What is the orientation of a ring?

My dear friend Alex Mennen and I had some fun this morning defining the orientation of a ring.

The motivating example is:

\(Z[x] \to Z\)
\(x \mapsto 1\)
\(x \mapsto -1\)

In the case of free abelian groups and null generated rings (i.e., there is no quotient ring that is a proper subring), asking for an orientation of the basis makes sense. It’s just a choice of generator.

What about non-free groups like \(\mathbb{Z}/p\)?

Well, any element of \(Z/p\), except \(0\), generates the ring, (e.g., the underlying elements of \(Z/3\) are of course \(0, 1, 2\); 1 = 1, 1+1 = 2, 1+1+1=0; 2 = 2, 2+2 = 1, 2+2+2 = 0). This is only true when \(p\) is prime (in the general case, where \(p\) is not prime, relatively prime elements can’t generate each other).

Take \(Z/p\), endow it with the discrete topology,

  1. remove the additive identity \((Z/p)\) {0}
  2. We now have \((p-1)\)-connected components. These are the possible choices of orientation.
  3. Pick an element of the ring {0}. This will be an element in a connected component, truly, it is a representative of a homotopy class of maps. (If we think of orientation simply as “which way is minus and which is plus,” in the case of \(Z[x] to Z\), \(x \mapsto -1\) might as well be \(x \mapsto -5\).)

What about things that are totally disconnected, like the p-adics?

As an example, let’s look at Z_3, with the following topology: start with a point, and draw three paths from the point, labeled “0,” “1,” “2,” at the end of each path, draw 3 paths, label those “0,” “1” and “2”, ad infinitum. A p-adic number is then on the fringe of this tree, and can be thought of as a string that describes the sequence of choices made from the point that we started with

path down the tree <=> p-adic number

following our recipe, we remove the origin, that is, all paths coming from the origin (which is on the fringe of this tree) …000000 (i.e., eliminate all paths that start with a 0). We now have 2 disconnected branches, that is, the branches 1 and 2. We choose either 1 or 2 as the orientation.

Perhaps this indicates, to deal with totally disconnected rings, we mod out by a maximal ideal and then we have a finite number of contractible components. But, I don’t know if this is true in general.

There is no consistent notion of orientation.

The natural path to look for is a humble addition law that expresses \(o(R_1 \otimes R_2)\) in terms of \(o(R_1)\) and \(o(R_2)\).

Alex pointed out that it cannot be done for rings, at least, not with our naive definition.

Z/4 and Z/6 both have 2-element orientation groups, but Z/4 tensor Z/4 is again Z/4, so it has a 2-element orientation group, and Z/4 tensor Z/6 is Z/2, which has a 1-element orientation group. Thus the orientation group of the tensor product of two rings is not determined by the orientation group of each ring.

We can see that orientation is not consistent even more clearly with this example and definition given by Alan Weinstein:

  1. On a finite set, define an “orientation” as an equivalence class
    of linear orderings, modulo even permutations. Is there a functorial way to associate to oddities on A and B oddities on their disjoint union (which I will denote simply by \(\cup\)) and their cartesian product which is consistent with the natural bijections \((A \cup B) \times C \rightarrow A \times C \cup B \times C\) and \(A \times (B \cup C) \rightarrow A \times B \cup A \times C\).
  2. As above, but replace finite sets by finite dimensional vector spaces, oddity by orientation, disjoint union by direct sum, and cartesian product by tensor product.

Let A have elements a and a’, B and C have one element each, b and c. Then the orders on \((A \cup B) \times C\) and \(A \times C \cup B \times C\) are related by one transposition.

Oh well, no consistent orientation after all :(.

Edit: Aaron Slipper commented that there is a very natural notion of the orientation of an ideal with an integral basis over a PID. Namely, every ideal is a sub module of full rank (as it contains a principal ideal of full rank). So the basis elements have a natural notion of orientation, in the same way that a basis of a vector space does.

