The String Coffee Table

Last time I ended with mentioning properties of schemes that come from varieties. Now I need to say a few words about what these properties mean and why this is true.

In particular, I need to say what (quasi-)projective schemes are.

1) Projective schemes.

Recall that a (quasi-)affine variety was defined to be an (open subset of an) irreducible closed subset of affine space $\mathbf{A}_k^n$.

There is a very popular slight generalization of this.

Instead of using affine space, we can use a projective space, something that only locally looks like an affine space.

Like $\mathbf{A}_k^n$ is the space of all $n$-tuples with entries in $k$, equipped with the Zariski topology, projective space $\mathbf{P}_k^n$ is the space of all lines through the origin in $\mathbf{A}_k^n$, equipped with a similar topology.

More concretely, $\mathbf{P}_k^n$ is the set of equivalence classes of $(n+1)$-tuples of elements of $k$, not all of them vanishing, under the relation

for all $\lambda \neq 0$ in $k$.

Polynomials on this space are polynomials on $\mathbf{A}_k^{n+1}$ whose sets of zeros respect this symmetry. This are the homogeneous polynomials.

The ring $S = k[X_0,\cdots, X_n]$ of all polynomials on $\mathbf{A}_k^n$ is graded, $S = \oplus_{d \in \mathbb{N}} S_d$, with $S_d$ being the space of homogeneous polynomials of degree $d$.

We can hence proceed as before in the affine situation if we add the adjective “homogeneous” to (almost) everything in sight.

First of all, algebraic subsets of projective space are precisely the sets $S(\{f_1,\cdots, f_r\})$ of common zeros of a collection of homogeneous polynomials.

The Zariski topology on $\mathbf{P}_k^n$ is that which has precisely the complements of these algebraic subsets as open sets.

And so on:

A projective algebraic variety is an irreducible algebraic subset in $\mathbf{P}_k^n$.

A quasi-projective algebraic variety is an open subset of an irreducible algebraic subset in $\mathbf{P}_k^n$.

In a sense, projective varieties are like a first tiny step from affine varieties to schemes, since every projective variety can be covered by affine varieties. Simply take the coordinate patches $U_i$ to be consisting of all those classes of tuples $(a_0,\cdots, a_n)$ with $a_i \neq 0$. These are isomorphic to $\mathbf{A}_k^n$ by

Hence, in particular, we can regard (quasi-)affine varieties just as well as irreducible closed subsets not of $\mathbf{A}_k^n$, but of $\mathbf{P}_k^{n+1}$. This allows us to sum up

$\bullet$ affine varietes

$\bullet$ quasi-affine varietes

$\bullet$ projective varietes

$\bullet$ quasi-projective varietes

all in one single definition:

Definition.([Hart, p. 15; bottom of p. 104]) A variety is a locally closed subset of projective space.

Fine. The goal now is to similarly move from affine schemes to projective schemes. So instead of talking extrinsically about coordinates, we want to talk intrinsically about ideals ([Hart, pp. 76]).

We can essentially read off all there is to say from the properties of the graded ring $S = \oplus_{d \in \mathbb{N}} S_d$, with $S_d$ containing the homogenous polynomials of grade $d$.

So let $S$ now be any graded ring. A homogeneous ideal in $S$ is any ideal which is a direct sum of homogeneous subspaces.

We want to copy the construction of the affine scheme $\mathrm{Spec}A$ for any ring $A$ now using the graded ring $S$ and replacing arbitrary ideals with homogeneous ideals everywhere.

So define the projective spectrum $\mathrm{ProjSpec}S$ of the graded ring $S$ to be

$\bullet$ as a set the collection of all homogeneous prime ideals of $S$,

$\bullet$ equipped with the topology whose closed sets are precisely the sets $V(\mathfrak{a})$ of all homogeneous prime ideals containing some homogeneous ideal $\mathfrak{a}$,

$\bullet$ together with the sheaf of regular homogeneous functions which is defined as for affine spectra, but with homogeneous ideals used everywhere.

