A Mind for Madness

Musings on art, philosophy, mathematics, and physics


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Sheaf of Witt Vectors 2

Recall last time we talked about how we can form the sheaf of Witt vectors over a variety {X} that is defined over an algebraically closed field {k} of characteristic {p}. The sections of the structure sheaf form rings and we can take {W_n} of those rings. The functoriality of {W_n} gives us that this is a sheaf that we denote {\mathcal{W}_n}. For today we’ll be define {\Lambda} to be {W(k)}.

Recall that we also noted that {H^q(X, \mathcal{W}_n)} makes sense and is a {\Lambda}-module annihilated by {p^n\Lambda} (recall that we noted that Frobenius followed by the shift operator is the same as multiplying by {p}, and since Frobenius is surjective, multiplying by {p} is just replacing the first entry by {0} and shifting, so multiplying by {p^n} is the same as shifting over {n} entries and putting {0}‘s in, since the action is component-wise, {p^n\Lambda} is just multiplying by {0} everywhere and hence annihilates the module).

In fact, all of our old operators {F}, {V}, and {R} still act on {H^q(X, \mathcal{W}_n)}. They are easily seen to satisfy the formulas {F(\lambda w)=F(\lambda)F(w)}, {V(\lambda w)=F^{-1}(\lambda)V(w)}, and {R(\lambda w)=\lambda R(w)} for {\lambda\in \Lambda}. Just by using basic cohomological facts we can get a bunch of standard properties of {H^q(X, \mathcal{W}_n)}. We won’t write them all down, but the two most interesting (of the very basic) ones are that if {X} is projective then {H^q(X, \mathcal{W}_n)} is a finite {\Lambda}-module, and from the short exact sequence we looked at last time {0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n \rightarrow \mathcal{W}_{n-1}\rightarrow 0}, we can take the long exact sequence associated to it to get {\cdots \rightarrow H^q(X, \mathcal{O}_X)\rightarrow H^q(X, \mathcal{W}_n)\rightarrow H^q(X, \mathcal{W}_{n-1})\rightarrow \cdots}

If you’re like me, you might be interested in studying Calabi-Yau manifolds in positive characteristic. If you’re not like me, then you might just be interested in positive characteristic K3 surfaces, either way these cohomology groups give some very good information as we’ll see later, and for a Calabi-Yau’s (including K3’s) we have {H^i(X, \mathcal{O}_X)=0} for {i=1, \ldots , n-1} where {n} is the dimension of {X}. Using this long exact sequence, we can extrapolate that for Calabi-Yau’s we get {H^i(X, \mathcal{W}_n)=0} for all {n>0} and {i=1, \ldots, n-1}. In particular, we get that {H^1(X, \mathcal{W})=0} for {X} a K3 surface where we just define {H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)} in the usual way.


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Sheaf of Witt Vectors

I was going to go on to prove a bunch of purely algebraic properties of the Witt vectors, but honestly this is probably only interesting to you if you are a pure algebraist. From that point of view, this ring we’ve constructed should be really cool. We already have the ring of {p}-adic integers, and clearly {W_{p^\infty}} directly generalizes it. They have some nice ring theoretic properties, especially {W_{p^\infty}(k)} where {k} is a perfect field of characteristic {p}.

Unfortunately it would take awhile to go through and prove these things, and it would just be tedious algebra. Let’s actually see why algebraic geometers and number theorists care about the Witt vectors. First, we’ll need a few algebraic facts that we haven’t talked about. For today, we’re going to fix a prime {p} and we have an {\mathbf{important}} notational change: when I write {W(A)} I mean {W_{p^\infty}(A)}, which means I’ll also write {(a_0, a_1, \ldots)} when I mean {(a_{p^0}, a_{p^1}, \ldots)} and I’ll write {W_n(A)} when I mean {W_{p^n}(A)}. This shouldn’t cause confusion as it is really just a different way of thinking about the same thing, and it is good to get used to since this is the typical way they appear in the literature (on the topics I’ll be discussing).

There is a cool application by thinking about these functors as representable by group schemes or ring schemes, but we’ll delay that for now in order to think about cohomology of varieties in characteristic {p} and hopefully relate it back to de Rham stuff from a month or so ago.

In addition to the fixed {p}, we will assume that {A} is a commutative ring with {1} and of characteristic {p}.

We have a shift operator {V: W_n(A)\rightarrow W_{n+1}(A)} that is given on elements by {(a_0, \ldots, a_{n-1})\mapsto (0, a_0, \ldots, a_{n-1})}. The V stands for Verschiebung which is German for “shift”. Note that this map is additive, but is not a ring map.

We have the restriction map {R: W_{n+1}(A)\rightarrow W_n(A)} given by {(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})}. This one is a ring map as was mentioned last time.

