Functors of points
The functors of points of $\AA^n$, $\AA^n \setminus \{0\}$ and $\PP^n$ (Tag FE-SCHREPFUNCTORS)
Let \(R\) be a ring and consider functors on the category of \(R\)-schemes.
(Affine space)
The functor \(X \mapsto \Gamma(X, \OO_X)^{n}\), sending a morphism \(f \colon X' \to X\) to \((a_i) \mapsto (f^\sharp a_i)\), is represented by \(\AA^n_R = \Spec R[x_1, \ldots, x_n]\), with universal element \((x_1, \ldots, x_n)\).
Over \(R = \ZZ\) this is the functor \(X \mapsto \Gamma(X, \OO_X)^n\) on all schemes, represented by \(\Spec \ZZ[x_1, \ldots, x_n]\): \(n\) variables, not \(n+1\).
(Punctured affine space)
\(\AA^n_R \setminus \{0\}\) denotes the open complement of \(V(x_1, \ldots, x_n)\). It represents the functor sending \(X\) to the set of \((a_1, \ldots, a_n) \in \Gamma(X, \OO_X)^n\) that are unimodular: the \(a_i\) generate the unit ideal sheaf, that is, for every \(x \in X\) some \(a_i(x) \neq 0\) in \(\kappa(x)\).
The condition is pointwise: on \(X = \Spec k[t]\), the pair \((t, 0)\) is not the zero pair, but both entries vanish at \(t = 0\), so it is not a point of \(\AA^2_k \setminus \{0\}\). When \(X = \Spec A\) it says \((a_1, \ldots, a_n) = A\).
(Projective space)
\(\PP^n_R\) represents the functor sending an \(R\)-scheme \(X\) to the set of isomorphism classes of tuples \((\mathcal{L}, s_0, \ldots, s_n)\) with \(\mathcal{L}\) an invertible sheaf on \(X\) and \(s_0, \ldots, s_n \in \Gamma(X, \mathcal{L})\) generating \(\mathcal{L}\), where \((\mathcal{L}, s_i) \cong (\mathcal{L}', s'_i)\) if there is an isomorphism \(\mathcal{L} \to \mathcal{L}'\) carrying each \(s_i\) to \(s'_i\). The universal tuple is \((\OO(1), x_0, \ldots, x_n)\), and a tuple on \(X\) corresponds to the morphism \(f \colon X \to \PP^n_R\) with \(f^* \OO(1) \cong \mathcal{L}\) and \(f^* x_i = s_i\) [@Har10a, Theorem II.7.1]; the construction is the gluing in D-OPENSUBFUNCTOR.
If every invertible sheaf on \(X\) is trivial, for instance \(X = \Spec A\) with \(A\) local, a tuple is a unimodular \((a_0, \ldots, a_n) \in \Gamma(X, \OO_X)^{n+1}\) up to multiplication by a unit of \(\Gamma(X, \OO_X)\). So the map \(\AA^{n+1}_R \setminus \{0\} \to \PP^n_R\) sends \((a_0, \ldots, a_n)\) to \((\OO_X, a_0, \ldots, a_n)\), and over a field \(k\) its \(k\)-points are the classical description of \(\PP^n(k)\) as nonzero \((n+1)\)-tuples up to \(k^\times\).
The functor of \(\PP^n\) sends a scheme \(T\) to the set of isomorphism classes of pairs \((\mcl;s_0,\ldots,s_n)\) of an invertible sheaf \(\mcl\) on \(T\) and global sections generating it. It is covered by the open subfunctors on which \(s_i\) generates \(\mcl\), each represented by \(\AA^n\), and gluing these gives its representability.
Zariski sheaves, open subfunctors, and representability by gluing (Tag D-OPENSUBFUNCTOR)
For a scheme \(X\), write \(h_X = \Hom(-, X) \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) for its functor of points (PR-SCHFOP).
(Zariski sheaf)
A functor \(F \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) is a Zariski sheaf if for every scheme \(Y\) the presheaf \(U \mapsto F(U)\) on the open subsets \(U \subseteq Y\), with restriction maps \(F(U) \to F(U')\) induced by the inclusions \(U' \subseteq U\), is a sheaf of sets on \(Y\).
