Definition 6.1.1. An affine site is a strict metric site B with finite limits in which any object is separated and quasicompact. Definition 6.1.2. An algebraic
site is a strict metric site C with finite limits together
with a basis B (called an affine basis)
having the following properties (a), (b) and (c):
Example 6.1.3.
All the basic algebraic sites considered in (1.1)
are affine or algebraic sites:
Note that all these sites satisfy (6.1.2.d) (For Sch see [H, p.116]) Example 6.1.4. An algebraic variety over an algebraically closed field k is a separated and quasicompact prevariety over k. Denote by Var/k the category of algebraic varieties over k. Var/k is a standard site with finite limits. It is an algebraic site with AVar/k as an affine basis. For the remainder of this section we assume C is an algebraic site with an affine basis B. Any object of C which is isomorphic to an object of B is called an affine object. Enlarging B if necessary we may assume that B consists of all the affine objects. If X is any object of C, an effective open subset U of X is called affine if the open effective subobject U is affine. (6.1.2.b) and (6.1.2.c) implies that any separated morphism, monomorphism, regular monomorphism, etc., in B is also a separated morphism, monomorphism, regular morphism in C. We first study separated morphisms in an algebraic site C. Proposition 6.1.5. Suppose f: Y ® X is a bicontinuous morphism. Then for any x Î f(Y) there exists an affine open neighborhood U of x such that V = f^{1}(U) is affine. Proof. Replacing X by an affine neighborhood W of x and Y by f^{1}(Y), we may assume that X is affine. If x Î f(Y), there exists a unique y in Y such that f(y) = x. Suppose V is an open affine neighborhood of y in Y; then f(V) is an open neighborhood of x in f(Y)), thus there exists an open neighborhood U' of x such that U' Ç f(Y) = f(V); hence f^{1}(U') = V. Denote by f' the restriction of f to V. Then f': V ® X is a morphism of affine objects. Suppose U is an open affine neighborhood of x contained in U'. Then f^{1}(U) = f'^{1}(U) = V ×_{X} U is affine. Proposition 6.1.6. Suppose X is an object such that for any pair U, V of affine open subsets of X, U È V is separated, then X is separated. Proof. The collection of all U È V forms an open diagonal cover of X by (5.3.9.b), hence the assertion follows from (5.3.6 and 5.3.10). Proposition 6.1.7. A morphism f: X ® S is separated if and only if for any open affine subset V of S, f^{1}(V) is separated. Proof. Suppose f is separated. Then f^{1}(V) is separated over the affine object V which is separated. Thus f^{1}(V) is separated (5.3.6.b). Conversely, if f^{1}(V) is separated, then it is separated over S (5.3.6.a). Since the collection of all such f^{1}(V) forms a diagonal covering of X (5.3.9.a), thus X is separated over S (5.3.10). Corollary 6.1.8. A morphism f: X ® S is separated if and only if for any pair of open affine subsets U, V of X such that f(U) and f(V) are contained in an affine open subset of S, U È V is separated. Proof. By (6.1.7) f is separated if and only if for any affine open subset W of S, f^{1}(W) is separated, i.e., if for any pair U and V of open affine subsets of f^{1}(W), U È V is separated (6.1.6). Lemma 6.1.9. Suppose X is covered by two affine open subsets U and V. Then X is separated if and only if the regular monomorphism t: U Ç V ® U × V induced by the inclusion morphisms i: U Ç V ® U and j:U Ç V ® V is a closed special morphism. (Note that the condition implies that U Ç V is affine if (6.1.2.d) holds for C since U × V is affine.) Proof. First we note that t: U Ç V = U ×_{X} V ® U × V is a regular monomorphism (5.1.8). X is separated if and only if the image of the special morphism D: X ® X × X is closed (6.1.2.c). The object X × X is covered by three open subsets U × U, V × V and U × V. Since U, V are affine, they are separated (6.1.1). Thus D(U) = D(X) Ç U × U is closed in U × U, and D(V) = D(X) Ç V × V is closed in V × V. It follows that D(X) is closed in X × X if and only if t(U Ç V) = D(U Ç V) Ç U × V is closed. Proposition 6.1.10. Suppose f: X ® S is a morphism and S is separated. Suppose V is an affine open subset of S and U an affine open subset of X. There is a closed special morphism U Ç f^{1}(V) ® U × V (and U Ç f^{1}(V) is affine if (6.1.2.d) holds for C). Proof. Suppose (X × S, p, q) is the product. The subobject U Ç f^{1}(V) is the fibre product of the graph G_{f} and p^{1}(U) Ç q^{1}(V) over X × S. But p^{1}(U) Ç q^{1}(V) coincides with the fibre product U × V (4.2.7), thus it is an affine object. Since G_{f} ® X × S is a closed special morphism (thus universally closed (5.2.6)), its extension U Ç f^{1}(V) ® p^{1}(U) Ç q^{1}(V) is a closed special morphism. Next we study quasiseparated and quasicompact morphisms in an algebraic site C. Proposition 6.1.11. Suppose f: X ® Y is a morphism. Then f is quasicompact if and only if for any affine open cover {V_{i}} of Y, f^{1}(V_{i}) is quasicompact for each i. Proof. Suppose {V_{i}} is an affine open cover of Y such that each f^{1}(V_{i}) is quasicompact. Let {U_{ij}} be a finite affine open cover of f^{1}(V_{i}). For each open affine subset Z of V_{i}, f^{1}(Z) is then the union of the affine open subsets f^{1}(Z) Ç U_{ij} = Z ×_{Y} U_{ij}, hence quasicompact. The open effective subsets Z of Y such that f^{1}(Z) is quasicompact thus form a basis; hence f is quasicompact by (5.4.9.a). The other direction is obvious. Proposition 6.1.12.
