Rabinowitsch in Translation

One of our projects in number theory led us to thinking about the class number problem, which has a story too long and interesting to recount here. See the survey of Goldfeld from an old AMS Bulletin to get an overview. Briefly, the question is about the degree to which unique factorization holds in quadratic extensions of the rational numbers.  In any case, one of the first important results in the program is an old theorem of G. Rabinowitsch (Rabinowitz), which gives a testable criterion for whether the number field \Q(\sqrt{D}), with D a negative integer, possesses unique factorization. In modernish language (cf. Theorem 6 in the linked-to document), keeping this framework we have:

Theorem (Rabinowitsch, 1913): The field \Q(D) is a unique factorization domain (UFD) if and only if x^2-x+ m is prime for all x\in \{1,2, \dots, n\}, where D=1-4m.

The paper was published in German in Crelle’s journal over a century ago, and was somewhat hard to find on its own. We could not locate any other source where the content has been rewritten since then, so we translated and typeset the article (with the aid of various online translation tools). To preserve the spirit and style of the writing, some outdated and perhaps idiosyncratic (to author or translator, as the case may be) jargon has been allowed to survive. These designations are hopefully all made clear within the article so it is readable. Some notation has been modified for clarity.

Perhaps one day this material can be condensed and fully modernised. The article consists mostly of pleasant and clean elementary number theory, situated near the headwaters of one of the important achievements of 20th century number theory. It seems worthy of further propagation, so we post here our translation (no warranty included).

 

Spectral Sequences III

Bigraded Algebrae

McCleary introduces the concept of a differential graded algebra in section 1.3 (Definition 1.6, p. 11). These are algebras (over a field k), which tend to be \N-graded, and importantly carry with them a map d called a differential which is k-linear, shifts the degree of elements (in the grading) up by one, and satisfies a “Leibniz rule:”

    \[d(a\cdot a')=d(a)\cdot a'+(-1)^{\deg(a)} a\cdot d(a')\]

for a, a' in our algebra A^*. This is a twisted version of what is usually called Leibniz’ rule in calculus (which is basically just product rule), which coincides with how the differential works in the algebra of differential forms.

This idea is easily extended to the notion of a differential bigraded algebra (E^{*,*}, d), where now the elements are \N^2 graded (for the time being, later we’ll have \Z^2), but d remains a total-degree 1 mapping. That is,

    \[d: \bigoplus_{p+q=n} E^{p,q} \lra \bigoplus_{r+s=n+1}E^{r,s},\]

and d still satisfies the Leibniz rule

(1)   \begin{equation*} d(e\cdot e')=d(e)\cdot e'+(-1)^{p+q}e\cdot d(e') \end{equation*}

where e\in E^{p,q}.

A standard construction is to form a bigraded algebra by tensoring two graded algebras together. This would work with just component-wise multiplication, but to get a working differential that satisfies our version of the Leibniz rule 1 as well, we introduce an extra sign: we mean, supposing (A^*,d) and (B^*, d') are differential graded algebras, then we can assign E^{p,q}:=A^p\ox B^q, and furthermore

(2)   \begin{equation*}  (a_1\ox b_1 )\cdot (a_2\ox b_2):= (-1)^{(\deg a_2)(\deg b_1)}a_1a_2\ox b_1 b_2.\end{equation*}

Then if we define a differential d_\ox on E^{*,*} by

(3)   \begin{equation*}  d_\ox(a\ox b)=d(a)\ox b + (-1)^{\deg a} a \ox d'(b), \end{equation*}

then d_\ox satisfies the Leibniz rule 1. It is clarifying to check this, so we’ll record it here. Switching notation a bit, we will write (-1)^{\ab{a}} instead of (-1)^{\deg a}. To satisfy 1 we need

    \begin{align*}d_\ox((a_1\ox b_1)\cdot (a_2 \ox b_2))& =  d_\ox (a_1\ox b_1)\cdot (a_2 \ox b_2) + \\ &(-1)^{\ab{a_1}+\ab{b_1}} (a_1\ox b_1) \cdot d_\ox(a_2\ox b_2) \end{align*}

we then apply 3 to the individual terms on the right side above to get

    \begin{align*} d_\ox((a_1\ox b_1)\cdot (a_2 \ox b_2)) = [d(a_1)\ox b_1+((-1)^{\ab{a_1}} a_1 \ox d'(b_1)]\cdot (a_2\ox b_2)+\\ (-1)^{\ab{a_1}+\ab{b_1}} (a_1\ox b_1) \cdot [d(a_2)\ox b_2+(-1)^{\ab{a_2}} a_2\ox d'(b_2)]. \end{align*}

Now applying the multiplication rule 2 and distributing, we find

(4)   \begin{align*}  d_\ox((a_1\ox b_1)\cdot (a_2 \ox b_2)) = (-1)^{\ab{a_2}\ab{b_1}}d(a_1)a_2\ox b_1b_2 +(-1)^{\ab{a_1}+\\\ab{d'(b_1)}\ab{a_2}}a_1a_2\ox d'(b_1)b_2 + (-1)^{\ab{a_1}+\ab{b_1}+\ab{d(a_2)}\ab{b_1}} a_1 d(a_2)\ox b_1b_2 +\\ (-1)^{\ab{a_1}+\ab{a_2}+\ab{b_1} +\ab{b_1}\ab{a_2}} a_1a_2 \ox b_1d'(b_2). \end{align*}