Edit: Something that has continued to confuse me is the notion of having an \(h\)-orientation, where \(h\) is a cohomology theory. The name “orientation” comes from choosing an orientation on \(S^2\), and looking at how a (multplicative complex-orientable) cohomology theory encodes this in the target category. For example, Thom’s interpretation of orientation (via the Thom isomorphism) in \(\text{AbGrp}\) seems to be: a choice of which element in \(h^2(S^2)\) corresponds to the generator of the coefficient ring \(h^*\) — since that \(S^2\) is a Moore space and iso to \(CP^1\).

Edit: My previous confusion wrt complex orientation has been resolved here.

Group Contractions for Elliptic Curves

When you construct a particular sheaf over an elliptic curve and then continuously vary the elliptic curve, what happens to the sheaf? I’m not sure, so I am first trying to understand what group contractions mean for elliptic curves.

It’s easiest to talk about group contractions at the level of Lie algebras. We can always pass from the Lie algebra to the Lie group by exponentiation, and then quotient out by a discrete subgroup as needed, so no loss there.

A group contraction (in Lie theory) happens when we modify commutators to make them depend on t. And here is the modification I have in mind:
\([X, Y] = tZ\)
\([Y, Z] = X\)
\([Z, X] = Y\)
for all \(t\) not less than 0.

When \(t = 1\), this is the Lie algebra of the rotation group, which may be identified with the rotations of the unit 2-sphere.

Now, as \(t \to 0\), the above Lie algebras are, for each t, the Lie algebras of geometric symmetries of the 2-sphere of radius 1/t.

There’s the usual moduli stack of elliptic curves, and then we have the “compactified” one, and where we add points that are not actually represented by elliptic curve but are “degenerate limit cases.”

Over C all of this is easy to see: elliptic curves are given by \(C/\Gamma\), where \(\Gamma\) is a lattice. The lattice is spanned by vectors v, w. Assume you make w longer and longer, the limit can be sensibly interpreted as \(C/v\) (which is an infinitely long cylinder: if you increase one of the two radii of a torus towards infinity, you get just an infinitely long tube).

But what is \(C/v\)? WLOG we can assume \(v=2\pi i\) (or any other value). Now \(\exp: C \to C\) sends \(z+v\) to the same thing as z and takes image in \(C\){0}, so it induces a map \(C/v \to C\){0}. That map is a bijection. The corresponding group structure on \(C\){0} is just multiplication. Similarly, we can send both v and w to infinity and get back the additive formal group.

In the characteristic p case, wikipedia tells me that over \(F_3\), an elliptic curve has height 2 iff its j-invariant = 0 in \(F_3\).

So an example of a height 2 curve in \(F_3\) is \(y^2 = x^3 + x\) whereas \(y^2 = x^3 + x^2 + x\) is height 1. This is a stupid example for putting a parameter t can be put on an elliptic curve so that it goes from a height 1 to height 2 case.

Here we could put t as the coefficient of \(x^2\), as in: \(y^2 = x^3 + tx^2 + x\)

A slightly more general form for such contractions, assuming that all elliptic curves have an affine model like \(y^2 = x(x-1)(x-\lambda)\) (I don’t understand how this works yet), we could take an elliptic curve and add \(x(x-1)t\), and for some value of t, it should be supersingular.

\(y^2 = x(x-1)(x-\lambda)\)
\(= x^3 + (1-\lambda)x^3 + \lambda x\)

and so when we add \(x^2t + xt\) to one side
then \(y^2=x^3 + (1-\lambda+t)x^2 + (\lambda-t)x\).

How do I do this more generally over any field that we might care to put an elliptic curve over? Does this give us the ability to define group contractions over the whole moduli stack?

Thanks to Achim Krause for speaking with me about the complex case, Alex Mennen for suggesting that we add \(x(x-1)t\), Laurens Gunnarsen for explaining the concept of a group contraction, and Aaron Slipper for urging me to ask Laurens to explain the concept of a group contraction. All errors are mine alone.

Complex Analysis: Poles, Residues, and Child’s Drawings

Thanks to Laurens Gunnarsen for his superb pedagogy and for this amazing explanation on the incredible depth of connections springing from the Sperner lemma. All errors are mine not his. This started with a chain of events, sitting in on number theory seminars and encountering Abel’s differentials of the first and second kind, interest in the dessin, and led up to asking Laurens:

How do I understand poles and residues?

