For closed subschemes \(Y,Z\subseteq X\), \(Y\times_XZ\) is the scheme-theoretic intersection; for \(f\colon X\to Y\) and \(y\in Y\), \(X\times_Y\Spec\kappa(y)\) is the fibre over \(y\); for a field extension \(L/k\) and a \(k\)-scheme \(X\), \(X\times_{\Spec k}\Spec L\) is the base change \(X_L\).
The fibre product of schemes (Tag D-SCHFPR)
For morphisms \(X \to S\) and \(Y \to S\), the fibre product \(\fiberprod{X}{S}{Y}\) is a scheme over \(S\) with projections to \(X\) and \(Y\), universal among schemes mapping compatibly to both: \[ \Hom_S(T, \fiberprod{X}{S}{Y}) \cong \Hom_S(T, X) \times \Hom_S(T, Y) . \]
(Existence)
Fibre products exist in \(\Sch\). For affines the anti-equivalence turns the limit into a colimit of rings: \[ \fiberprod{\Spec A}{\Spec R}{\Spec B} \cong \Spec (A \tensor_R B) . \] The general case is obtained by covering \(X\), \(Y\), \(S\) by affines, building the affine pieces, and gluing.
The underlying set of \(\fiberprod{X}{S}{Y}\) is in general not the fibre product of the underlying sets. \(\fiberprod{\Spec \CC}{\Spec \RR}{\Spec \CC} \cong \Spec (\CC \tensor_\RR \CC) \cong \Spec (\CC \times \CC)\) has two points, while the fibre product of the underlying sets has one. For \(K = \FF_p(t)\) over \(L = \FF_p(t^p)\), \(\fiberprod{\Spec K}{\Spec L}{\Spec K} \cong \Spec K[x]/(x-t)^p\) has one point and a nonreduced structure sheaf.
On affine schemes the construction is computed by a tensor product and then glued. The underlying set of a scheme-theoretic fibre product need not be the fibre product of the underlying sets: \(\Spec\CC\times_{\Spec\RR}\Spec\CC=\Spec(\CC\otimes_\RR\CC)\cong\Spec(\CC\times\CC)\) has two points.
Fibres
The scheme-theoretic fibre of a morphism (Tag PR-SCHFIB)
For \(f: X \to Y\) and \(y \in Y\) with residue field \(\kappa(y) \da \OO_{Y,y}/\mfm_y\), the fibre is \[ X_y \da \fiberprod{X}{Y}{\Spec \kappa(y)} . \] Its underlying space is \(\inverseof{f}(y)\), but it carries a scheme structure over the field \(\kappa(y)\).
For \(\AA^1\slice k \to \AA^1\slice k\), \(t \mapsto t^2\), the fibre over a nonzero \(a\) is \(\Spec k[t]/(t^2 - a)\), which is two reduced points when \(a\) is a square in \(k\) and one point with residue field \(k(\sqrt a)\) otherwise. The fibre over \(0\) is \(\Spec k[t]/(t^2)\), one point of length \(2\). For \(\Spec \ZZ[i] \to \Spec \ZZ\) the fibre over \((p)\) is \(\Spec \FF_p[x]/(x^2+1)\): two points, one point, or a double point according to \(p \bmod 4\).
Fibres are a principal application of fibre products. The set-theoretic preimage loses the two things that make fibres behave: the multiplicity at a branch point, and the residue field extension at a point that splits only after base change. Both are visible in the examples, and in each the fibre has \(k\)-dimension \(2\) for every base point — this is the constancy that a naive count of preimages fails to see.
Flatness is the condition that controls this kind of fibrewise variation and preserves scheme-theoretic multiplicity under base change.
For \(\Spec k[x]\to\Spec k[t]\), \(t\mapsto x^2\), with \(\operatorname{char}k\neq2\), the fibre over \(t=0\) is the double point \(\Spec k[x]/(x^2)\). For \(\Spec\ZZ[i]\to\Spec\ZZ\), the fibre over \((3)\) is \(\Spec\FF_9\), one point whose residue field has degree \(2\) over \(\FF_3\).
Base change
Base change, and properties stable under it (Tag D-SCHBC)
For \(X \to S\) and a morphism \(T \to S\), the base change of \(X\) to \(T\) is \(X_T \da \fiberprod{X}{S}{T}\), with its projection \(X_T \to T\).
A property \(P\) of morphisms is stable under base change if \(X \to S\) having \(P\) implies that \(X_T \to T\) has \(P\) for every \(T \to S\). It is local on the base if \(f\colon X \to S\) has \(P\) whenever there is an open cover \(\ts{V_i}\) of \(S\) with each \(\inverseof{f}(V_i) \to V_i\) having \(P\).
Extension of the ground field is base change along \(\Spec L\to\Spec k\) for a field extension \(L/k\); a \(k\)-scheme \(X\) is geometrically connected, or geometrically integral, if \(X\times_kL\) is connected, or integral, for every field extension \(L/k\). Connectedness is not stable under this base change: \(\Spec \CC\) is connected, and \(\Spec\CC\times_{\RR}\Spec\CC=\Spec(\CC\tensor_\RR\CC)\cong\Spec\CC\sqcup\Spec\CC\) is not. The fibre of \(X\to Y\) over \(y\) is the base change along \(\Spec \kappa(y) \to Y\).