To check the rule holds, we perform this computation by instead multiplying first and then applying the differential. That calculation looks like

    \begin{align*} d_\ox((a_1\ox b_1)\cdot (a_2 \ox b_2)) &= d_\ox((-1)^{\ab{a_2}\ab{b_1}} a_1 a_2 \ox b_1b_2) \\ &=(-1)^{\ab{a_2}\ab{b_1}}[d(a_1a_2)\ox b_1b_2+ (-1)^{\ab{a_1}+\ab{a_2}}a_1a_2\ox d'(b_1b_2)] \\ &=(-1)^{\ab{a_2}\ab{b_1}}[(d(a_1)a_2+(-1)^{\ab{a_1}}a_1d(a_2)\ox b_1b_2 +\\ &(-1)^{\ab{a_1}+\ab{a_2}}a_1a_2\ox(d'(b_1)b_2+(-1)^{\ab{b_1}}b_1d'(b_2))]  \qquad \\ &=(-1)^{\ab{a_2}\ab{b_1}}d(a_1)a_2\ox b_1b_2+(-1)^{\ab{a_1}+\ab{a_2}\ab{b_1}} a_1d(a_2)\ox b_1b_2+ \\ &(-1)^{\ab{a_1}+\ab{a_2}+\ab{a_2}\ab{b_1}} a_1 a_2\ox d'(b_1) b_2 +\qquad\\ & (-1)^{\ab{a_1}+\ab{a_2}+\ab{b_1} +\ab{a_2}\ab{b_1}}a_1a_2\ox b_1d'(b_2). \end{align*}

Finally, remarking that \ab{d'(b_1)}=\ab{b_1}+1 and \ab{d(a_2)}+1=\ab{a_2} shows that terms of the last line above match with those of 4, so everything checks out and (A^*\ox B^*, d_\ox) becomes a differential bigraded algebra.

A Chain Rule

Given the length and detail of section 1.3, surprisingly we find no glaring errors in this section, but the use of the differential becomes somewhat muddled in calculation in section 1.4. Again, perhaps as an undesirable side effect of the fact that we remain at the “informal stage,” it’s always difficult to keep track of what assumptions we’re working with in each example. Case in point, example 1.H, p. 20. The paragraph preceding definition 1.11 seems to indicate that all graded algebras are assumed to be graded commutative — at least for the rest of the section, one guesses, though the language is vague. Let’s try this here with a bit more force.

Assumption: All graded algebras are graded commutative for the rest of the post. This is to say, for all x,y in any A^*, we have x\cdot y =(-1)^{\ab{x}\ab{y}} y\cdot x. Now let’s have a look at the example. We suppose a spectral sequence of algebras (E_r^{*,*}, d_r) with E_2^{*,*}\cong V^*\ox W^*, converging to the graded algebra H^* which is \Q in degree 0 and \{0\} in all others.  The example asserts that if V^* is a graded commutative polynomial algebra in one generator/variable, then W^* is a graded commutative exterior algebra in one generator, and vice versa.

The first confusion appears in a restatement of the Leibniz rule near the bottom of page 20, except this time there are tensors involved. This appears to be a mixed use/abuse of notation, which was slightly different in the first edition of the book, but not more consistent. The idea is as follows. V* and W^* embed into V^*\ox W^* under the maps v \mapsto v \ox 1 and w \mapsto 1\ox w.  Then one can also write an element w\ox z \in V^*\ox W^* (mind the inexplicable inconsistent choice of letters) as

(5)   \begin{equation*}  w \ox z = (-1)^0 (w\ox 1)\cdot (1 \ox z)= (w\ox 1) \cdot (1 \ox z) \end{equation*}

since the degree of 1 is zero in each graded algebra. Note that this also allows us to regard V^*\ox W^* as graded commutative with the tensor product as multiplication between pure V^* and pure W^* elements, writing

    \[z\ox w:=(1\ox z) \cdot (w \ox 1)=(-1)^{\ab{z}\ab{w}} (w\ox z). \]

One can apply Leibniz rule to the product in 5 so that if V^*\ox W^* comes with a differential d, we get

    \[d(w\ox z) = d((w\ox 1) (1\ox z)) =d(w\ox 1)(1\ox z)+(-1)^{\ab{w}}(w\ox 1) d(1\ox z).\]

The thing is we really need not write the tensor product w\ox 1; it is just as correct to write w on it’s own, as we often do with polynomial algebras and so on. Then the above can be written instead as

    \[d(w\ox z) = d(w)\ox z +(-1)^{\ab{w}} w \ox d(z) \]

as McCleary does near the bottom of page 20. What makes this confusing is that up to this point we had only seen differentials acting on tensors by defining the bigraded differential from tensoring two differential graded algebras together, seen above. In this context, the differential of the bigraded algebra must act on an element of the algebra coming from V^* \ox W, it cannot act on just one side of the tensor. What’s different here is that the tensor product is actually the multiplication operation on each page of the spectral sequence. Thus, the restatement of the familiar rule with new notation.