By understanding Riemann-Roch. But, first of all, you should know the zeroes and the poles of an analytic function, together with the residues of the function at each, pretty much tell you all you need to know about the function.

This is a little bit like saying that if you know the zeroes of a rational function — which is to say, the zeros of the polynomial that is its numerator — and the poles of a rational function — which is to say, the zeroes of the polynomial that is its denominator — then you basically know the rational function.

The roots of a polynomial determine it up to an overall multiple. That’s about all this idea involves, at the level of rational functions.

And meromorphic functions are only very slightly more complex things than rational functions, as it turns out.

A polynomial is determined by its roots (up to a scalar). When you add in multiplicity data, it’s completely determined. This multiplicity data is what the residues give you.

If two polynomials have the same roots with the same multiplicities, then they are indeed proportional.

Sidenote: This follows from the fundamental theorem of polynomial arithmetic.

Every polynomial (in one variable?) factors uniquely into \((x – a)^n\) factors, with the a and n varying from factor to factor. Well, so you know how uniqueness works, right? You have to exclude 1 and -1 from the primes, if you want uniqueness.

So, meromorphic functions don’t have to be rational functions, but they very nearly are.

What is the condition defining meromorphicity? It’s really just that the function be a conformal map wherever it is defined, which is a local differential condition. Essentially just the Cauchy-Riemann equations. Just a constraint on the first partial derivatives at a point, really.

So you impose that constraint at all points where it’s possible to do so, and the resulting functions are called meromorphic.

They’re like analytic functions, except that they may have singularities at finite points. As opposed to analytic functions, which only have singularities at infinity.

Look at it this way: \(\sin(x)\) is analytic, but it isn’t a polynomial. \(\tan(x)\) is meromorphic, but it isn’t rational.

Analytic functions are like generalized polynomials. \(\sin(x)\) is like a polynomial, in that it has isolated roots, and is (almost) uniquely determined by those roots.

We have, in fact, Euler’s product formula,
\\(\sin(x) = x[(1 – (x/\pi)^2][(1 – (x/2\pi)^2][(1 – (x/3\pi)^2]… \\), which is like the expression

\\(a + bx + cx^2 + … + mx^n = a(1 – x/r)(1 – x/s) … (1 – x/t)\\)
where \(r, s, …, t\) are the \(n\) roots of the polynomial of degree \(n\) on the left-hand side.

Of course I do not mean to constrain these roots by insisting that no two of them may coincide. I want, on the contrary, to allow coincidences of that sort.

Sidenote: A general degree-n polynomial factors, at least over the complex numbers, at least over the complex numbers it does. In complete generality, you have all kinds of annoyances. But in special situations you can get around them. I believe there is indeed a p-adic version of Riemann-Roch, but I don’t know exactly what it is. I mean p-adic analysis is pretty cool though. Probably the best example is really the first one, namely, Dirichlet’s theorem. Dirichlet proved that if a, b are relatively prime, then the sequence a + bn, as n ranges over the natural numbers, contains infinitely many primes. And indeed that the density of primes in this sequence is just 1/b times the density of primes in the integers. Here’s an example of a nice theorem in p-adic analysis: if {a_n} is a sequence which is such that \(a_n \to 0\) as \(n \to \infty\), then Sum(a_n) exists. (p-adically, that is.)

Riemann-Roch asserts a sort of mismatch between poles and zeroes for meromorphic functions on a Riemann surface, with this mismatch due to the Euler characteristic.

Riemann-Roch basically tells you that the genus of a Riemann surface is detectable by examining the meromorphic functions defined on it. This is possible for essentially for the same reason that the average Gaussian curvature of a surface is determined by and determines the genus.

That is, the same mechanism is at work in Riemann-Roch as is at work in Gauss-Bonnet.

We might say that instead of total positive and negative curvature we have number of poles.

Poles, on the one hand, and zeroes, on the other.
Poles are zeroes of the reciprocal of the function.