Like the spectrum $\mathrm{Spec} A$ for $A = k[X_1,\cdots,X_n]$ is the affine scheme corresponding to affine space $\mathbf{A}_k^n$, the projective spectrum $\mathrm{ProjSpec} S$ for $S = k[X_0,X_1,\cdots,X_n]$ is the projective scheme corresponding to projective space $\mathbf{P}_k^n$.

It is obvious what a quasi-projective scheme should be, namely something that is like an open subset in some $\mathrm{ProjSpec}S$. The right formulation of schemes that are “like open subsets” is

Definition. ([Hart, p. 85]) A morphism of schemes $F : (X,\mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a closed immersion if

$\bullet$ on the underlying topological spaces $f : X \to Y$ is a homeomorphism onto a closed subset;

$\bullet$ and the morphism of sheaves $f^\sharp : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is surjective (on fibers, I guess).

So, like a general variety over $k$ is any locally compact irreducible subset of projective space $\mathbf{P}_k^n$, we say

Definition. ([Hart, p. 103 (here only for the simple subcase needed in prop. 4.10)]) A quasi-projective scheme over $k$ is a scheme over $k$ $$(X,\mathcal{O}_X) \to \mathrm{Spec} k$$ which factors by a closed immersion through the projective scheme $$\cdots = (X,\mathcal{O}_X) \overset{\text{closed imm.}}{\to} \mathrm{ProjSpec} k[X_0, \cdots, X_n] \to \mathrm{Spec} k \,.$$

2) Some more properties of schemes.

Schemes coming from varieties turn out to be in particular integral, seperated and of finite type. Here some words on what all these properties mean (following [Hart, section 3]).

First of all, a remark on the notion of dimension.

How do we measure the dimension of some space using a purely algebraic measurement apparatus?

Roughly, we check if its points are subsets of 1-dimensional closed subsets, which are subsets of 2-dimensional closed subsets, which are subsets of 3-dimensional, subsets, and so on. The maximum $n$ we find this way is the dimension of our space.

The sequence of inclusions constructed this way should terminate if our space is to have finite dimension, which is all we want to consider at the moment. Since Emmy Noether was the first to realize the importance of such finiteness conditions in algebraic geometry, objects of finite dimension in this sense are called noetherian.

So, a topological space is noetherian ([Hart, p. 5]) if any sequence of inclusions of closed subsets

stabilizes at some point, such that from there on all $X_i$ are equal (known as the ascending chain condition).

Definition. ([Hart, p. 5]) The dimension of a topological space $X$ is the length minus 1 of the longest chain of inclusions of distinct irreducible closed subsets $X$.

Algebraically the situation is the same, but the other way around. Closed subsets $X$ correspond to ideals $I(X)$ (of polynomials vanishing on $X$), and larger closed subset correspond to smaller ideals.

So, a (coordinate) ring is a noetherian ring if it contains only finitely long chains of distinct (prime?) ideals.

For instance, a maximum chain of distinct prime ideals in the affine space $A_k^3$ would be

which is indeed of length $3 +1 = 4$, as expected. It describes the point with coordinates $(1,5,999)$ sitting inside the line $t \mapsto (1,5,t)$ which again is a subset of the surface $(s,t) \mapsto (1,s,t)$, which, finally, is a subset of the total space.

Hence, we can just as well speak about the dimension of a ring (called the Krull dimension), which is (one less than) the maximum length of chains of inclusions of distinct prime ideals in that ring.

This, then, allows us to easily transport the notion of dimension over to the world of schemes, simply by looking at the the Krull dimension of the coordinate rings of its affine subschemes (see [Hart, p. 83, p. 86]).

This notion of dimension reproduces the expected notion of dimension in standard cases of schemes. But on general schemes there can be counterintuitive phenomena regarding their dimension (see [Hart, Caution 3.2.8, p. 87]).

Example. Let the ring $A = \mathbb{N}$ be that of the natural numbers. Its prime ideals are all those generated by prime numbers (hence the name!), which are all maximal ideals, and the 0-ideal. So $\mathrm{Spec}(A)$ is a space that consists of countably many discrete points together with a generic open point which “touches” all of them. Every maximal chain of inclusions of distinct prime ideals looks like

for $p \gt 0$ a prime number. Hence this collection of discrete points actually has dimension $1$, in the above sense, not $0$.