Nevertheless, the next equality is also a bit confounding at first, partly because McCleary, goes back to writing the extra 1 in the tensor, suggesting that we need to pay attention to its effect. He says that if d_i(1\ox u) =\sum_j v_j\ox w_j, then

(6)   \begin{equation*}  d_i(1\ox u^k) = k \lt(\sum_j v_j \ox ( w_ju^{k-1})\rt) \end{equation*}

which looks sort of reasonable as it resembles something like a chain rule, d(u^k)=k u^{k-1} d(u). It is presented as if it should follow immediately from the Leibniz rule stated before. But this seems weird when the degree of u is odd. To be totally transparent about this, let’s illustrate the case where k=2, suppressing the subscript on the differential again, but maintaining the tensorial notation.

    \begin{align*} d(1\ox u^2) & =d((1\ox u)(1\ox u)) \\ & = d(1\ox u) (1\ox u) +(-1)^{\ab{u}} (1\ox u) d (1\ox u) \\ &= \sum_j (v_j \ox w_j)(1\ox u) + (-1)^{\ab{u}} \sum_j (1\ox u) (v_j \ox w_j) \\ &=\sum_j v_j\ox w_j u + (-1)^{\ab{u}} \sum_j (-1)^{\ab{u}\ab{v_j}} v_j \ox u w_j \\ &=\sum_j v_j\ox w_j u + (-1)^{\ab{u}} \sum_j (-1)^{\ab{u}\ab{v_j}+\ab{u} \ab{w_j}} v_j \ox  w_j u \\ &=\sum_j v_j\ox w_j u + (-1)^{\ab{u}} \sum_j  v_j \ox  w_j u \\ \end{align*}

where the last line follows since v_j\ox w_j has total degree \ab{u}+1, so the sign inside the sum there has exponent \ab{u}(\ab{u}+1) which is even. We see that if u has odd degree then, these terms cancel and we get 0. So you say “wait a minute, that’s not right, I wan’t my chain rule looking thing” until you eventually realize that if u has odd degree, since it’s sitting in a graded commutative algebra, u^2 is actually zero! And the same goes for all higher powers of u. Then, d(u^2)=d(0)=0 makes complete sense. Meanwhile, if u has even degree, the terms will pile up with positive sign and we get the chain rule looking thing that was claimed. So the statement 6 is in fact true, though it really breaks down into two distinct cases.

Going forward in the example, McCleary only really seems to use the chain rule (liberally mixing in the described sort of abuse of notation) on terms of even degree, so it’s tempting to think that it only applies there, but it is sort of “vacuously true” in odd degree as well. Oh well. Onwards.

Spectral Sequences II

Two Stripes

The next thing to address in McCleary is an apparent mistake on p. 9 of section 1.2. Here we again assume a first quadrant spectral sequence converging to a graded vector space H^*. This is mentioned at the beginning of the section, but it’s easy to forget that when a bold-faced and titled example (1.D) seems to be presenting a reset of assumptions, rather than building upon prior discussion. Furthermore, in this example, McCleary seems to be working again with the assumption from example 1.A that F^p+kH^p=0 for k>0. On the other hand, this can be seen as a consequence of the fact that our spectral sequence is limited to the first quadrant, provided the filtration is finite in the sense of Weibel’s Homological Algebra, p. 123 (F^mH^s=0 for some m). But then it would be unclear why McCleary took this as an additional assumption rather than as a consequence of prior assumptions in the first case. : /

The new part of this example is the assumption that E^{p,q}_2=0 unless q=0 or q=n, so all terms of the spectral sequence are to be found just in two horizontal stripes. In particular E_\infty^{p,q} is only possibly non-zero in these stripes, and since these correspond to filtration quotients, the filtration takes a special form.

First, we might look at the filtration on H^s where 0\leq s \leq n-1.  Note that the spectral sequence terms that give information about H^s are those along the diagonal line where p+q=s.  Since s\leq n-1, the only place where anything interesting might happen is when this line crosses the p-axis, i. e. when q=0. This forces p=s, so the only possible nonzero filtration quotient is

    \[E_\infty^{s,0}=F^sH^s/F^{s+1}H^s=F^sH^s=F^0H^s=H^s \]

working with the assumption that F^{s+1}H^s=0. So on the one hand, we get no interesting filtration of H^s for s<n, but on the other hand we can see exactly what it is from the spectral sequence limit.

Now we treat the case of H^{n+p}, where p\geq 0. I find this awkward notation again, preferring to reserve p for a pure arbitrary spectral sequence index, but since we are trying to address the mistake in this notation, we should keep it for now. The filtration of this vector space/cohomology is interesting when q=n and q=0, where the quotients are given by

    \[E^{p,n}_\infty=F^pH^{p+n}/F^{p+1}H^{p+n}\quad\text{and}\quad E^{p+n,0}_\infty=F^{p+n}H^{p+n}/0.\]

Every where else, successive quotients are 0, meaning the filtration looks like…

    \[0\sus F^{n+p}H^{n+p}= \dots =F^{p+1}H^{n+p} \sus F^pH^{n+p}=\dots=F^1H^{n+p}\sus H^0{n+p}\]

In the filtration on page 9, McCleary puts one of the (possibly) non-trivial quotients at F^nH^{n+p} instead of at F^pH^{n+p} where it should be.  That’s all I’m saying.

