Firstly, sorry for the blog silence! Christmas, New Year, birthday and exams all got in the way… but now we’re finally onto this Zariski site post I’ve been going on about!

If you remember some of the stuff from this post about sheaves on sites, and this post about the functor of points of a scheme , then you’re good to go here. We introduced the functor of points of a scheme $X$, namely the representable presheaf $h_X = \text{Mor}(-,X)$, and looked at how it relates to the solutions of Diophantine equations in the affine case $X = \text{Spec}(A)$. We also saw here how if $X$ is a $K$-scheme then the set of “rational points”  $X(K)$ of $X$ identifies with $h_X (\text{Spec}(K))$ and with the sections of the morphism $X\rightarrow \text{Spec}(K)$. And then I mentioned somewhere that it is possible to “identify” a scheme with its functor of points.

What this really means is that we can “embed’ the category of schemes into the larger category of presheaves on the category of schemes (this is just the Yoneda embedding), but we can also completely characterise when a presheaf on the category of schemes is the functor of points of some scheme. To do this, we had to introduce the notion of what a sheaf on a category was, so we needed Grothendieck (pre)topologies. It will turn out that a presheaf on the category of (affine) schemes is the functor of points of a scheme if and only if it is a sheaf on the Zariski site and a certain “covering condition” is satisfied.

To explain why I have put the word “affine” in brackets above, we’ll prove the following nice lemma:

Lemma: Let $R$ be a ring, and $\textbf{Sch}/R$ the category of schemes over $R$. The functor

$h: \textbf{Sch}/R \rightarrow [R\textbf{-Alg}, \textbf{Set}], \quad X\mapsto h_X = \text{Mor}_R (-,X)$

is full and faithful, where the presheaf $text{Mor}_R (-,X)$ denotes the presheaf on the category of $R$-schemes.

This looks very like the usual Yoneda embedding except that the functor category here is slightly different. We’re sending each $R$-scheme $X$ to its functor of points $h_X$, but this time not considered as a presheaf on the category of all $R$-schemes but now just as a presheaf on the subcategory of affine $R$-schemes; in other words, as a covariant functor from the category of $R$-algebras to sets. The lemma says that each functor of points is completely determined by its action on affine $R$-schemes, which in some way just reflects the fact that all schemes are just glued together from affine schemes.

Proof of the lemma: Let $S = \text{Spec}(R)$ and let $h_X$ denote the restriction of the presheaf $\text{Mor}_S (-, X)$ to the category of affine $R$-schemes. We want to show that every natural transformation $\phi: h_X \mapsto h_{X'}$ comes from a unique morphism of $R$-schemes $f: X\rightarrow X'$. To do this, let $X = \cup_{i} X_i$ be a cover of $X$ by affine schemes, and let

$j_i : X_i \hookrightarrow X$

denote the inclusions (which are morphisms of $R$-schemes). Then $f_i:=\phi_{X_i} (j_i)\in h_{X'} (X_i)$ is a morphism $X_i \rightarrow X'$. Then using the “glueing” of the topological spaces, continuous maps and structure sheaves, we get a unique morphism of schemes $f: X\rightarrow X'$ such that the restriction of $f$ to $X_i$ is $f_i$.

You can check that the natural transformation $\phi: h_X \rightarrow h_{X'}$ is the image of $f$ under $h$, so $h$ is a full functor. Now again using that schemes are glued up from affine schemes, we see that if we have two morphisms $f,g: X\rightarrow X'$ that differ then they must differ on one of the affine covers $X_i\subseteq X$. Then by the construction above, the induced natural transformations $h_f, h_g: h_X \rightarrow h_{X'}$ will not be equal because they differ on their component at $X_i$. Thus $h$ is a full functor. This completes the proof.

So a (relative) scheme is determined by the values its functor of points takes on the category of affine (relative) schemes, which really is just the dual of the category of rings. This gets around the awkward circular attempt to define a scheme as a certain type of presheaf on the category of schemes, because we already know what the category of rings is. When we eventually categorise schemes as certain sheaves on the category of affine schemes (the dual of the category of rings) we won’t need to already know what a scheme is because everything goes through purely categorically using that the category of affine schemes is dual to rings, and we can define topologies on this dual category.

