schemes

Definition

A scheme is a locally ringed space $(X,{\mathcal{O}}_X)$ that is locally isomorphic to an affine scheme. Explicitly, this means:

1. $(X,{\mathcal{O}}_X)$ is a locally ringed space.
2. There exists an open cover $\{U_i{\}}_{i\in I}$ of $X$ such that for each $i$, the restriction $(U_i,{\mathcal{O}}_X{|}_{U_i})$ is isomorphic (as a locally ringed space) to an affine scheme $(\mathrm{Spec}(R_i),{\mathcal{O}}_{\mathrm{Spec}(R_i)})$ for some commutative ring $R_i$.

Affine Schemes

An affine scheme is a locally ringed space isomorphic to the spectrum of a ring, denoted $\mathrm{Spec}(R)$.

The points of $X=\mathrm{Spec}(R)$ are the prime ideals of $R$. The structure sheaf ${\mathcal{O}}_X$ is determined by the localization properties. For any prime ideal $\mathfrak{p}\in X$, the stalk of the structure sheaf at $\mathfrak{p}$ is the local ring:

$${\mathcal{O}}_{X,\mathfrak{p}}\cong R_{\mathfrak{p}}$$

where $R_{\mathfrak{p}}$ is the localization of $R$ at the prime ideal $\mathfrak{p}$.

Basic Properties

Connectedness

A scheme $X$ is connected if its underlying topological space is connected. This corresponds to the ring $R$ (if $X$ is affine) having no nontrivial idempotents $e$ such that:

$$e^2=e\text{and}e\ne 0,1$$

Reduced Schemes

A scheme $X$ is reduced if for every open set $U\subset X$, the ring of sections ${\mathcal{O}}_X(U)$ has no nilpotent elements. That is, for all $f\in {\mathcal{O}}_X(U)$:

$$f^n=0⟹f=0$$

If $X=\mathrm{Spec}(R)$, then $X$ is reduced if and only if the nilradical of $R$ is zero: $\mathfrak{N}(R)=\sqrt{(0)}=0$.

Integral Schemes

A scheme is integral if it is both reduced and irreducible. For an affine scheme $\mathrm{Spec}(R)$, this is equivalent to $R$ being an integral domain.

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Structure Morphisms

For any scheme $X$, there exists a unique morphism to the terminal object in the category of schemes, $\mathrm{Spec}(\mathbb{Z})$.

$$X⟶\mathrm{Spec}(\mathbb{Z})$$

If $X$ is a scheme over a field $k$, there is a structure morphism $f:X\to \mathrm{Spec}(k)$. For a closed point $x\in X$, the residue field is a finite extension of $k$:

$$\kappa (x)={\mathcal{O}}_{X,x}/{\mathfrak{m}}_x$$

where $[\kappa (x):k]<\infty$.