This situation is modeled on a spectral sequence for sphere bundles i.e. bundles where the fibers are spheres of a given dimension. The stripes coincide with the fact that a sphere \mathbb{S}^n has nontrivial cohomology only at H^n and H^0. This sort of computation is famous enough that it has a name: the Thom-Gysin sequence (or just Gysin sequence).

As a final remark on section 1.2, McCleary says that the sequence in example 1.C is the Gysin sequence. Example 1.C doesn’t exist, we mean example 1.D : )

Spectral Sequences I

A goose chase through homological algebra etc. has led us to start reading McCleary’s A User’s Guide to Spectral Sequences. The book seems like a nice introduction for those that know their way around graduate topology and geometry, but haven’t yet encountered cause to pull out this extra machinery to compute (co)homology. First published by Spivak’s Publish or Perish Press in the 80’s, a second edition was released by Cambridge University Press in 2000. Though it sounds like the second edition is rather improved, there seem to be a number of mistakes remaining which may frustrate those trying to learn a notoriously complicated subject for the first time. Pending an official list of errata, we may as well collect some of them here.

Section 1.1 – Notation

 

The first comment worth making is regarding some confusing notation, largely an overuse of the letter E. The first use comes on p. 4 (Section 1.1), given a graded vector space H^* and a filtration F^* of H^*, by defining

    \[E_0^p(H^*) := F^p(H^*)/F^{p+1}(H^*). \]

Here, the symbol E_0^p seems to designate an endofunctor on (graded) vector spaces — it eats one and gives back another; transporting morphisms through the filtration and quotient shouldn’t be a problem either. It isn’t really clear what the 0 subscript is supposed to indicate at this point, but the reader sits tight expecting the truth to be revealed.

However, on the very same page, McCleary twists things by making the assignment

(1)   \begin{equation*}   E^{p,q}_0 := F^pH^{p+q} / F^{p+1} H^{p+q} , \end{equation*}

where F^pH^r : = F^pH^*\cap H^r, the r\ts{th} graded piece of the p\ts{th} filtration. Now, with the extra index q, E^{p,q}_0 is a vector space on it’s own. The notation doesn’t indicate reference to H^*, though in this case it really depends on H^*. For instance, McCleary indicates that we should write something like

    \[ E^p_0(H^*)=\bigoplus_q E^{p,q}_0.\]

The definition immediately afterwards (Definition 1.1) indicates E^{p,q}_r is to be used to designate a vector space in a spectral sequence which is irrespective of any H^* for all r\geq 1. The typical way to relate a spectral sequence \{E^{*,*}_r, d_r\}, to a graded vector space H^* is the situation of convergence (Definition 1.2, p. 5) where instead

    \[E_\infty^{p,q} \cong E_0^{p,q}(H^*).\]

The right hand side above has nothing to do with the spectral sequence E^{*,*}_r (since we take r\geq 1 in our definition), it is just an instance of the definition from equation 1… but with distinct use of notation… oh. So on the one hand, E^{p,q}_0 should be a standalone vector space, like the other E_r^{p,q}‘s, but also it needs to come from an H^* so one should really write E_0^{p,q}(H^*) as in Definition 1.2. Wha? Shoot. Couldn’t we have used like an A instead or something?

Perhaps there is good reasoning for all of this to be discovered once we get further in. Also, it seems so far that initial terms are usually E_2^{*,*}. Why not E_1^{*,*}? And why don’t we allow 0-pages? In these cases the differentials would be vertical and horizontal (resp.) instead of diagonal, which feels less interesting somehow, though this doesn’t seem like it would be totally frivolous… TBD.

Splicing Short Exact Sequences

Finishing out the first section, we address what seems to be a typo in example 1.A (p. 6). McCleary’s expository style consists of many statements which are not obvious, though usually not difficult to work out. This is perhaps for the best, as the community seems to indicate that the only real way to learn spectral sequences (make that: all math?) is by working them out. Nevertheless, it is a bit discouraging to find yourself at odds with the author at the first example…

We have assumed a first quadrant spectral sequence with initial term E_2^{*,*} converging to H^* with a filtration satisfying F^{p+k}H^p=\{0\} for all k> 0. Then we have a filtration on H^1 in particular, given by

    \[ \{0\}\sus F^1H^1\sus H^1, \]

since, by the assumption, F^{2}H^1=\{0\} etc., and F^0H^1=H^1 by definition. By convergence, then,

    \[E_\infty^{1,0}=F^1H^1/F^2H^2=F^1H^1,\]

so E_\infty^{1,0} is a submodule of H^1. But also because E_r^{1,0} lies on the p-axis (depicted as what is usually the x-axis) and our spectral sequence has only first quadrant terms, d_r(E_r^{1,0}) must be the zero map for all r. Furthermore, E_r^{1,0} is too close to the q-axis to get hit by any differential d_r, thus E_2^{1,0} survives as the kernel of every d_r, mod the image of a d_r from a zero vector space in the second quadrant. This is all to say

    \[E_2^{1,0}=E_\infty^{1,0}\cong F^1H^1.\]

We then have part of the short exact sequence McCleary gives for H^1 as

    \[ 0 \lra E_2^{1,0}\lra H^1 \lra H^1/F^1H^1 \lra 0 .\]

How can we describe the third term using the spectral sequence? Well, from our definitions, H^1/F^1H^1=F^0H^1/F^1H^1\cong E_\infty^{0,1}. The book seems to be indicating that E_\infty^{0,1}=E_2^{0,1} but this is not necessarily the case! It also doesn’t make sense with how the short exact sequences are spliced later on.