In some sense, it boils down to Poincare-Hopf, which in turn may be thought of as a sort of elaborate consequence of Sperner’s lemma.

The Sperner lemma is the combinatorial root of almost all the theorems of (classical real and complex) analysis with a topological flavor. It really ought to be required reading for everybody interested in any topological applications in analysis! Here‘s an elegant and well-illustrated paper on it.

As the paper above shows, you take an arbitrary compact surface (e.g, the surface of the earth) and you remove points and make branch cuts until you have the surface decomposed into a (typically small) number of components to which Sperner’s lemma applies.

That is, it’s decomposed into topological discs with boundaries on which the behavior of something or other of interest (e.g., a vector field) may be regarded as giving a “Sperner coloring” there, and in the interior.

It looks like the Cauchy residue theorem follows from the Sperner Lemma.

Yes, that’s right: the Cauchy theorem is pretty much just an artful elaboration of the Sperner lemma. Of course there is a little bit more to it. One needs to say, as in the paper I just shared with you, how the analytic structure on the surface gives rise to a Sperner coloring on some simplicial approximation to the surface.

This looks similar to a dessin d’enfant. Given a dessin, can I tell with certainly which polynomial it represents? How?

Yes. This, in fact, is what Grothendieck found so impressive about dessins. Well, Grothendiek and, of course, Belyi. They determine a Riemann surface pretty much completely.

The crucial thing to appreciate is that a dessin is essentially a recipe for making a Riemann surface.

And it is possible to make a Riemann surface with only combinatorial information of the sort provided by a dessin because all Riemann surfaces with genus at least 2 may be given a geometry of constant negative curvature, which is what people call the Uniformization Theorem.

Once you have a geometry of constant negative curvature on your Riemann surface, you can triangulate it, and each triangle will resemble a triangle in the upper half plane.

Globally the triangles need not be connected as would be a triangulation of the upper half plane, of course. But each individual triangle is just an ordinary hyperbolic triangle.

So then all you need is incidence data. Which triangles are connected how to which others?

This sort of thing is precisely what you get from a so-called Belyi function. Which, in turn, is specified by a dessin.

A Belyi function basically assigns the value 0 to each vertex of the simplicial complex determining the Riemann surface, the value 1 to the midpoint of each edge, and the value infinity to each face center.

So you can tell how many vertices the simplicial complex has just by looking at the inverse image of 0. You can also tell how these vertices are connected, and which sets of edges bound faces.

And so forth, and so on.

Essentially, a Belyi function keeps track of the vertices, edges, and faces of a triangulation by hyperbolic triangles of a compact Riemann surface of sufficiently large genus (g is 2 or more). These things are also specified by a dessin.
From the dessin you can write down the Belyi function, and conversely.

Klein does all this very explicitly in several very illustrative cases. He noticed, but did not assert as a theorem, the remarkable theorem of Belyi.

It is an extraordinary theorem, that any Riemann surface that may be mapped meromorphically and surjectively to the Riemann sphere in such a way that the mapping is branched only over 0, 1 and infinity must in fact be an algebraic surface.

That is, a Riemann surface with this property is the zero set of a polynomial WITH ALGEBRAIC COEFFICIENTS.

Amazing, really, that you can conclude that essentially number-theoretical fact from these complex analytic and geometric data.

Grothendieck called Belyi’s theorem miraculous. He was stunned by it. Amazed. He maintained that he had never been so impressed by any mathematical result, before or since. It essentially gives a combinatorial means of investigating the absolute Galois group!

What is the difference between homotopy and coherent homotopy?

This question had been bugging me for a while, and I have been unable to find a source that is suited to the beginning topologist. Eric Peterson kindly answered this for me, and I found his explanation so astoundingly beautiful that I wish to share on the off chance that you, dear reader, will similarly appreciate this visually rich narrative. All errors are mine and not his.

