What is a scheme?

 Conference topic

This is from a problem session at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Tyler Lawson for explaining the material.

Definition: Affine schemes are the category $\Ring^{op}$. An object $R\in \Ring$ becomes an object $\Spec(R)$ in affine schemes, and a ring map $R\to S$ becomes a map $\Spec(S)\to \Spec(R)$, where $\Spec$ denotes the set of prime ideals.

We try to think of $Spec(R)$ as a geometrical object.

Example: Let $k$ be a field and consider the ring
\[
R = k[x_1,\dots,x_n] / (f_1(x_1,\dots,x_n),\dots,f_r(x_1,\dots,x_n)).
\]
$\Spec(R)$ is supposed to be a substitute for the set of solutions to a system of equations
\begin{align*}
f_1(x_1,\dots,x_n) & = 0,\\
\vdots \hspace{.7cm}\\
f_r(x_1,\dots,x_n) & = 0.
\end{align*}

The scheme $\Spec(R)$ has a more precise definition. It consists of a set, a topology, and a sheaf. 

1. Set: The underlying set of the scheme $\Spec(R)$ is the set of prime ideals of $R$. For example:

• if $R = \C[x]$, then the prime ideals are $(x-\alpha)$ and $(0)$;
• if $R = \C[x,y]$, then the prime ideals are $(x-\alpha,y-\beta)$, irreducible polynomials $(f(x,y))$, and $(0)$.

2. Topology: For every ideal $I\subset R$, the set $V(I) = \{P\subset R$ prime, $P\supset I\}$ is a closed set. Note that

\[
\bigcup_{n=1}^N V(I_n) = V\left(\bigcap_{n=1}^N I_n\right)
\hspace{1cm}\text{and}\hspace{1cm}
\bigcap_{\alpha\in I} V(I_\alpha) = V\left(\sum_{\alpha\in A} I_A\right).
\]
Geometrically, the closed sets are sets of points where one or more identities (like $f(x)=0$) can hold. For example, if $R=\C[x]$, then we have three different closed set types: $\Spec(C[x])$, $\emptyset$, or a finite union of $(x-\alpha_1,\dots, x-\alpha_n)$. Solutions to equations can be one of the following types below.

3. Sheaf: Let $X$ be a set with a topology. $\mathcal O_X$ is the sheaf for which:

• to each open set $U\subseteq X$ we get a ring $\mathcal O_X(U)$;
• to each containment $V\subseteq U\subseteq X$ of open sets, there exists a restriction map $\res_{UV}:\mathcal O_X(U)\to \mathcal O_X(V)$;
• the restriction maps are compatible, in the sense that $\res_{VW}\circ \res_{UV} = \res_{UW}$.

This is called the structure sheaf of $X$.

Say $R$ is our ring, $\Spec(R)$ our set of primes, and we have some open set $U\subseteq \Spec(R)$. We like to think of it in the following way:

• elements of $R$ are functions;
• elements of $\Spec(R)$ are points where we can evaluate a function $f\in P$ (or where the function vanishes);
• subsets $S\subset R$ are the sets $\{f\in R\ :\ f$ only vanishes at points outside $U\}$.

Note that $S$ is closed under multiplication. We localize $R$ at $S$ to get a set
\[
S^{-1}R = \left\{\left[\frac fs\right]\ :\ f\in R, s\in S\right\},
\]
for which $\mathcal O_X(U) = S^{-1}R$ (good enough for today's purposes). Now we have a triple $(\Spec(R),\tau,\mathcal O_X)$, for $\tau$ the Zariski topology, which we call a locally ringed space.

Definition: A scheme is a space $X$ with a topology and a sheaf of rings that is locally isomorphic to $\Spec(R)$.

Since the sheaf has the space $X$ and the topology (through the open sets) encoded in it, we may think of a scheme as a special type of sheaf. Also, isomorphism is meant in the category of locally ringed spaces.

Proposition: Morphisms of schemes $\Spec(R)\to \Spec(S)$ are the same as ring maps $S\to R$.