Let’s address the first claim first. Because E_r^{0,1} lies on the q-axis, and the differentials point “southeast” towards the empty fourth quadrant, d_r(E_r^{0,1}) is the zero map for any r\geq 3, but it can’t be hit by anything so we have now

    \[E_\infty^{0,1}=E_3^{0,1} = \frac{\ker d_2 : E_2^{0,1}\to E_2^{2,0} }{\im d_2: E_2^{-2,0}\to E_2^{0,1}}. \]

The denominator is the image of a map from a zero vector space, so it is zero, and thus E_\infty^{1,0} is a subspace of E_2^{0,1}, but this latter space can be larger! This is all to say, the short exact sequence for H^1 is misprinted, and should go

(2)   \begin{equation*} 0 \lra E_2^{1,0}\lra H^1 \lra E_\infty^{0,1} \lra 0 . \end{equation*}

One can confirm this by examining the SES given just below, where we see E_\infty^{0,1} injecting into E_2^{0,1}:

(3)   \begin{equation*} 0 \lra E_\infty^{0,1}\lra E_2^{0,1}\overset{d_2}{\lra} E_2^{2,0}\lra E_\infty^{2,0}\lra 0. \end{equation*}

This is a standard decomposition of the map d_2 in the middle: for any morphism \phi:A\to B (in an abelian category at least, we suppose) there is a SES

    \[0\lra\ker\phi\lra A \overset{\phi}{\lra} B\lra B/\im \phi \lra 0 .\]

It remains to see that E_\infty^{2,0}\cong E_2^{2,0}/\im d_2. Because of where E_r^{2,0} sits on the p-axis, it is again the kernel of d_r for all r. Further, it can only possibly be hit by d_2, so in fact E_3^{2,0} survives through all further terms to give the desired equality

    \[E_\infty^{2,0}=E_3^{2,0} = E_2^{2,0}/\im d_2.\]

To splice all this together, we recall that we can connect

    \[\dots\oset{s}{\lra} L\overset{\alpha}{\lra} M\lra 0 \quad \text{and} \quad 0 \lra M \overset{\beta}{\lra} N \oset{t}{\lra} \dots \]

as

    \[\dots \oset{s}{\lra} L \overset{\gamma}{\lra}  N \oset{t}{\lra} \dots \]

where \gamma=\beta\circ\alpha. We maintain exactness since \ker \gamma=\ker \alpha=\im s and \ker t = \im \beta = \im \gamma.

Performing this surgery on sequences 2 and 3 yields the main exact sequence claimed by the example, namely

(4)   \begin{equation*} 0 \lra E_2^{1,0}\lra H^1\lra E_2^{0,1}\overset{d_2}{\lra} E_2^{2,0}\lra E_\infty^{2,0}\lra 0. \end{equation*}

Stay tuned for more clarifications from Chapter 1.

Hilbert Schemes IV: A Definition

At this point in the project, we begin to understand why so many books and papers are written in dry, terse, often difficult mathematical prose. Discursiveness which is beneficial is risky, time consuming and hard to pull off! As we continue to hurtle preliminary posts, maybe a bit of motivation is in order.

At the highest level, far-out, hardcore algebraic geometry is finding application in strange places these days. We will just cite the three instances which are close to the heart of mathlab: Hodge Theory in Combinatorics, Geometric Complexity Theory, and the Macdonald Positivity Conjecture where Hilbert schemes played a starring role. Understanding this last one is what we are heading for in these posts. Moving on…

So let’s just do this the way where you put a loaded definition out front and then try to tease it apart, following Nakajima‘s exposition, repeated in seminar.

Definition Let {X} be a projective scheme over an algebraically closed field {k}. The contravariant Hilbert functor {\mathcal{H}ilb_X:\mathop\mathrm{Sch}\rightarrow\mathop\mathrm{Sets}} is the functor that associates to a scheme {Y} the set of closed subschemes {Z\subset X\times Y} such that the natural morphism from {Z} to {Y} is flat.

From the top, we are choosing an algebraically closed field {k} so the “projective” here is with respect to {\mathop{\mathbb P}^n_k= \mathop\mathrm{Proj} k[x_0,\dots,x_n] = \mathop\mathrm{Spec} k \times \mathop{\mathbb P}^n_{\mathbb Z}}, as described in the last post. We would like to take this opportunity to express our regret for not really describing {\mathop\mathrm{Proj}} or the fiber product in these posts even though we earlier gestured towards thoroughness. Hopefully the references serve.