The Zariski Topology

Let $\mathcal{C}$ denote the category of affine schemes (i.e. the dual of the category of rings). To specify what the Zariski pretopology is on $\mathcal{C}$ we need to say what the covering families are:

Suppose $\left\{\text{Spec}(A_i)\rightarrow \text{Spec}(S)\right\}$ is a family of morphisms in $\mathcal{C}$. This family is a covering family for the Zariski pretopology iff

1. Each ring $A_i$ is the localisation of $S$ at a single element $s_i \in S$;
2. The morphism $\text{Spec}(A_i)\rightarrow\text{Spec}(S)$ is the functorial inclusion induced by the localisation map $S\rightarrow A_i = S[s_i^{-1}]$;
3. There exists a finite set $f_i \in S$ such that $s_1 f_1 + \dots + s_n f_n = 1$.

The first condition means we want to think of each $s_i$ as a function on $\text{Spec}(S)$; the second condition means we want the inclusion $\text{Spec}(A_i)\hookrightarrow \text{Spec}(S)$ to be the subset of the space on which the function $s_i$ is nonzero; the third condition expresses that these functions should form a “partition of unity“, a useful geometric condition.

You can check that these satisfy the axioms for a pretopology on $\mathcal{C}$ given in this post – clearly, the family $\left\{id_{\text{Spec}(S)}\right\}$ is a covering family induced by localisation at $1\in S$; the pullback condition holds by looking at the images of these localised elements under homomorphisms; the final condition holds via composition of localisations.

This defines a pretopology on $\mathcal{C}$ denoted $J_{\text{Zar}}$, the Zariski pretopology. This induces a unique Grothendieck topology on the category, and we denote the site $(\mathcal{C}, J_{\text{Zar}})$ by $\textbf{Aff}_{\text{Zar}}$, the affine Zariski site. A sheaf on this site is then a functor

$F: \textbf{Aff}_{\text{Zar}} \rightarrow \textbf{Set}$

such that for every covering family

$\left\{\text{Spec}(A_i) = X_i \rightarrow\text{Spec}(S) = X\right\}$

the set $F(X)$ is the equaliser of the two maps $\prod F(X_i) \rightrightarrows \prod F(X_i \times_X X_j)$ induced by restrictions, where the pullback in the category of affine schemes is defined as

$X_i \times_X X_j = \text{Spec}\left(A_i \otimes A_j \right)$

i.e. the spectrum of the tensor product of the $S$-algebras $A_i$ and $A_j$.

Open subfunctors

We’re nearly ready to state the classification theorem for schemes as functors; we just need one final notion:

Suppose that $\alpha: G\hookrightarrow F$ is a subfunctor, where both are functors from rings to sets. We say that $G$ is an open subfunctor if whenever $X = \text{Spec}(R)$ is an affine scheme and $\psi: h_X \rightarrow F$ is a natural transformation, the morphism

$h_X \times_F G \rightarrow h_X$

is isomorphic to the natural transformation $h_Y \rightarrow h_X$ induced by the inclusion $Y\hookrightarrow X$ of an open subscheme.

One can show that the open subfunctors of a functor of points $h_X$ are precisely the functors of points of open subschemes of $X$.

Characterisation of schemes amongst all functors

We’ve seen that the whole category of schemes embeds inside the category of presheaves on the category of affine schemes. Let’s now see what that embedded category looks like:

Functor classication theorem: Let $F: \textbf{Aff}^{\text{op}} \rightarrow\textbf{Set}$ be a presheaf on the Zariski site (i.e. a functor from rings to sets). Then $F$ is the functor of points of a scheme $X$ if and only if

1. $F$ is a sheaf on the Zariski site $\textbf{Aff}_{\text{Zar}}$;
2. there exists rings $R_i$ and open subfunctors $\alpha_i : h_{R_i} \rightarrow F$ such that for each field $K$, $F(\text{Spec}(K))$ is the union of the images of the sets $h_{R_i} (K)$ under the $\text{Spec}(K)$-components of the natural transformations $\alpha_i$.

This is actually not hard to prove – just using that schemes are glued up from affine schemes. Although it’s abstract it’s somehow closer to home than viewing schemes as geometric spaces, because we understand algebra (rings) well in the abstract sense, and we understand categorical algebra well too – the geometry of schemes then emerges as categorical algebra on the category of rings. I’ll try to emphasise the two points of view throughout this blog, although the “ringed space” viewpoint will undoubtedly serve us better until we start doing more advanced stuff, where it is possible to change the Zariski site to “finer” sites like the étale site having better properties. It’s also necessary to use the functor of points viewpoint for anything involving stacks, and generalisation of schemes, and this particular arm of algebraic geometry makes heavy use of this “geometry-through-algebra” idea.

Next time I’m leaving all this abstract nonsense and going back to doing some real geometry with schemes, starting from the basics.