The schemes that the classical picture cannot hold

(The double point)

\(\Spec k[\varepsilon]/(\varepsilon^2)\) has one point and a two-dimensional ring of functions. It is the scheme-theoretic intersection of the parabola \(y = x^2\) with the line \(y = 0\), and it is what distinguishes tangency from transversality: the reduced intersection is a single point either way.

The function \(\varepsilon\) is nonzero, but its value at the unique point, its image in the residue field \(k[\varepsilon]/(\varepsilon) = k\), is \(0\); so \(\varepsilon\) and \(0\) are different functions with the same values at every point.

Its \(k\)-points valued in a \(k\)-algebra compute tangent vectors: a morphism \(\Spec k[\varepsilon]/(\varepsilon^2) \to X\) is a point of \(X\) together with an element of \(T_p X\).

(Two points that are not the same size)

\(\Spec \ZZ\) has a closed point for each prime and a dense generic point \((0)\). Its residue fields are \(\FF_p\) and \(\QQ\), so the “functions” on it take values in different fields at different points.

(Values of a regular function on $D(6) \subseteq \Spec \ZZ$)

The distinguished open set \(D(6) \subseteq \Spec \ZZ\) consists of the generic point \((0)\) and the points \((p)\) for primes \(p \neq 2, 3\), and \(\OO_{\Spec \ZZ}(D(6)) \cong \ZZ[1/6]\). So \(\varphi = 5/6\) is a regular function on \(D(6)\), and its value at a point is its image in the residue field there.

• At \((0)\), the value is \(5/6 \in \QQ = \kappa((0))\).

• At \((p)\) with \(p \neq 2, 3\), the value is \(\bar{5} \cdot \bar{6}^{-1} \in \FF_p = \kappa((p))\). Hence \(\varphi\) vanishes at \((5)\), and \(\varphi((11)) = \bar{5} \cdot \bar{2} = \bar{10} \in \FF_{11}\), since \(\bar{6} \cdot \bar{2} = \bar{1}\) in \(\FF_{11}\).

(A nonreduced multiple structure)

\(\Spec k[x]/(x^2)\) and \(\Spec k[x]/(x)\) have the same topological space. The first is the fibre of \(\AA^1 \to \AA^1\), \(t \mapsto t^2\), over the branch point, and its length \(2\) is what makes the degree of that map constant.