Moving on, an arbitrary scheme is said to be projective (over {k}) if it is of the form {\mathop\mathrm{Proj} S} for {S} a finitely generated graded {k}-algebra. A subset {W} of a scheme {V} is said to be a closed subscheme iff:

  • The inclusion map {i: W\hookrightarrow V} is an affine morphism, meaning that for each open affine subset {\mathop\mathrm{Spec} B \subset V}, the inverse image {i^{-1}(\mathop\mathrm{Spec} B)=\mathop\mathrm{Spec} A} for some ring {A}. “The inverse image of affines is affine.”
  • Additionally, for each of these pairs, the map {B\rightarrow A} is surjective.

\underline{Note}: In case {W} is not a subset of {V} but we still have a map {W\rightarrow V} which satisfies the above properties, that map is said to be a closed embedding (or, less tastefully, a closed immersion) of schemes.

Back in the definition, the map from {Z} to {Y} is the composition of the inclusion {i:Z\hookrightarrow X\times Y} and the projection map {\mathop\mathrm{pr_Y}:X \times Y \rightarrow Y}. Remember, this isn’t set-wise projection, our product is in the category of schemes, where we can also think of it as the fibered product over the terminal object, {\mathop\mathrm{Spec} {\mathbb Z}}. It is guaranteed to exist as in section 9.1 of The Rising Sea.

Finally we come to flatness, which is sort of exciting because this is the first time I’ve come across it in the wild since studying for my algebra qual, but still sort of mysterious. Apparently Mumford agrees, even as he reassures us that it is “technically the answer to many prayers,” in The Red Book (III.10). Remember that an {A} module {M} is said to be flat if tensoring with {M} preserves short exact sequences. More loosely, flat modules are pretty close to (i.e. are direct limits of) free modules.

That the map {f:Z\rightarrow Y} is flat means that for all {z\in Z}, the induced map on the stalks {f^\sharp:\mathcal{O}_{Y,f(z)}\rightarrow \mathcal{O}_{Z,z}} makes {\mathcal{O}_{Z,z}} into a flat {\mathcal{O}_{Y,f(z)}}-module. For motivation on why flatness is the “right” condition in these sorts of circumstances, see [EH] II.3.4.

An important theorem of Grothendieck says that {\mathcal{H}ilb_X} with {X} as above is representable by a scheme which is “locally of finite type” called a *gasp* HILBERT SCHEME! A version of the proof with additional discussion and background is here. It is also explained in section 9 there that we can weaken the requirements so that {X} need only be quasi-projective. In fact, one of the most interesting cases for us will be where {X} is {{\mathbb A}^2} ({{\mathbb C}^2} in particular). Further on, we will refine this by introducing Hilbert polynomials. Yay!

Hilbert Schemes III

Having established that to each scheme {X} we can associate its functor of points {h_X}, we continue our introduction by mentioning some eminent examples.

First the projective space {\mathop{\mathbb P}^n}. This is sort of the basic example of a moduli space, which is also what were after with Hilbert Schemes. While {\mathop{\mathbb P}^n} (over your ring of choice) has a few different constructions, it’s points correspond to other geometric objects: lines in the affine space {{\mathbb A}^{n+1}}. This is the essence of moduli, and their value for “counting” objects of a given type.

We also have a description of the functor of points for projective space, {h_{\mathop{\mathbb P}^n}}, recalling that {h_{\mathop{\mathbb P}^n}(X)=\mathop\mathrm{Mor}(X, \mathop{\mathbb P}^n)} for a given scheme {X}. See Vakil, Theorem 16.4.1, or also Eisenbud and Harris [EH], Theorem III-37

Theorem Morphisms {X\rightarrow \mathop{\mathbb P}^n} are in bijection with the data (up to isomorphism) of {(\mathcal{L},s_0,\dots,s_n)} where {\mathcal{L}} is a line bundle (invertible sheaf) on {X} and the {s_i} are global sections which do not vanish simultaneously.

By “up to isomorphism” here we mean that the value of the section {s_i} at a given point on {X} is depends on the trivialization used, but it will differ according to multiplication by a scalar which multiplies all {s_0,\dots,s_n} simultaneously. Thus, the ensemble of all sections together effectively gives a point in {\mathop{\mathbb P}^n} (or a fiber of the tautological bundle, if you prefer), provided they are not all zero. Ranging over all of {X} we get a morphism of schemes.

You might wonder which projective space we are working with here, i.e. over which field/ring. As will usually be the case, it doesn’t much matter because maps behave nicely under base change, but if we want to be as general as possible we should probably use {{\mathbb Z}}. This is because {{\mathbb Z}} is the initial object in the category {\mathop\mathrm{CRing}}, which makes it the final object in the category of affine schemes. Initialness comes from the fact that there is a unique morphism from {{\mathbb Z}} to any other commutative ring with identity which just takes 1 to itself.

