$ under the universal map ${\mathcal{O}_S\rightarrow \Omega}$.

Given a connection ${\rho}$, we get homomorphisms for all ${i}$, ${\rho_i: \Omega^i\otimes\mathcal{E}\rightarrow \Omega^{i+1}\otimes \mathcal{E}}$. These are given by ${\rho_i(\omega\otimes e)=d\omega \otimes e+(-1)^i \omega\wedge \rho(e)}$.

The notation is just the one that makes sense: ${\rho(e)\in \Omega\otimes \mathcal{E}}$, so it looks like ${\tau\otimes \epsilon}$. So we define ${\omega\wedge \rho(e)=\omega\wedge(\tau\otimes \epsilon)}$ to be ${(\omega\wedge \tau)\otimes \epsilon\in \Omega^{i+1}\otimes \mathcal{E}}$.

Now we define the curvature of the connection ${K:\mathcal{E}\rightarrow \Omega^2\otimes \mathcal{E}}$ to be the map ${\rho_1\circ \rho}$. The curvature is related to the other ${\rho_i}$ by an easy check ${\rho_{i+1}\circ \rho_i(\omega\otimes e)=\omega\wedge K(e)}$.

This gives some sort of meaning to the curvature now. If the curvature is ${0}$, then the natural de Rham-like sequence we get from a connection by stringing together the ${\rho_i}$ as follows ${0\rightarrow \mathcal{E}\stackrel{\rho}{\rightarrow} \Omega\otimes\mathcal{E}\stackrel{\rho_1}{\rightarrow} \Omega^2\otimes \mathcal{E}\rightarrow \cdots}$ is an honest complex that we can take cohomology with respect to, since ${\rho_{i+1}\circ \rho_i=0}$.

When this happens we call the connection ${\rho}$ integrable. Now let ${\mathcal{D}er_k(\mathcal{O}_S)}$ be the sheaf of germs of ${k}$-derivations of ${\mathcal{O}_S}$ into itself. From the fact that the module of differentials is a representing object, we get that as a sheaf of ${\mathcal{O}_S}$-modules, ${\mathcal{D}er_k(\mathcal{O}_S)\simeq \mathcal{H}om_{\mathcal{O}_S}(\Omega, \mathcal{O}_S)}$.

Let ${\mathcal{E}nd_k(\mathcal{E})}$ be the sheaf of germs of ${k}$-linear endomorphisms of ${\mathcal{E}}$. Given any connection ${\rho}$ on ${\mathcal{E}}$ we get an induced ${\mathcal{O}_S}$-linear map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ as follows. Let ${\delta}$ be a derivation, then it corresponds to a map ${D: \Omega\rightarrow \mathcal{O}_S}$.

So consider the composition ${\overline{D}:\mathcal{E}\rightarrow \Omega\otimes \mathcal{E}\rightarrow \mathcal{O}_S\otimes \mathcal{E}\simeq \mathcal{E}}$, where the first is the connection and the second is ${D\otimes Id}$. This gives the map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ as ${\delta\mapsto \overline{D}}$.

Lastly for today, note that we get a nice relation between ${D}$ and ${\overline{D}}$ as follows ${\overline{D}(fe)=D(f)e+f\overline{D}(e)}$ and that any map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ satisfying this relation comes from a unique connection on ${\mathcal{E}}$.

Today was just a bunch of notation and definitions, but next time it should get more interesting.

## The Exterior Derivative

I need to stop putting blogging off until night, because then I just say I’ll do it the next morning, and then I just say I’ll put it off until night, etc. This is the main source of why I haven’t posted anything for awhile.

I think we only need one more tool out of our differential forms bag. Define $d:\Omega^k(M)\to \Omega^{k+1}(M)$ to be the unique $\mathbb{R}$-linear operator satisfying

1) If $\omega\in\Omega^k(M)$ and $\eta\in\Omega^l(M)$, then $d(\omega\wedge\eta)=d\omega\wedge\eta + (-1)^k\omega\wedge d\eta$.
2) $d\circ d\equiv 0$
3) If $f\in\Omega^0(M)=C^\infty(M)$ and $X$ is a smooth vector field, then $df(X)=Xf$.

Note that this last part is sort of the motivation for this operator. We have that 0-forms are just smooth functions, and that this operator gives the differential of the function. So it is a generalization of the differential to all forms. It is non-trivial to show such an operator exists and is unique, but I’ll define an operator that does the trick next, and leave the checking that it actually works as an exercise.

The above definition tells me almost nothing about how to actually compute what the operator does. Let’s look at it in coordinates. Let $\omega=\sum_J \omega_Jdx^J$, where $\omega_J$ are smooth functions, and the multi-indexes are in some natural order. Then $\displaystyle d\left(\sum \omega_Jdx^J\right)=\sum d\omega_J\wedge dx^J$, where we already said that the operator on 0-forms just gives the differential.

Thus we get $\displaystyle d \left( \sum \omega_Jdx^{j_1}\wedge \cdots \wedge dx^{j_k}\right) =\sum_{J}\sum_{i}\frac{\partial \omega_J}{\partial x^i} dx^i\wedge dx^{j_1}\wedge \cdots \wedge dx^{j_k}$.

There are so many interesting things we can go on to say about this, but none are relevant to near future posts.

I’ll just briefly mention a few. The condition about $d\circ d\equiv 0$ actually gives us a cochain complex $\Omega^0(M)\to \Omega^1(M)\to \Omega^2(M)\to \cdots$. If we look at the cohomology of this, we have what is known as de Rham cohomology.

Another interesting bit is that if we take our manifold to be $\mathbb{R}^3$, then this operator gives us lots of our familiar calc 3 operators (in slightly modified form). For example, the components of the exterior derivative of a 1-form, gives the curl of the components of the 1-form treated as a vector field. Also, the 2-form to 3-form computation similarly gives you the divergence.

One thing that might come up later is that the exterior derivative commutes with pullbacks. By this I mean that if $F:M\to N$ is a smooth map, then $F^*(d\omega)=d(F^*\omega)$. One way to prove this is to show it holds for smooth functions, and then induct and use the formula $d(\omega\wedge\eta)=d\omega\wedge (-1)^k\omega\wedge d\eta$.

The last sort of interesting thing to mention is that there is a coordinate independent form of the exterior derivative, but I’m pretty sure I’ve never actually used it.