The String Coffee Table

The main goal of my exposition shall be to

$\bullet$ describe how schemes are a generalization of varieties

$\bullet$ or, more precisely, how varieties are a special case of schemes,

$\bullet$ or, more precisely, how the category of varieties is a subcategory of that of schemes,

$\bullet$ or, to be really precise, how there is a fully faithful functor

from the category of varieties over a field $k$, into that of schemes over a field $k$.

In order to be even able to state this result, I will need to say a few words about what varieties and schemes actually are.

Doing so already goes a long way towards proving the above statement, since the definition of a scheme is precisely motivated by the desire to generalize that of a variety.

What is it all about?

The main premise of algebraic geometry (as the study of varieties and schemes is called) is to regard (generalized) spaces as spectra.

If you feel like a physicist, you might want to regard this as a way of understanding observables like positions in terms of spectra of certain operators.

This point of view is clearly more pronounced in Alain Connes’ notion of spectral geometry (“NCG”), which is to algebraic geometry roughly like Riemannian geometry is to topology.

In both cases, the starting point is the basic observation that the information contained in ordinary spaces may be encoded in the (rings and/or algebras of) functions on these spaces.

In fact, both fit into the following dictionary

As the table indicates, algebraic geometry is motivated by the desire to understand the geometry of the spectrum of rings of polynomials. In this case, this “spectrum” is essentially the space of zeros of these polynomials. A more precise definition is given below.

Hence, (historically) much of the motivation for algebraic geometry comes from pure number theory, and the desire to understand it in geometrical terms. Accordingly, some of the more prominent examples of algebraic “spaces” are extremely exotic from the point of view of ordinary geometry.

Of course, the nice thing is that much more genuinely geometric objects, like complex manifolds, are also examples of schemes, which allows us to consider all these phenomena on the same footing.

So what is a variety?

Like a manifold is defined as being anything that locally looks like $\mathbb{R}^n$, an

affine variety is defined to be anything that looks like the set of common zeros of a collection of polynomials.

Example. Let $A = \mathbb{C}[X]$ be the ring of polynomials in $X$ with complex coefficients. Let $f = X - 1$ be one such polynomial. Its set of zeros $Z(\{f\}) = \{1\}$ contains precisely one element, the number $1$. This a point, which is an example of an affine variety.

More generally, every polynomial $f$ with complex coefficients is of the form $f = (X-a_1)(X-a_2)\cdots (X-a_n)$, for $a_1, \dots, a_n$ fixed complex numbers. So the set of zeros $Z(\{f\}) = \{a_1,\dots,a_n\}$ is the disjoint collection of these points. This, too, is an affine variety.

But we said above that we want to think of spaces in terms of the functions living over them. So we need to find a sensible notion of functions on sets of common zeros of a collection of polynomials.

Assume the polynomial defining our affine variety is the 0-polynomial in $k[X_1, X_2, \cdots X_n]$, for $k$ some algebraically closed field (for instance $k = \mathbb{C}$). Then the set of zeros $Z(\{0\}) = k^n$ is everything. In this context, this set is denoted $A_k^n$ and given the fancy name affine $n$-space over $k$.

The suitable ring of functions over $A_k^n$ is clearly nothing but the ring of polynomials $k[X_1, X_2, \cdots X_n]$ itself. Moreover, we see that the points of $A_k^n$ are in 1-1 correspondence with the prime ideals of $A$.

Essentially the same remains true if we look at an affine variety inside $A_k^n$, i.e. at the set $Z(\{f_1,\cdots, f_r\})$ of common zeros of some plynomials $f_1,\cdots, f_r \in k[X_1,\cdots X_n] = A$.

From the point of view of the “space” $X := Z(\{f_1,\cdots, f_r\})$, every polynomial in $A$ which vanishes on $X$ is indistuingishable from the zero polynomial. Equivalently, any two polynomials which coincide on $X$ are indistinguishable from within $X$.