Then, {\mathop\mathrm{Spec}({\mathbb Z})} is also final in {\mathop\mathrm{Sch}} because we have this ring morphism (which is an {\mathop\mathrm{Aff}} morphism in reverse) into all of the open affine covering neighborhoods of a given scheme, and these glue up nicely. So, whether we construct {\mathop{\mathbb P}^n_{\mathbb Z}} from a graded ring as {\mathop\mathrm{Proj} Z[x_0,\dots,x_n]} or as the gluing of a bunch of {{\mathbb A}^n_{\mathbb Z}=\mathop\mathrm{Spec}{\mathbb Z}[\frac{x_0}{x_i},\dots,\frac{x_n}{x_i}]}, we again have a canonical scheme morphism {\mathop{\mathbb P}^n_{\mathbb Z}\rightarrow\mathop\mathrm{Spec}{\mathbb Z}}. This together with {A\rightarrow\mathop\mathrm{Spec}{\mathbb Z}} lets us construct the fiber product {A\times_{\mathop\mathrm{Spec} {\mathbb Z}}\mathop{\mathbb P}^n_{\mathbb Z} =: \mathop{\mathbb P}^n_A}; see [EH] III.2.5 for a more thorough treatment of all of this. I’m still essentially just recording the things I had to remind myself of in jumping into this so it’s maybe more garbled than pedagogical.

[EH] here also gives the description of {\mathop\mathrm{Mor}(X, \mathop{\mathbb P}^n)} as subsheafs of {\mathcal{O}^{n+1}} which are locally of rank {n}, which we think of as the kernels of the surjection from the direct sum sheaf {\mathcal{O}^{n+1} \twoheadrightarrow \mathcal{L}}, surjection being guaranteed by the non-vanishing of at least one of the sections above (at every point). This leads naturally into our other key example, the Grassmannian.

According to the Yoneda philosophy (which is often attributed to Grothendieck as well) we should actually think of the Grassmannian (and the special case of projective space) not just as the “space of {k}-planes in {n}-space,” but as a moduli space for certain bundles (locally free sheaves in AG-speak) and their sections. Really, it is a functor which can be applied to other schemes, and returns the collection of morphism which define the scheme that represents the functor. : S

The change with the Grassmann functor {G(k,n)} is that now we look at surjections from {\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}} where {\mathcal{Q}} is locally free of rank {k}. At neighborhoods in a locally trivializing cover, we identify {\mathcal{Q}} with {\mathcal{O}^k} and we can treat {\phi} as a {k\times n} matrix. The Grassmannian will be covered by open sets which are the nonvanishings of the determinant of the {\binom{n}{k}} maximal minors. This leaves the other {k(n-k)} entries of the matrix free, so they can vary and together determine a map {X\rightarrow{\mathbb A}^{k(n-k)}} in the way the sections {s_0,\dots,s_n} gave a map to {\mathop{\mathbb P}^n} before. Thus, we can cover {G(k,n)} by a bunch of affine spaces which glue together. In the parlance, these are open subfunctors represented by the schemes {{\mathbb A}^{k(n-k)}}.

It’s worth noting that the full Grassman functor {G(k,n)} is represented by a bona fide scheme of the form {\mathop\mathrm{Proj}({\mathbb Z}[\dots, x_I,\dots]/I_{k,n})} where {I} denotes a subset of {[n]=\{1,\dots,n\}} of size {k}, which indexes the variables, an {I_{k,n}} is a particular homogeneous ideal given by the Plücker equations. So we have a more traditional geometric object to think of as well. We may come back to this in more detail later if we need to. For now it suffices to say that applying the {k^{th}} exterior power functor {\wedge^k} to {[\phi:\mathcal{O}_X^{n}\twoheadrightarrow \mathcal{Q}]} gives a surjection from a locally free sheaf of rank {\binom{n}{k}} onto a line bundle. So by the prior discussion, we have a map from the Grassmannian into {\mathop{\mathbb P}^{\binom{n}{k}-1}} which is the Plücker embedding.

Phew, okay I think in the next post we might actually define a Hilbert Scheme.

Hilbert Schemes II

Next, we introduce the Hilbert scheme functorially following notes by Maclagan. See also Bertram’s course notes, who points out that we (as in Grothendieck) are trying to develop a category-theoretic framework for counting objects with a given property. First we should say a little bit about morphisms of schemes.

To do this, one should indicate that schemes (like manifolds) actually carry the additional structure of a locally ringed space. This means that when you take a kind of (filtered co-)limit on the structure sheaf {\mathcal{O}_X} over all the open neighborhoods containing some point {p\in X}, then we get a local ring {\mathcal{O}_{X,p}}. In this case of an affine scheme {\mathop\mathrm{Spec}{R}} it works out that {\mathcal{O}_{\mathop\mathrm{Spec} R,p}} is actually the localization {R_p}, as {p} is a prime ideal of {R}, and localization satisfies exactly the universal property prescribed by the colimit.

First, remember that schemes are ringed spaces, and that a morphism of ringed spaces {f:X\rightarrow Y} is a continuous map of topological spaces together with a morphism of sheaves from {\mathcal{O}_Y} to the push-forward {f_*\mathcal{O}_X}. This induces a map on stalks

\displaystyle f^\sharp:\mathcal{O}_{Y,q} \rightarrow \mathcal{O}_{X,p}

where {f(p)=q} and supposing {f^\sharp} sends the maximal ideal of one to the other, then we have a morphism of locally ringed spaces, and it is THIS, my friends, which will be a scheme morphism when the locally ringed spaces happen to be schemes… oh hell, this can’t be useful for anyone as exposition; just seriously go skim the first six chapters of The Rising Sea to clear things up. It’s only 200 pages.