Suppose X is an object. The following properties are equivalent:
Proof. The properties (b) and (c) are equivalent
by the definition of a quasicompact morphism; (c) implies (d) trivially.
Corollary 6.1.13. Any object X whose underlying space is locally noetherian is quasiseparated. If this is so then any morphism f: X ® Y is also quasiseparated. Proposition 6.1.14.
(a) Suppose f: X ®
Y is a quasicompact Smorphism. Then for any base extension g: S'
® S, f_{S'}:
X_{S'} ®
Y_{S'} is quasicompact. Thus any quasicompact morphism is
universally quasicompact.
Proof. Clearly (a) implies (b) and (c). To prove (a), suppose f is quasicompact. To prove that f_{S}' is quasiseparated, we may assume S = Y. Let (X ×_{S} S', p, q) be the fibre product. Consider an affine open set U of S' such that g(U) Í V for an affine set V of S. Such open sets U forms a basis for S', thus by (5.4.9.a) it suffices to prove that q^{1}(U) = f^{1}(V) ×_{V} U is quasicompact. Since f and V are quasicompact, f^{1}(V) is quasicompact, thus it is a finite union of affine open sets {W_{i}}, hence q^{1}(U) = f^{1}(V) ×_{V} U is a finite union of affine objects W_{i} ×_{V} U. Thus q^{1}(U) is quasicompact. This proves the assertion. Proposition 6.1.15. Suppose the composition g.f of two morphisms f: X ® Y, g: Y ® Z are quasiseparated, and f is quasicompact and surjective. Then g is quasiseparated. Proof. The morphism D_{g}f = jD_{f} (see the proof of (5.3.4.c)) is quasicompact by assumption. Proposition 6.1.16. Suppose f: X ® Y, g: Y' ® Y are two morphisms and g is quasicompact and surjective. If f_{Y'}: X_{Y'} ® Y' is quasicompact (resp. quasiseparated), then f is quasicompact (resp. quasiseparated). Proof. For simplicity we shall write X'
for X_{Y'} and f' for f_{Y'}.
Proposition 6.1.17. Suppose the composition gf of two morphisms f: X ® Y, g: Y ® Z is quasicompact. If g is quasiseparated, or the underlying space of X is locally noetherian, then f is quasicompact. Proof. (a) First consider the case that g
is quasiseparated. By (5.1.12) f
= G_{f}.p_{2}
where p_{2}: X ×_{Z} Y
® Y is the projection,
and p_{2} = (gf) ×_{Z} 1_{Y}.
Since by assumption gf is quasicompact, p_{2} is
so by (6.1.12.b). Since g is quasiseparated,
D_{g}: Y
® Y ×_{Z}
Y is quasicompact, using the diagram of (5.1.7)
(exchange X and Y and let S = W)) we see that
G_{f} is
quasicompact (6.1.12.a). Thus f is quasicompact
(5.4.5.a).
Proposition 6.1.18.
Assume (6.1.2.d) holds for C.
Proof. First we prove the assertion under the assumption
that Y is separated. Since X is quasicompact, X
is a finite union of open affine sets U_{i}. Since (6.1.2.d)
holds for C, we may apply (6.1.10) to see
that for any affine open set V of Y, f^{1}(V)
Ç U_{i}
is open affine, hence quasicompact, thus f^{1}(V)
is quasicompact. The assertion then follows from (6.1.11).