Lastly, we have the Frobenius endomorphism {F: W_n(A)\rightarrow W_n(A)} given by {(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)}. This is also a ring map, but only because of our necessary assumption that {A} is of characteristic {p}.

Just by brute force checking on elements we see a few relations between these operations, namely that {V(x)y=V(x F(R(y)))} and {RVF=FRV=RFV=p} the multiplication by {p} map.

Now on to the algebraic geometry part of all of this. Suppose {X} is a variety defined over an algebraically closed field of characteristic {p}, say {k}. Then we can form the sheaf of Witt vectors on {X} as follows. Notice that all the stalks of the structure sheaf {\mathcal{O}_x} are local rings of characteristic {p}, so it makes sense to define the Witt rings {W_n(\mathcal{O}_x)} for any postive {n}. Now just form the natural sheaf {\mathcal{W}_n} that has as its stalks {(\mathcal{W}_{n})_x=W_n(\mathcal{O}_x)}.

Note that forgetting ring structure and thinking only as a sheaf of sets we have that {\mathcal{W}_n} is just {\mathcal{O}^n}, and when {n=1} it is actually isomorphic as a sheaf of rings. For larger {n} the addition and multiplication is defined in that strange way, so we no longer get an isomorphism of rings. Using our earlier operations and the isomorphism for {n=1}, we can use the following sequences to extract information.

When {n\geq m} we have the exact sequence {0\rightarrow \mathcal{W}_m\stackrel{V}{\rightarrow} \mathcal{W}_n\stackrel{R}{\rightarrow}\mathcal{W}_{n-m}\rightarrow 0}. If we take {m=1}, then we get the sequence {0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n\rightarrow \mathcal{W}_{n-1}\rightarrow 0}. This will be useful later when trying to convert cohomological facts about {\mathcal{O}_X} to {\mathcal{W}}.

We could also define {H^q(X, \mathcal{W}_n)} as sheaf cohomology because we can think of {\mathcal{W}_n} just as a sheaf of abelian groups. Let {\Lambda=W(k)}, then since {\mathcal{W}_n} are {\Lambda}-modules annihilated by {p^n\Lambda}, we get that {H^q(X, \mathcal{W}_n)} are also {\Lambda}-modules annihilated by {p^n\Lambda}. Next time we’ll talk about some other fundamental properties of the cohomology of these sheaves.


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Schemes

There was talk about schemes in the comments of my last post, so after reviewing what I’ve already posted about, I decided I may as well package it all up nicely in a brief post so that I’m allowed to use the term freely from now on.

First, recall the sheaf structure we already have. For any ring, R, we have the associated topological space Spec(R) and the sheaf of rings \mathcal{O}. Then the stalk for p\in Spec(R) is \mathcal{O}_p\cong R_p. Also, \mathcal{O}(D(f))\cong R_f for any f\in R.

Let’s extrapolate what was the important structure here. We really have a topological space and a sheaf of rings on it. We call this a ringed space. Morphism in this category are a pair (f, g): (X, \mathcal{O}_X)\to (Y, \mathcal{O}_Y), where f:X\to Y is continuous, and the sheaf structure is preserved, i.e. g: \mathcal{O}_Y\to f_*\mathcal{O}_X is a map of sheaves of rings on Y.

A ringed space is called a locally ringed space if each stalk is a local ring. I’m not sure how technical I should be about the definition of a local homomorphism. Essentially, we want to preserve localness on the homomorphisms induced on the stalks by the sheaf homomorphism. So a homomorphism is local if the preimage of the maximal ideal in one go to the maximal ideal in the other.

So without proof I’ll just state that a homomorphism of rings \phi : A\to B induces a natural morphism of locally ringed spaces (contravariantly), and conversely, given A and B, any morphism of locally ringed spaces Spec(B)\to Spec(A) is induced by a ring hom A\to B. The first statement essentially follows from laying down definitions, but it is not trivial. The second one requires some more thought.

Now we define a scheme. An affine scheme is a locally ringed space that is isomorphic to the spectrum of some ring. A scheme is a locally ringed space in which every point has an open neighborhood U such that (U, \mathcal{O}_X\Big|_U) is an affine scheme. Morphisms are in the locally ringed sense.

The easiest example would be a field, where the topological space is a point and the structure sheaf is the field back again. If we step the dimension up by one (and require the field to be algebraically closed for sake of example), then Spec (k[x])\cong \mathbb{A}_k^1

I may or may not return to elaborate. I sort of want to consolidate the algebra I’ve learned this quarter through a series of posts before doing anything else along the algebraic geometry side of things.


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The Structure Sheaf of a Variety

Alright, so I’m still taking this really round about way to the Nullstellensatz, but someday I’ll get there.

For those of you that know about sheaves, some of the things I’ve been talking about should be looking vaguely familiar. We haven’t fully gotten there yet, but that is what today is about.

I won’t explicitly define what a general sheaf is, but of course there is always wikipedia or a textbook if you really want to know.