(Open subfunctor)
Let \(F, G \colon \Sch^{\mathrm{op}} \to \mathsf{Set}\) be functors. A natural transformation \(\iota \colon G \to F\) exhibits \(G\) as an open subfunctor of \(F\) if for every scheme \(X\) and every natural transformation \(h_X \to F\), the fibre product \(h_X \times_F G\) is represented by a scheme \(U\) and the induced morphism \(U \to X\) is an open immersion. A family of open subfunctors \(\{G_i \to F\}\) is an open cover of \(F\) if for every \(h_X \to F\) the open subschemes \(U_i \subseteq X\) representing \(h_X \times_F G_i\) cover \(X\).
(Representability by gluing)
Let \(F\) be a Zariski sheaf with an open cover by open subfunctors \(\{G_i \to F\}_{i \in I}\) such that each \(G_i\) is represented by a scheme \(X_i\). Then \(F\) is represented by a scheme \(X\) covered by open subschemes isomorphic to the \(X_i\).
The functor \(\PP^n\) of FE-SCHREPFUNCTORS is a Zariski sheaf because line bundles and their sections glue. For \(0 \leq i \leq n\) the subfunctor \(G_i\) of pairs \((\mathcal{L}, s_0, \ldots, s_n)\) in which \(s_i\) generates \(\mathcal{L}\) is open, its fibre product with \(h_X\) being represented by the open set \(X_{s_i} = \{x \in X : s_i(x) \neq 0\}\). Each \(G_i\) is represented by \(\AA^n_\ZZ\), via \((\mathcal{L}, s) \mapsto (s_j/s_i)_{j \neq i}\), and the \(G_i\) cover because generating sections have no common zero, so the theorem constructs \(\PP^n_\ZZ\) by gluing \(n+1\) copies of \(\AA^n_\ZZ\).
Grothendieck topologies
Sites, sheaves on a site, and topoi (Tag D-GROTHTOP)
Let \(\mathcal{C}\) be a category with fibre products. A Grothendieck topology on \(\mathcal{C}\) assigns to each object \(U\) a set \(\operatorname{Cov}(U)\) of families of morphisms \(\{U_i \to U\}_{i \in I}\), the coverings of \(U\), such that:
every isomorphism \(V \to U\), as a one-element family, is a covering of \(U\);
if \(\{U_i \to U\}_i\) is a covering and \(V \to U\) is any morphism, then \(\{U_i \times_U V \to V\}_i\) is a covering of \(V\);
if \(\{U_i \to U\}_i\) is a covering and \(\{U_{ij} \to U_i\}_j\) is a covering of \(U_i\) for each \(i\), then the composites \(\{U_{ij} \to U\}_{i,j}\) form a covering of \(U\).
A site is a category with fibre products together with a Grothendieck topology.
(Sheaf on a site)
A presheaf of sets on a site \(\mathcal{C}\) is a functor \(F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{Set}\). It is a sheaf if for every covering \(\{U_i \to U\}_i\) the diagram \[F(U) \to \prod_i F(U_i) \rightrightarrows \prod_{i, j} F(U_i \times_U U_j)\] is an equalizer, where the two parallel maps are induced by the projections \(U_i \times_U U_j \to U_i\) and \(U_i \times_U U_j \to U_j\). Sheaves of abelian groups, rings or modules are defined in the same way with values in those categories. The sheaves of sets form a full subcategory \(\Sh(\mathcal{C}) \subseteq \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathsf{Set})\).
(Topos)
A topos is a category equivalent to \(\Sh(\mathcal{C})\) for some site \(\mathcal{C}\).
For a topological space \(X\), let \(\operatorname{Op}(X)\) be the category of open subsets with inclusions, where \(V \times_U W = V \cap W\), and let the coverings of \(U\) be the families \(\{U_i \subseteq U\}\) with \(\bigcup_i U_i = U\). The sheaves on this site are the sheaves on \(X\) in the sense of D-RCCFY, and \(\Sh(X)\) is a topos. The étale and fppf topologies on a category of schemes are the sites of D-ETFPPF.