Hence, the relevant space of polynomial functions on $X$ is not quite the original $A$, but the quotient $A/I(X)$ of $A$ by the ideal $I(X)$ of all those polynomials which vanish on $X$, i.e. we identify two polynomials in $A$ if their difference lies in $I(X)$.

We call

the affine coordinate ring of $X$.

The only possibly troubling aspect of all this is that we seem to be about to build a theory of geometry which relies heavily on coordinates.

But this is an illusion.

As a first step towards getting rid of this illusion, we do what we have done before in $A$, but now inside $A(X)$: we look at subsets of functions that vanish at certain points. Such subsets are always ideals, in that the product of any function with one that vanishes at a given point also vanishes at that point. So we define

Definition. Given an affine variety $X \subset A_k^n$, and given a point $P \in X$, we denote by $$\mathfrak{m}_P \subset A(X)$$ the maximal ideal (and hence prime ideal) of coordinate functions on $X$ which vanish at $P$.

Good. So far this is just a bunch of fancy terminology. But next, here is a little result which is, morally speaking, at the very heart of the spectral way of thinking about geometry.

Fact. ([Hart, Th. 3.2]) Let $X \subset A_k^n$ be an affine variety with coordinate ring $A(X)$. Then the assignment of points $P$ in $X$ to maximal ideals $\mathfrak{m}_P$ in $A(X)$ $$P \mapsto \mathfrak{m}_P$$ is a bijection.

So we see that an affine variety can alternatively be described as something that locally looks like a collection of prime ideals of some ring of coordinate functions. This observation seamlessly leads to the generalization that we are after.

All we have to do is to pass from collections of prime ideals in coordinate rings $A(X)$ to prime ideals in arbitrary rings $A$. The collection of all such prime ideals is called the spectrum of $A$.

And what is a scheme?

In brief,

a scheme is anything that locally looks like the spectrum of some ring.

If the scheme even globally looks like the spectrum of some ring, it is called an affine scheme.

Stated in this somewhat sloppy way, this is the most straightforward idea in the world. The only technical subtlety that needs to be taken care of is the precise definition of what it means for some object to locally look like some other entity.

But there is a standard, general nonsense answer to this. Sheaves.

The idea is this:

1) We equip our collections of points/prime ideals with a topology in order to be able to say what an open neighbourhood of a point/prime ideal could be.

2) Given that, we assign to each open set something like the corresponding coordinate ring of that set, pretty much (details below) like we did assign coordinate rings $A(X)$ to affine variaties $X$ above.

3) We do this in such a way that restricting to smaller subsets corresponds suitably to a restriction of the corresponding coordinate rings.

Let’s first see how this works for our motivating example, the affine varieties.

This is a slight variation on the construction of the coordinate ring above.

So let $X$ be any affine variety in $A_k^n$. Consider arbitrary functions $f : X \to k$ from the variety into the ground field $k$.

Definition. ([Hart, p. 15]) A function $f : X \mapsto k$ is regular if in the vicinity of every point of $X$ it looks like a quotient in $A = k[X_1,\cdots X_n]$. More precisely, if every point $x \in X$ has a neightbourhood $U$ such that $f|_U = \frac{g_U}{h_U}$ with $g_U$ and $h_U$ both in $A$ and $h_U$ nonvanishing on $U$.

Clearly, the last condition on $h$ is necessary for the condition to make any sense. For the following generalization of this statement to a more algebraic formulation, notice that we can equivalently demand that $h_U$ is not an element of any of the ideals $\mathfrak{m}_p$ for $p \in U$.

Let us denote the collection of all regular functions on $X$ by $\mathcal{O}(X)$. As the notation suggests, we can similarly associate to every open subset $U \subset X$ the regular functions $\mathcal{O}(U)$ on $U$. This assignment gives a sheaf of rings.

For that to make sense, I need to say what an open set in an affine variety looks like.