Let’s think back to what we had before. We define what we called k[V] the coordinate ring on the algebraic set V. So now we do the natural thing, we look at the field of fractions of k[V] which we will denote k(V). You should say, “Wait a minute!” at this point, since we might have some “zero denominators.” So let’s hold off on actually defining this until we’ve built the way to work around the problem.

So as a set, f \in k(V) is something of the form f=g/h, where g, h \in k[V]. So it is a fraction of polynomials, or a rational function. The problem is that it is not defined at zeros of h. Luckily, zeros of polynomials are all we’ve been studying and talking about for awhile.

Call f \in k(V) regular at a point P \in V if there is a representation f=g/h such that h(P) \neq 0. In fact for any h \in k[V] we can define a set corresponding to where it can be in the denominator, i.e. V_h=\{P \in V : h(P) \neq 0\}. Note that this is just the principal open set we defined earlier for the Zariski topology, but now it seems to have vital use.

Let’s now define the local ring of V at P to be \mathcal{O}_{V, P}=\{f \in k(V) : \ f \ regular \ at \ P\}. Clearly this is a subring of k(V). The not as obvious fact is that it is actually local. If you want to check, the unique maximal ideal is the set of elements of the form f/g where f(P)=0 and g(P) \neq 0. So now some things are shaping up, since we have an object defined for sets and have a ring of functions at a point.

What would really be exciting is if this construction which seemed ad hoc by taking everything in the field of fractions and throwing out things that don’t work, actually turned out to be a nice localization of the ring. Define the ideal \overline{M}_P=\{ f \in k[V] : f(P)=0\}. So this is technically what we were calling \overline{I({P})} before. (The line meaning that we aren’t in k[x_1, \ldots, x_n] anymore, we’re in k[V]=k[x_1, ldots , x_n]/I(V). So this is is a maximal ideal and hence prime, so we can localize at it.

Exactly what we were hoping for actually does happen, i.e. k[V]_{\overline{M}_P}=\mathcal{O}_{V, P}. In words, the localization of the coordinate ring at \overline{M}_P.

Now for any open set U \subset V we define \mathcal{O}(U)=\{ f \in k(V) : f \ regular \ on \ U\}. And for convenience \mathcal{O}_V(\emptyset)={0}. So not only is \mathcal{O}_V(U) a ring, it is a k-algebra. This set of rings with the restrictions we defined last time form the structure sheaf \mathcal{O}_V, and the local ring \mathcal{O}_{V, P} is the stalk of the sheaf at P with the elements as the germ of functions at P.

So I’ll leave you with a nice way to rephrase some older posts: we should now think of k[V]=\mathcal{O}(V), and \mathcal{O}(V_h)=k[V][h^{-1}]=k[V]_h.

Severely edited: Sorry, some weird bug took out every backslash of this post rendering it incomprehensible. I’m really glad I decided to glance at it randomly.


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A closer look at Spec

Let’s think about what is going on in a different way. So now let’s think of f \in R elements of the ring as functions with domain Spec(R). We define the value of the function at a point in our space f(P) to be the residue class in R/P. This looks weird at first, since the image space depends on the point that you are evaluating the function.

Before worrying about that too much, let’s see if we can get this notion to match up with what we did yesterday. We have the nice property that f(P)=0 if and only if f \in P. (Remember that even though we think of f as a function, it is really an element of the ring).

Define for any subset of the ring S the zero set: Z(S)=\{P\in Spec(R): f(P)=0, \forall f \in S\}. Now from what I just noted in the previous paragraph, we get that these are just precisely the elements of Spec(R) that contain S, i.e. the closed sets of the Zariski topology. Thus we can define our basis for the Zariski topology to be the collection of D(f)=Spec(R)\setminus Z(f).

We also will want what is “an inverse” to the zero set. We want the ideal that vanishes on a subset of Spec. So given Y\subset Spec(R), define I(Y)=\{f \in R : f(P)=0, \forall P\in Y\}. Now this isn’t really an inverse, but we get close in the following sense:

If J\subset R is an ideal, then \displaystyle I(Z(J))=\sqrt{J}. Taking the ideal of the zero set is the radical of the ideal. And the radical has two equivalent definitions: \displaystyle \sqrt{J}=\cap_{P\in Spec(R), P\supset J} P=\{a\in R : \exists n\in \mathbb{N},  a^n\in J\}.

If we take the ideal and zero set in the other order we get that Z(I(Y))=\overline{Y} : the closure in the Zariski topology.

We can abstract one step further and put a sheaf on D(f). Note that for any f\in R we have that \{1, f, f^2, \ldots\} is a multiplicative set, so we can localize at it. Since I haven’t talked at all about sheaves, I’m not sure if I want to go any further with this, so maybe I’ll do some more examples next time and possibly start to scratch this surface.

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