Now we have objects and morphisms so we can talk about the category {\mathop\mathrm{Sch}} and functors therefrom. We are interested in the confusingly named functor of points for a given scheme {X}, which is really the one that comes up in many versions of the Yoneda Lemma. The idea is that if category is locally small, meaning the collection of morphisms (arrows) {\mathop\mathrm{Mor}(A,B)} between any two objects forms a set, then we can understand an object by the morphisms going into (or out of) it. Bearing in mind that a contravariant functor between categories {F: \mathcal{C}\rightarrow \mathcal{D}} is the same as a covariant functor {\mathcal{C}^{op}\rightarrow \mathcal{D}}, we have the following.

Definition Suppose {X} is an object of {\mathop\mathrm{Sch}}, the category of schemes. Then there is a contravariant functor {h_X: \mathop\mathrm{Sch}\rightarrow \mathop\mathrm{Set}} such that on objects we have {h_X(Y)=\mathop\mathrm{Mor}(Y,X)}, and for a given morphism {f:Y\rightarrow Z} we have

\displaystyle \begin{array}{rcl} h_X(f):& \mathop\mathrm{Mor}(Z,X) \rightarrow \mathop\mathrm{Mor}(Y,X) \\ & g \mapsto g\circ f \end{array}

In general, functors that come from such {h_X}‘s are said to be representable. Elements of the set {\mathop\mathrm{Mor}(Y,X)} are called the Y-valued points of X, which is weird, because on sight they are just maps. Further, by laziness, in case {Y=\mathop\mathrm{Spec A}} for some ring (or field) A, then we will call theses maps the “{A}-valued points of X.”

Endnote: for an interesting discussion on whether scheme theory can be taught using just {\mathop\mathrm{MaxSpec}}, the maximal ideals of a ring, instead of the primes, that then digresses into thinking of schemes as their functors of points instead of the whole locally ringed space shebang, as well as other AG/Berkeley cultural inside jokes by a bunch of dudes, see the post and extensive comments here. Other interesting background can be found on nLab.

Hilbert Schemes I

In this series of posts, we will report and editorialize the goings-on of the algebraic geometry (AG) and combinatorics learning seminar. The goal is to learn something about Hilbert Schemes and the relevant combinatorics.

First, since we are working with the category of schemes, we should indicate what those are and where to find out more about them. This feels like joining the chorus or preaching to the choir, but it’s hard to overstate the value of Vakil’s The Rising Sea/FOAG, for the background category and scheme theory: an innovation in mathematical text writing.

Okay so a scheme is like a variety, which is to say it’s like a manifold, which is to say it’s locally like other spaces we know and love and go under the general banner affine. The central metaphor here is that it comes from gluing simpler things together along the places they match up, and that the information about the spaces themselves really comes from information about the functions on the space… Ahem, made precise by the following gesture at a definition which really takes scores of pages to build up properly, and relies on a multitude of other definitions:

Definition 1 A Scheme is a ringed space {(X, \mathcal{O}_X)} such that every point in {X} has a neighborhood {U} such that {(U, \mathcal{O}_X |_U)} is an affine scheme.

{\mathcal{O}_X} is a sheaf (of rings). That is, it’s a (contravariant) functor from the category of open subsets of {X}, sometimes written {\mathop\mathrm{Op}(X)}, to {\mathop\mathrm{CRing}}, the category of commutative rings (with identity) and their homomorphisms. As elsewhere in life, if you find the category theoretic phrasing more obnoxious than illuminating, please ignore and seek the alternative. Or spend some time discovering the joy of cats, and join us. : D

So what’s an affine scheme? Turns out the affine scheme category is nothing but {\mathop\mathrm{CRing}^{op}}! You know, the category with the same objects but with the arrows reversed? OK, I shut up. Just take your favorite ring {R}. The collection of prime ideals of {R} is (hopefully) a set, denoted {\mathop\mathrm{Spec}(R)}, which can be given a topology by taking sets of the form

\displaystyle D(f)=\{\mathfrak{p}\in\mathop\mathrm{Spec}(R) \mid f\not\in \mathfrak{p}\}

as basis for the open sets, where {f} is some element of {R}. We get the sheaf (also called a structure sheaf) for an affine scheme by taking

\displaystyle \mathcal{O}_{\mathop\mathrm{Spec}{R}}(D(f))=R_f,

where the right hand side denotes localization at the multiplicative subset generated by {f}.

Why use the prime ideals? I guess because localization works nicely for prime ideals. But also if we have an algebraically closed field {K}, then Hilbert’s Nullstellensatz gives an identification of the points in {K^n} with maximal ideals of the polynomial ring {K[x_1,\dots,x_n]}.

I always groan when I see people spend a page introducing like a semester’s worth of algebraic geometry, but, well, now I’ve done it too. Maybe this is another one of those rites of passage. Next up: some representable functors.