Example: In the Zariski topology, take $U\subseteq \Spec(k[x,y])$. Locally $U$ looks like it is covered by rings, though that may not be the case globally. Indeed:

Example: Consider projective space $\P^2$, where $[x:y:z] = [\lambda x: \lambda y:\lambda z]$. We may write
\[
\begin{array}{r c c c c c c}
\P^2 & = & U_0 & \cup & U_1 & \cup & U_2. \\
& & [1:y:z] & & [x:1:z] & & [x:y:1] \\
& & \Spec(k[y,z]) & & \Spec(k[x,z]) & & \Spec(k[x,y])
\end{array}
\]
How can we express $U_0\cap U_1$? This is left as an exercise.

The Hodge decomposition, diamond, and Euler characteristics

 Seminar topic

Recall the sheaf of $r$-differential forms $\Omega^r_X$ on $X$ (with $\Omega^r_X(U) = \{fdx_{i_1}\wedge \cdots \wedge dx_{i_r}\ :\ f\ $is well-defined on $U\}$ and such sums) and the structure sheaf $\mathcal O_X$ on $X$ (with $\mathcal O_X(U) = \{f/g\ :\ f,g\in k[U],\ g\neq 0\ $on$\ U\}$). Then we may consider the sheaf cohomology of $X$, with values in $\Omega^r_X$ or $\mathcal O_X$.

Definition: Let $X$ be a smooth manifold of dimension $n$. The $(p,q)$th Hodge number is $h^{p,q}=\dim(H^{p,q})$, where $H^{p,q} = H^q(X,\Omega^p_X)$. These numbers are arranged in a Hodge diamond as below.

The Hodge diamond has a lot of repetition - by complex conjugation, we get that $h^{p,q}=h^{q,p}$, so it is symmetric about its vertical axis. By the Hard Lefschetz theorem (or the Hodge star operator, or Poincare duality), we get that $h^{p,q}=h^{n-q,n-p}$, so it is symmetric about its horizontal axis.

Proposition: Let $X$ be a Kähler manifold (note that all smooth projective varieties are Kähler) of dimension $n$. Then the cohomology groups of $X$ decompose as
\[
H^k(X,\C) = \bigoplus_{p+q=k}H^{p,q}(X),
\]
for all $0\leqslant k\leqslant 2n$. This is called the Hodge decomposition of $X$.

This decomposition immediately gives all the Hodge numbers for $\P^n$, knowing its cohomology. For a manifold of complex dimension $n$, there are several numbers and polynomials that may be defined. These are:
\begin{align*}
\chi_{top}(X) & = \sum_{i=1}^{2n}(-1)^i \dim(H^i(X,\C)) & \text{the (topological) Euler characteristic} \\
\chi^p(X) & = \sum_{q=0}^{n-1}(-1)^qh^{p,q} & \text{the chi-$p$ characteristic} \\
\chi_y(X) & = \sum_{p=0}^{n-1}\chi^py^p & \text{the chi-$y$ characteristic}
\end{align*}
Note the Euler characteristic is the alternating sum of the rows of the Hodge diamond, and the chi-$p$ characteristic is the alternating sum of the left-right diagonals of the diamond.

Example: In the case $X$ is a hypersurface in projective $n$-space $\P^n$ defined by a degree $d$ polynomial,
\[
\chi_y = [z^n]\frac{1}{(1+zy)(1-z)^2}\cdot\frac{(1+zy)^d-(1-z)^d}{(1+zy)^d+y(1-z)^d}.
\]
Since every row except the middle row of the Hodge diamond of a hypersurface is known (as it comes from the Hodge diamond of $\P^n$ by the Lefschetz hyperplane theorem), this expression gives all the unknown numbers. This particular formula is a simplification  of Theorem 22.1.1 in Hirzebruch, which itself comes from the Riemann--Roch theorem.

References: Huybrechts (Complex Geometry: An Introduction, Chapters 3.2, 3.3), Hirzebruch (Topological Methods in Algebraic Geometry, Appendix 1, Section 22)