Responsible for this is that funny “generic point” $(0)$, which somehow glues all the discrete points to something 1-dimensional.

3) Still more properties of schemes:

$\bullet$ a scheme is integral if all its fibers $\mathcal{O}_X(U)$ are integral domains, meaning that all its coordinate rings are free of zero divisors.

$\bullet$ a scheme is reduced if all its coordinate rings $\mathcal{O}_X(U)$ have no nilpotent elements.

(Nilpotent elements in rings correspond to “infinitesimal distances” in the associated geometry. Hence one can understand “reduced” in the sense that all infinitesimally close points are identified. I think.)

$\bullet$ a scheme is locally noethrian (“locally finite dimensional”) if it can be covered by affine subschemes $\mathrm{Spec} A_i$ which come from noetherian rings.

$\bullet$ a morphism of schemes $f : X \to \mathrm{Spec} R$ is of finite type if the pre-image of $f$ can be covered by by finitely many open affine subsets $\mathrm{Spec}A_i$, where each $A_i$ is a finitely generated $R$ algebra.

Note that $A_i$ is an $R$-algebra in any case. The crucial condition is that it be finitely generated.

Since we are concerned with the case of schemes over a field $k$, i.e. morphisms $f : X \to \mathrm{Spec} k$, we say these are of of finite type if this morphism satisfies the above condition.

$\bullet$ A scheme is seperated if it has the same abstract behaviour as a Hausdorff topological space. Since the Zariski topology is never Hausdorff, this requires a little sophistication ([Hart, p. 96]). But for the moment I shall just skip this.

Recall that we are interested in understanding the nature of the image of the functor $t : \mathrm{Var}(k) \to \mathrm{Sch}(k)$, which sends any variety to the collection of its irreducible closed subsets (equivalently: to the collection of prime ideals of its coordinate ring).

Now, it is a fact ([Hart, prop. 4.3, p. 25]) that every variety $X$ can be covered by a finite number of open affine subvarieties. This means ([Hart, ex. 3.2.1, p. 84]) that $t(X)$ can be covered by a finite number of open affine schemes $\mathrm{Spec}A_i$, with $A_i$ being

$\bullet$ an integral domain (because it is of the form $A = A/I(U_i)$ with $I(U_i)$ a prime ideal),

$\bullet$ a finitely generated $k$-algebra (because it comes from a ring of polynomials),

$\bullet$ hence noetherian (its spectrum is a finite dimensional space).

Moreover, since any variety $X$ is a locally closed subset of projective space, it follows that $t(X)$ is a quasi projective scheme. This again implies ([Hart, theor. 4.9, p. 103]) that it is of finite type and seperated.

Given all that, we suspect that the result that we are after (at least that’s what I am after) is this

Fact.([Hart, prop. 4.10, p. 104]) The image of the category of varieties over $k$ in the category of schemes over $k$ is exactly the subcategory of quasi-projective integral schemes. All these schemes are, in particular, integral, seperated and of finite type.

Indeed, that’s true. That everything in the image of $t$ is a quasi-projective integral scheme is essentially obvious, given that this are precisely the properties inherited from the very definition of varieties.

What remains to be shown is only the surjectity of $t$ on this subcategory, i.e. that every quasi-projective integral scheme really comes from some variety.

This follows after one notices that for every closed subscheme $Y$ of $\mathrm{ProjSpec} k[X_0,\cdots,X_n]$ the set of closed points is a dense set in $Y$, which (apparently) imlies that it is irreducible and hence in fact a variety.

Maybe more on all these technical details later. Now I have at least mentioned all the necessary ingredients.

Maybe in closing, just one remark on what the theorem buys us.

Consequence: abstract varieties.

One upshot of all this is that we get an intrinsic definition of what a variety is, one that does not rely on constructing it as a set of common zeros of a collection of polynomials.

Namely, we have that a variety over $k$ is a quasi-projective integral scheme over $k$.

Better yet, we may now generalize and remove the condition on quasi-projectivity: an abstract variety is an integral seperated scheme of finite type over an algebraically closed field $k$.