Coherence is essentially about the existence of diagram categories. For instance, suppose you have some homotopy class \(A \wedge A \to A\), which I’d like you to think of as a multiplication map on \(A\). You can think of this as a homotopy class

\\(S^0 \to F(A \wedge A, A)\\)

in the appropriate function spectrum. Then, given any map of this signature, you can build two maps \(A \wedge A \wedge A \to A\) out of it:

\(A \wedge A \wedge A = A \wedge (A \wedge A) \to A \wedge (A) = A \wedge A \to A\), and
\(A \wedge A \wedge A = (A \wedge A) \wedge A \to (A) \wedge A = A \wedge A \to A\),

corresponding on the level of function spectra to a map

\\(F(A \wedge A, A) \to F(A \wedge A \wedge A, A)^{\partial \Delta^0}\\)

where the right-hand object has the distinguished property

\(S^0 \to F(A \wedge A \wedge A, A)^{\partial \Delta^1} = (\partial \Delta^1_+ \wedge S^0) \to F(A \wedge A \wedge A, A)\)
\(= (\partial \Delta^1_+) \to F(A \wedge A \wedge A, A) = (S^0 \vee S^0) \to F(A \wedge A \wedge A, A)\)

i.e., it selects two homotopy classes.

The assertion that the multiplication is associative is the same as saying that these two homotopy classes lie in the same path component of the function space, and a witness to this fact is a map \(h\) in

\\(S^0 \xrightarrow{h} F(A \wedge A \wedge A, A)^{\Delta^1} to F(A \wedge A \wedge A, A)^{\partial \Delta^1}\\)

which upon restriction to the endpoints of the 1-simplex recovers the two-different-associations map specified above. Then, given such an \(h\), you can follow a similar procedure with \(F(A \wedge A \wedge A \wedge A, A)\) to build an element of

\\(S^1_+ \to F(A \wedge A \wedge A \wedge A, A)\\)

that, naively, could be nonzero. This is distinctly different from the algebraic case, where there are still function objects, but they’re all sets, and hence they will never have any \(\pi_1\), and so this map is always fillable. With spaces or spectra (or anything else homotopical), this is no longer guaranteed, and so you have to assure yourself that you can perform this filling. And then, once you’ve chosen such a filling, you can use it to build an element of \(\pi_2\) of the next thing…

There are different ways to bundle this formalism, but all of them rely essentially on the existence of these “diagram categories”


where \(C\) is some suitable category (like: a category enriched in spaces or an (\(\infty\), 1)-category, as is the case for Spectra) and \(X\) is some indexing object (like: a simplicial set, as is the case for \(\Delta\)). The two big names are “quasicategories” (which give you a little more than this but are generally useful) and “derivators” (which give you exactly this and are considerably less popular for general use). Continuing the example above, an associative algebra object is then a map

\\(\Delta \to C\\)

(or a point in \(C^\Delta\)) which restricts on objects to the tensor powers of some fixed object in \(C\) (cf. of Higher Algebra for a taste).

In any case, “coherent” is meant to express that the process is self-referential: each choice of previous homotopy influences choice of the next homotopy (and, indeed, even the ability to choose one at all). Note also that the process is complicated dramatically when considering commutative algebra objects, since then you have all these shuffle maps to keep track of.

This summary is sort of separate, though, from what coherence is useful for. The low-level summary is that ring spectra in the homotopy category have homotopy classes of maps between them (which homotopy-commute with the ring structure maps), but structured ring spectra have function spaces (or spectra) between them (which describe how to slide a maps between these \(\Delta\)-indexed objects around).

So, if you want to do homotopy theory with ring objects, you don’t have access to that until you start saying structured things.

The high-level summary of the Elmendorf-Kriz-Mandell-May book is that coherent ring spectra are the spectra for which you can build a theory of algebra. Coherent ring spectra have categories of modules (with function spaces between them, so this is a statement about homotopies), those categories have suitably monoidal tensor products (where e.g. the associativity of the tensor product is also controlled by homotopies), and the tensor products are controlled by spectral sequences grounded in homological algebra (i.e., there are machines interchanging doing homotopy with algebraic objects for doing algebra with homotopical objects). So, if you have some algebraic question (about formal group laws) and some homotopical objects (like \(MU\) and \(BP\)), then this is the sort of framework you’d want to design in an effort to repeat algebraic proofs, applied to these fancier objects.