Essentially, closed sets are sets of common zeros of polynomials and open sets are hence the complements thereof. More precisely, we have the following definitions:

Definition.([Hart, p. 2])

$\bullet$ an algebraic subset of $A_k^n$ is a set $C$ of common zeros of a collection of polynomials $C = Z(\{f_1,f_2,\cdots\})$.

$\bullet$ the Zariski topology on $A_k^n$ is given by taking the open subsets to be the complements of the algebraic subsets.

$\bullet$ the Zariski topology on an affine variety $X$ in $A_k^n$ is simply the subset topology induced by the Zariski topology on $A_k^n$.

(Notice that the Zariski topology is quite different from the topology one might image intuitively. For instance, it is not Hausdorff. As far as I understand, it justifies itself by being well suited for the algebraic formulation.)

For moving from varieties to schemes, all we need to do is to take these definitions and rephrase them in a more algebraic way, that still applies if the ring of polynomials is replaced by any other ring.

So, given any ring $A$, we now define the collection $\mathrm{Spec} A$ of all its points, equipped with a certain topology.

Definition.([Hart, p. 70]) Given any ring $A$, its spectrum, $\mathrm{Spec}(A)$ is

$\bullet$ as a set, the collection of all prime ideals in $A$

$\bullet$ equipped with the topology in which closed sets are collections $V(\mathfrak{a})$ of all those prime ideals that contain a specified ideal $\mathfrak{a}$ of $A$,

$\bullet$ and equipped with the sheaf $\mathcal{O}_A$ of rings of regular functions on $\mathrm{Spec}A$, to be defined below.

Notice how this reproduces the definition for affine varieties above in the case where $A = k[X_1,\cdots, X_n]$ is a ring of polynomials, $\mathfrak{a} = I(X)$ the ideal of polynomials vanishing on an algebraic subset $X$ and $V(\mathfrak{a})$ the collection of all prime ideals consisting of polynomials that vanish at any given point of $X$.

There is now only one ingredient left, which we need to generalize from varieties to schemes: given an abstract spectrum of some ring $A$, we need to have a notion of regular functions on $\mathrm{Spec} A$.

Essentially, a regular function on the spectrum of a ring will, as for varieties, be a function on the spectrum which is locally given by the quotient of two elements of the ring.

The only technical subtlety to be taken care of is hence the precise notion of quotient in an arbitrary ring.

We need some basic notions of ring theory:

What distinguishes a ring from a field is of course that a ring need not have multiplicative inverses. But given some ring $A$, we may want to add in formal multiplicative inverses to at least some of its elements (for instance for forming the sort of quotients that we are after).

If we declare some element $f$ to be invertible, we clearly have, for consistency, also to take any powers $f^n$ to be invertible. More generally, we may invert any subset $S \subset A$ which contains the unit 1 and is closed under multiplication. Such a subset is called a multiplicative subset, for obvious reasons.

Inverting elements of $S$ in $A$ then means that we form formal fractions $\frac{a}{s}$ with $a \in A$ and $s \in S$ and multiply them componentwise $\frac{a}{s} \cdot \frac{a'}{s'} = \frac{aa'}{ss'}$, where the product in the numerator and in the denominator is that in $A$. Of course we want to identify fractions that are reducible. This is done in a fashion slightly more sophisticated than one might expect:

We take $\frac{a}{s} = \frac{a'}{s'}$ not just if $s' a - a' s = 0$, but if there exists any $s'' \in S$ such that

Passing to equivalence classes of fractions this way gives rise to a new ring denoted $S^{-1}A$ and called the localization of $A$ at $S$.

For our application, we always need two special cases of multiplicative systems.

1) For $\mathfrak{p}$ any prime ideal of $A$, $S = A - \mathfrak{p}$ is a multiplicative system. The corresponding localization is denoted

(It’s clear, but worth emphasizing, that $A_\mathfrak{p}$ is obtained by inverting everything that is not in $\mathfrak{p}$.)