Closed immersion, separated, proper, finite, flat, smooth, and affine are stable under base change and local on the base. Surjectivity is stable under base change. Dominance is not: the open immersion \(D(x)\to\AA^1_k\) is dominant, and its base change along the origin \(\Spec k\to\AA^1_k\) is \(\emptyset\to\Spec k\).
A \(k\)-scheme \(X\) is geometrically integral when \(X_{\bar k}\) is integral; the fibre over \(y\) is the base change along \(\Spec\kappa(y)\to Y\); a scheme over the fraction field \(K\) of a ring \(R\) spreads out over \(R\) when it is the base change of an \(R\)-scheme along \(\Spec K\to\Spec R\). Integrality is not stable under base change: \(\Spec\CC\) is integral over \(\RR\), and its base change \(\Spec(\CC\otimes_\RR\CC)\) is not.
The functor of points
The functor of points, and what small test schemes detect (Tag PR-SCHFOP)
For \(X \in \Sch\slice S\) and \(T \in \Sch\slice S\), a \(T\)-valued point of \(X\) is a morphism \(T \to X\) over \(S\). The assignment \(T \mapsto X(T) \da \Hom_S(T, X)\) is the functor of points of \(X\), and \(X \mapsto X(-)\) is fully faithful into \(\Sets^{\opcat{\Sch\slice S}}\) by Yoneda.
Morphisms \(\Spec L \to X\), for \(L\) a field, correspond to pairs: a point \(x \in X\) and a field embedding \(\kappa(x) \injects L\).
Morphisms \(\Spec k[\eps]/\eps^2 \to X\) for \(X\) over \(k\) correspond to pairs: a \(k\)-rational point \(x\), and a tangent vector in \(\dualof{(\mfm_x/\mfm_x^2)}\).
The functor of points is what makes the fibre product’s universal property usable, and it is how you recover the naive picture: \(X(k)\) for \(k\) algebraically closed is the classical variety, and \(X(\ZZ)\) or \(X(\QQ)\) is what a number theorist means by the solutions.
These two computations explain the standard test schemes: a field detects a point together with a residue-field embedding, while the dual numbers detect a tangent vector. Replacing \(k[\eps]/\eps^2\) by \(k[t]/t^n\) detects an \(n\)-th order arc, which is where jet schemes come from.
The usable slogan is that a scheme is determined by what maps into it, so constructing a scheme can be replaced by writing down a functor and proving it representable — this is how Hilbert and Picard schemes are defined.
The functor \(h_X=\Hom(-,X)\) sends fibre products to fibre products: \(h_{X\times_SY}(T)=h_X(T)\times_{h_S(T)}h_Y(T)\). For a \(k\)-scheme \(X\), \(\Hom_k(\Spec K,X)\) is the set of \(K\)-points, and \(\Hom_k(\Spec k[\varepsilon]/(\varepsilon^2),X)\) is the set of pairs of a \(k\)-point \(x\) and a tangent vector in \((\mathfrak m_x/\mathfrak m_x^2)^\vee\).
Fibres of families
Generic, special and central fibres; fat points; scheme-theoretic intersections (Tag D-SCHFIBRES)
Let \(f \colon X \to S\) be a morphism. If \(S\) is irreducible with generic point \(\eta\), the generic fibre is \(X_\eta = X \times_S \Spec \kappa(\eta)\). If \(S = \Spec R\) for a discrete valuation ring \(R\) with fraction field \(K\) and residue field \(k\), the generic fibre is \(X_K\) and the special fibre is \(X_k = X \times_R \Spec k\), the fibre over the closed point. When \(S\) is a pointed curve or the spectrum of a local ring, the fibre over the distinguished point \(0\) is called the central fibre.
(Fat points)
An \(n\)-fold point over a field \(k\) is a scheme \(\Spec R\) with one point, where \(R\) is a local \(k\)-algebra with \(\dim_k R = n\).
(Scheme-theoretic intersection)
For closed subschemes \(X_1, X_2 \subseteq Y\), the scheme-theoretic intersection is the fibre product \(X_1 \times_Y X_2\), a closed subscheme of \(Y\). If \(Y = \Spec R\) and \(X_i = \Spec R/J_i\), it is \(\Spec (R/J_1 \otimes_R R/J_2) = \Spec R/(J_1 + J_2)\).
The family \(X = V(y^2 - x^3 - t) \subseteq \AA^2 \times \AA^1_t \to \AA^1_t\) over a field of characteristic \(0\) has smooth general fibres and the cuspidal cubic \(y^2 = x^3\) as central fibre. The line \(y = 0\) and the parabola \(y = x^2\) in \(\AA^2\) meet in \(\Spec k[x,y]/(y, y - x^2) = \Spec k[x]/(x^2)\), a \(2\)-fold point, which records the tangency that the set-theoretic intersection \(\{0\}\) forgets.