The étale and fppf topologies, and the small étale site (Tag D-ETFPPF)
Let \(S\) be a scheme and \(\Sch_{/S}\) the category of \(S\)-schemes.
(Étale and fppf topologies)
A family of \(S\)-morphisms \(\{U_i \to U\}_i\) in \(\Sch_{/S}\) is an
étale covering if each \(U_i \to U\) is étale and \(\bigsqcup_i U_i \to U\) is surjective;
fppf covering if each \(U_i \to U\) is flat and locally of finite presentation and \(\bigsqcup_i U_i \to U\) is surjective.
With these coverings \(\Sch_{/S}\) is the big étale site \((\Sch_{/S})_{\mathrm{et}}\), respectively the big fppf site \((\Sch_{/S})_{\mathrm{fppf}}\).
(Small étale site)
The small étale site \(S_{\mathrm{et}}\) of \(S\) is the full subcategory of \(\Sch_{/S}\) whose objects are the étale morphisms \(U \to S\), with the étale coverings as coverings.
Every Zariski open covering is an étale covering, and every étale covering is an fppf covering, because étale morphisms are flat and locally of finite presentation. So every fppf sheaf is an étale sheaf, and every étale sheaf restricts to a Zariski sheaf on each \(S\)-scheme.
(Representable functors are sheaves)
For every \(S\)-scheme \(X\), the functor \(h_X = \Hom_S(-, X)\) is a sheaf on \((\Sch_{/S})_{\mathrm{fppf}}\), hence on \((\Sch_{/S})_{\mathrm{et}}\). For every quasicoherent \(\OO_S\)-module \(\mathcal{M}\), the functor \((f \colon U \to S) \mapsto \Gamma(U, f^* \mathcal{M})\) is an fppf sheaf.
Let \(n \geq 1\) be invertible on \(S\). On \(S_{\mathrm{et}}\), the multiplicative group \(\GG_m \colon U \mapsto \Gamma(U, \OO_U)^\times\) and its subgroup \(\mu_n\) of \(n\)-th roots of unity fit into the Kummer sequence \[1 \to \mu_n \to \GG_m \xrightarrow{\, u \mapsto u^n \,} \GG_m \to 1,\] which is exact on \(S_{\mathrm{et}}\): every unit \(u\) acquires an \(n\)-th root \(z\) on the étale covering \(\Spec_U \OO_U[z]/(z^n - u) \to U\). The sequence is not exact for the Zariski topology. For \(S = U = \Spec \CC[t, t^{-1}]\) and \(n = 2\), the unit \(t\) is not a square on any nonempty Zariski open subset of \(U\).
Descent
The statement that \(h_X\) is an fppf sheaf, and that quasicoherent sheaves glue along fppf coverings, rests on exactness of the Amitsur complex for a faithfully flat ring map.
Faithfully flat descent and the Amitsur complex (Tag T-FFDESCENT)
Let \(\varphi \colon A \to B\) be a ring map, and write \(B^{\otimes n} = B \otimes_A \cdots \otimes_A B\) with \(n\) factors.
(Amitsur complex)
For an \(A\)-module \(M\), the Amitsur complex of \(M\) along \(\varphi\) is \[0 \to M \xrightarrow{d^0} M \otimes_A B \xrightarrow{d^1} M \otimes_A B^{\otimes 2} \xrightarrow{d^2} M \otimes_A B^{\otimes 3} \to \cdots,\] where \(d^0(m) = m \otimes 1\) and, for \(n \geq 1\), \(d^n = \sum_{i=0}^{n} (-1)^i e_i\) with \[e_i(m \otimes b_1 \otimes \cdots \otimes b_n) = m \otimes b_1 \otimes \cdots \otimes b_i \otimes 1 \otimes b_{i+1} \otimes \cdots \otimes b_n .\]
(Exactness of the Amitsur complex)
If \(\varphi\) is faithfully flat, the Amitsur complex of every \(A\)-module \(M\) is exact. In particular the sequence \(0 \to M \to M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B\), with the two maps \(m \otimes b \mapsto m \otimes b \otimes 1\) and \(m \otimes b \mapsto m \otimes 1 \otimes b\), is an equalizer.
