I think I am thinking of (for instance) proposition 1.2(e) of this: https://math.stanford.edu/~vakil/0708-216/216class18.pdf

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Ha! “Proper Open Subschemes of a Proper Scheme Turn Out Not to Be”

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Some comments:

1) I think you have a typo in the second paragraph of ‘Connections to Langlands’. Namely, you are missing ‘up to isogeny’ in your first claim about connected -divisible groups of dimension and height being unique.

2) Maybe it’s worth mentioning what this formal group is. So, for height it’s , for height it’s with supersingular, and in general it’s one of the finite Witt schemes that, if we wanted to, we could figure out by slope considerations. OK, let’s do this. In your langue from the second talk we have that has height and dimension . Thus, it has to be . This is, not shockingly, related to the fact that the Shimura variety coming into play for Taylor-Harris’s proof is .

3) One makes precise the fact that is like the local analogue of a Shimura variety, I believe, by the statement that it’s the rigid generic fiber of the completion of the local ring of the Shimura variety at the associated point.

4) I think it’s more natural, and less scary if you don’t like adic spaces, to think about these ‘s as being rigid generic fibers of formal schemes which are, literally, just deformation rings of formal -modules with level structures.

5) One usually wants to get the whole action of (I think you meant by the way) on the cohomology of our space to get Jacquet-Langlands. One does this by considering the moduli problem of deformations but only requiring that the special fiber have a quasi-isogeny to our fixed formal group.

6) I think you mean to say “ is just with formal fiber “.

7) Pedantic point is that doesn’t act on the Shimura variety. In fact, (or ) doesn’t even act on the Shimura variety. It acts by ‘correspondences’ which, in turn, gives an *honest* action on *cohomology*.

8) As mentioned in the lecture, one needs to be careful about the direct analogy to higher . Namely, for and one can use and as the global Shimura varieties. Unfortunately, there is no Shimura variety for . Thus, one uses something like .

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Some comments:

1) Typo in your first paragraph. You say that the Dieudonné module is determined by its slopes when, of course, you mean that the Dieudonné isocrystal is determined by its slopes.

2) Typo ’50 treatise’ should perhaps be ‘Alex Youcis level tour de force of mathematical exposition’ or, maybe, ’50 page treatise’. 🙂

3) Just a pedantic comment, but “The above is nothing crazy. All we’ve done is axiomatized the notion of a tensor product and shown that the constructions we enjoy for vector spaces extend to this setting.” should perhaps be that you’ve axiomatized tensor products *and* duals.

4) Typo in your definition of fiber functor, unless you’re using some much more general notion than I am. Namely, fiber functors are supposed to exact and faithful.

5) Some comments about the confusiong of whether things belong over or , but we discussed this in your lecture.

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1) Connected and geometrically connected are the same thing for group schemes, so you can drop that pesky adjective when discussing them. In fact, if is, say, finite type and has a -point then connected implies geometrically connected. This is a nice exercise if you haven’t seen it before.

2) Minor pedantic point. The equivalence between connected -divisible groups and -divisible formal groups only holds (as far as I know) over a Noetherian local ring with with residue characteristic . You probably only were thinking over , but I’ll add this regardless.

3) A minor typo or, rather, ambiguity. You say that “Thus we cannot have a p-divisible group where both it and its dual are etale.”. Whereas the previous sentence as just talking about perfect fields in general, this sentence is only talking about perfect fields of characteristic . This observation also follows from the fact that if is étale and -torsion then, geometrically, it’s some product of ‘s whose Cartier dual is a product of some ‘s which are not étale.

4) Again, pedantic point since you sort of fix this in the next sentence, but, of course, being abelian is not equivalent to being a category of modules. An abelian category is a category of modules if and only if it has a ‘compact progenerator’ which, I believe, is somehow the ‘dual’ of your .

5) As we talked about in the seminar, I think you really want to change your (the Dieudonné ring) to be modulo relations. I think that the makes sense if and are supposed to be nilpotent on . This is the case if is a finite flat connected group scheme over . Since your goal is to build , for a -divisible group, as a limit of such things this is, perhaps, where the two notions meet.

6) I think that your third property of the Dieudonne module is slightly confusing (although correct!) and is one reason to use the contravariant Dieudonne module. Namely, it’s not hard to show that a scheme (with ) is étale if and only if the relative Frobenius map is an isomorphism. Thus, one would expect that is étale if is bijective. This is what does happen for the contravariant theory, but and get switched when one goes to the covariant theory. Just a thought.

7) To answer something we talked about in the lecture, we can explicitly describe the such that . Namely, they should be the truncated Witt scheme (or maybe the indices reversed since we’re covariant). You can read about them here: https://people.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf

8) Silly typo: in the last properties bullet the phrase ‘anf’ is, what I can only assume, a jive way of saying ‘and’. 🙂

9) With regards to your references, the all-encompassing theory of Fontaine is found in *Groupes -divisibles sur les corps locaux’.

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Many thanks for pointing this out and for your kind words!

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P.S. This course is (so far) an excellent summary of this material.

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There is a typo in your second paragraph. You either meant or .

The equality (as I know you know, but you didn’t make clear) only happens over . Also, over the algebraic closure there is also a nice description of, say, when is supersingular. In this case it’s the non-trivial extension of by itself. In fact, it’s evident that and since this can’t be or for obvious reasons, it must be . Similarly, the quotient is order and not or . So, you get that it is such an extension. One can check that there are three groups which are extensions of by itself—, , and something we’ll call . We know that can’t be the first two, and so it must be . One can do all of this much easier with Dieudonne theory, but since that is the topic of your next talk, I assumed you’d want to avoid that.

You should note that Cartier duality is, in general, contravariant. So, when you write your map comes from the multiplication by map .

I guess it’s worth saying, although it might be obvious, that you’re not really considering in the category of locally ringed spaces, but in the category of locally topologically ringed spaces. You should also mention why connectedness is important here.

Also, you should be careful when you say isogeny because it somehow calls to mind the wrong property of the morphism. The key point is that it’s locally free—that is a projective -module.

Also, although not strictly necessary, when it comes to thinking about things like -divisible groups, I think that it really is helpful to recall Serre’s fact that one can think of , for a projective abelian scheme, as being the sheaf given by . This immediately proves the claim that for any with dual map that . This can also be acheived by the generalized Weil pairing.

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