Given our identification of prime ideals with points in generalized spaces, we can understand the terminology localization of the ring $A$ geometrically: prime ideals $\mathfrak{p} = \mathfrak{m}_P$ correspond to sets of functions all vanishing at a given point $P$. Taking every function to be invertible that is not in this ideal, hence does not vanish at $P$ is like restricting all functions to the point $P$, since its precisely in this restriction that all functions not in $\mathfrak{m}_P$ become invertible. Hence inverting $A - \mathfrak{m}_P$ corresponds to localizing functions to a given point.

Accordingly, the ring $A_\mathfrak{p}$ is a local ring, namely one which contains precisely one maximal ideal. This maximal ideal is precisely $\mathfrak{p}$ itself.

2) Sometimes, though, we just have a single element $f \in A$ and feel like inverting it. So we also invert all its powers $f^n$, which together form a multiplicative system $S = \{f^n | n \in \mathbb{N}\}$. Maybe somewhat confusingly, the resulting ring of fractions

is also denoted by a subscript. Beware that now the subscript indicates what has to be inverted, while in the other example it indicated elements that were not to be inverted.

All right, so much for rings of fractions. Recall that I recalled this in order to be able to give an abstract definition of regular functions on a spectrum, which should locally look like such fractions.

More precisely, a regular function should locally look like a fraction whose denominator is locally nonvanishing, in some sense. Ultimately, this means that at every point, the regular function must be a quotient whose denominator is nonvanishing, at that point.

That we can easily rephrase algebraically:

Over every point $\mathfrak{p}$ (to be thought of as a prime ideal of functions vanishing at that point), we require the regular function to take values in the ring of fractions obtained by localizing $A$ to that point, as described above, i.e. in $A_\mathfrak{p}$. There, its value is a formal fraction of the form $\frac{g_p}{h_p}$, with $g_p \in A$ and $h_p \in A-\mathfrak{p}$.

Instead of rambling on, I could just as well state the precise

Definition. ([Hart, p. 70]) Over each open set $U$ of the spectrum $\mathrm{Spec} A$, the ring $\mathcal{O}(U)$ of regular functions on the spectrum, restricted to $U$, is the ring of all maps $$\array{ f &:& U &\to& \bigsqcup_{\mathfrak{p}\in U} A_\mathfrak{p} \\ && \mathfrak{p} &\mapsto& \frac{g_\mathfrak{p}}{h_\mathfrak{p}} \in A_\mathfrak{p} }$$ with $g_\mathfrak{p} \in A$ and $h_\mathfrak{p} \in A - \mathfrak{p}$.

I tried to indicate the simple idea behind this definition. In as far as it looks awkward, this is due to the fact that in the abstract (intrinsic) setup of schemes, we have no global, ambient coordinate ring in which everything takes place, like we had for affine varieties. As a consequence, a regular functions maps each point to a different ring.

But if we don’t like this, we can easily get something nice by forgetting some information.

Namely, in the above, we still distinguish, at every point of the spectrum, functions that coincide at that given point if they disagree away from that point. That’s not what you’d usually expect when comparing functions restricted to a point.

Hence, as we did above for polynomial coordinate functions on varieties, we may want to identify elements of $A_\mathfrak{p}$ if they differ by an element that “is zero” at $\mathfrak{p}$. As before, this corresponds to dividing out the prime ideal $\mathfrak{p}$ and passing to

Since this is a quotient by a maximal ideal it is in fact a field, and the same field for each $\mathfrak{p}$. When schemes and varieties are related below, this quotienting out by ideals allows us to identify regular functions on spectra with regular functions on affine varieties.

Anyway, for the moment we shall be content with recording that the assignment of regular functions to open sets of a spectrum

is a sheaf of commutative rings with units.

We hence find, that both affine varienties as well as spectra of rings come to us as pairs

$\bullet$ of a topological space $X$

$\bullet$ together with a sheaf $\mathcal{O}$ of rings on $X$.