(Descent data)
A descent datum for \(A \to B\) is a \(B\)-module \(N\) with an isomorphism of \(B \otimes_A B\)-modules \(\theta \colon B \otimes_A N \to N \otimes_A B\) satisfying the cocycle condition \(\theta_{13} = \theta_{23} \circ \theta_{12}\) as maps \(B \otimes_A B \otimes_A N \to N \otimes_A B \otimes_A B\), where \(\theta_{ij}\) is \(\theta\) acting on the factors in positions \(i\) and \(j\). For an \(A\)-module \(M\), the module \(M \otimes_A B\) has the canonical descent datum \(b \otimes (m \otimes b') \mapsto (m \otimes b) \otimes b'\).
(Faithfully flat descent)
If \(A \to B\) is faithfully flat, then \(M \mapsto (M \otimes_A B, \text{canonical } \theta)\) is an equivalence from \(A\)-modules to \(B\)-modules with descent data. A quasi-inverse sends \((N, \theta)\) to \(\{ n \in N : \theta(1 \otimes n) = n \otimes 1 \}\).
Replacing modules by quasicoherent sheaves and \(\Spec B \to \Spec A\) by a faithfully flat quasicompact morphism \(S' \to S\) gives descent of quasicoherent sheaves; applied to \(\Hom\) it shows that morphisms of schemes descend, which is the statement that \(h_X\) is an fppf sheaf in D-ETFPPF.
Torsors
Torsors under a sheaf of groups (Tag D-TORSOR)
Let \(\mathcal{C}\) be a site with a terminal object \(X\), and \(\mathcal{G}\) a sheaf of groups on \(\mathcal{C}\). A \(\mathcal{G}\)-torsor is a sheaf of sets \(\mathcal{P}\) on \(\mathcal{C}\) with a right action \(\mathcal{P} \times \mathcal{G} \to \mathcal{P}\) such that
there is a covering \(\{U_i \to X\}\) with \(\mathcal{P}(U_i) \neq \emptyset\) for every \(i\), and
the morphism \(\mathcal{P} \times \mathcal{G} \to \mathcal{P} \times \mathcal{P}\), \((p, g) \mapsto (p, pg)\), is an isomorphism of sheaves.
A morphism of \(\mathcal{G}\)-torsors is a \(\mathcal{G}\)-equivariant morphism of sheaves. The trivial torsor is \(\mathcal{G}\) acting on itself by right multiplication.
A \(\mathcal{G}\)-torsor \(\mathcal{P}\) is trivial exactly when \(\mathcal{P}(X) \neq \emptyset\), and every morphism of \(\mathcal{G}\)-torsors is an isomorphism. Isomorphism classes of \(\mathcal{G}\)-torsors trivialized on a covering \(\mathfrak{U} = \{U_i \to X\}\) are in bijection with the Čech cohomology set \(\check{H}^1(\mathfrak{U}, \mathcal{G})\), and all torsors with \(\check{H}^1(X, \mathcal{G}) = \varinjlim_{\mathfrak{U}} \check{H}^1(\mathfrak{U}, \mathcal{G})\); for \(\mathcal{G}\) abelian this is \(H^1(X, \mathcal{G})\).
On a scheme \(X\) with the Zariski topology, a \(\GG_m = \OO_X^\times\)-torsor is the sheaf of nowhere-vanishing local sections of an invertible sheaf \(\mathcal{L}\), so \(\Pic X \cong H^1(X, \OO_X^\times)\) (PR-DIVLB). Likewise \(\operatorname{GL}_n\)-torsors correspond to locally free sheaves of rank \(n\), through their sheaves of local frames.
Let \(k\) be a field and \(n\) invertible in \(k\). On \((\Spec k)_{\mathrm{et}}\), the Kummer sequence of D-ETFPPF and \(H^1((\Spec k)_{\mathrm{et}}, \GG_m) = 0\) (Hilbert’s Theorem 90) give \(H^1((\Spec k)_{\mathrm{et}}, \mu_n) \cong k^\times / (k^\times)^n\). The torsor corresponding to \(a \in k^\times\) is \(\Spec k[z]/(z^n - a)\) with \(\mu_n\) acting by \(z \mapsto \zeta z\).