In fact, these sheaves have the special property (as I vaguely indicated above) that the stalk over each point is a local ring (and hence something that we may interpret as a ring of coordinate functions regarded at a given point).

So let’s give this structure a name:

Definition.([Hart, p. 72]) A ringed space is a a pair $(X, \mathcal{O}_X)$ consisting of a topological space $X$ and a sheaf $\mathcal{O}_X$ of rings on $X$.

If each stalk is a local ring, this is called a locally ringed space.

For example, supermanifolds are naturally regarded as ringed spaces, whose rings are the rings of “bosonic” and “fermionic” coordinate functions over them ($\to$).

There is an obvious notion of morphisms between ringed spaces: a morphism

is a pair $(f,f^\sharp)$ of a continuous map $f$ and a morphism of sheaves $f^\sharp$

There is a more or less obvious refinement of this definition to morphisms of locally ringed spaces.

($f_*$ denotes the pushforward of sheaves along $f$. For more on pushing and pulling of sheaves see this recent entry.)

To conclude, all these considerations, we have found

$\bullet$ that sets of common zeros of polynomials come with regular function on them such that they form a locally ringed space.

$\bullet$ By more or less straightforwardly generalizing this construction away from rings of polynomials to arbitrary rings, we have equipped the collection of prime ideals $\mathrm{Spec} A$ of any arbitrary ring $A$ with a notion of regular functions on it and hence, too, with the structure of a locally ringed space.

The first item describes (up to some conditions) affine varieties. The second is a generalization thereof and called an affine scheme.

Essentially. This is a slight lie. Since there is a category of ringed spaces, we want to be careful with identifying them on the nose. So instead of saying that an affine variety is a spectrum $\mathrm{Spec} A$, we say that an affine variety is anything ismorphic (in the category of locally ringed spaces) to $\mathrm{Spec} A$, for some $A$.

When we are even more relaxed about identifying structures, we will identify by isomorphisms only locally. A scheme is something that locally looks like an affine scheme:

Definition.([Hart, p. 74])

$\bullet$ An affine scheme is a locally ringed space which is isomorphic to the spectrum of some ring

$\bullet$ A scheme is a locally ringed space in which every point has a neighbourhood that is an affine scheme.

In particular, schemes form a subcategory of that of locally ringed spaces.

The answer to the following question is now almost obvious:

How are varieties a special case of schemes?

The way schemes were defined as a generalization of varieties makes this almost a tautology. But in order to do it properly one has to take care of the technicalities.

Fact (roughly)([Hart, prop. 2.6]) Varieties are fully faithfully embedded into the category of schemes.

The idea of the proof is this: given any variety $(X, \mathcal{O}_X)$ with coordinate ring $A(X)$, we construct an isomorphism

from the variety to the spectrum of its coordinate ring.

This isomorphism consists of

$\bullet$ the homeomorphism

which sends every point $p$ of the variety to the prime ideal $\mathfrak{m}_p$ of regular functions that vanish at this point,

$\bullet$ the isomorphism of sheaves

which simply sends every regular function $s$ on $\mathrm{Spec}A(X)$ (which, recall, takes values in a disjoint union of local rings instead of in a field) to the regular function $f^\sharp(s)$ on $X$ obtained by identifying over each point sections that agree on that point.

As I have tried to discuss in some detail above, this corresponds to quotienting over each point by $\mathfrak{m}_p$.

That this makes good sense is a consequence of the following

Fact.([Hart, prop. 2.2]) On the locally ringed space $(\mathrm{Spec}A,\mathcal{O}_{\mathrm{Spec}A})$, the stalk $\mathcal{O}_{\mathrm{Spec}A,\mathfrak{p}}$ over any point $\mathfrak{p} \in \mathrm{Spec}A$ is isomorphic to the local ring $A_\mathfrak{p}$, $$\mathcal{O}_{\mathrm{Spec}A,\mathfrak{p}} \simeq A_\mathfrak{p} \,.$$ There are more details, which I have glossed over here.