Definition 4.3.1. A morphism f: Y ® X of ringed space (resp. local ringed spaces) is called an immersion if f is bicontinuous, and for any x Î Y, the induced map f_x^#: O_X,f(y) ® O_Y,y of stalks is surjective. 

Remark 4.3.2. A morphism f is an immersion if and only if f is bicontinuous and the underlying morphism f(f): f(Y) ® f(X) of ringed sets (resp. local ringed sets) is an immersion. 

Remark 4.3.3. (a) The composite of two immersions is an immersion. 
(b) If X is a local ringed space, then the canonical morphism X_red d X from the associated geometric space X_red of X to X is an immersion. 
(c) Suppose f: X ® Y is an immersion of local ringed spaces, then the induced morphism f_red: X_red ® Y_red of the associated geometric spaces is an immersion. 
(d) Any effective morphism in the category of ringed (resp. local ringed, resp. geometric) spaces is an immersion. This is obvious for ringed and local ringed spaces. Suppose U is a subset of a geometric space, then the effective morphism U_red ® X is the composition U_red ® U ® X where U ® X is effective in LSp. Since U_red ® U and U ® X are immersions, U_red ® X is an immersion by (a). 

Proposition 4.3.4. An immersion f: X ® Y of ringed spaces (resp. local ringed spaces, resp. geometric spaces) is a monomorphism in RSp (resp. LSp, resp. GSp). 

Proof. Suppose f: X ® Y is an immersion of ringed spaces. Suppose g, h: Z ® X are two morphisms of ringed spaces such that gf = hf. Since f is bicontinuous, the underlying map of g and h must be the same, i.e. g(z) = h(z) for any z Î Z. Therefore it suffices to prove that g_z^# = h_z^#: O_X,x ® O_Z,z for any z ® Z, where x = f(z) = g(z). Since f is an immersion, f_x^#: O_Y,f(x) ® O_X,x is surjective, hence g_z^#f_x^# = h_z^#f_x^# implies that g_z^# = h_z^# as desired. This proof also works for local ringed spaces. For geometric spaces we may replace the stalks by the residue fields of the stalks. 

Proposition 4.3.5. Suppose f: X ® S, g: Y ® S are morphisms of ringed spaces. Let (X ×_S Y, p, q) be the fibre product of X and Y over S in the site of ringed spaces. Suppose f is an immersion. Then  
(a) q: X ×_S Y ® Y is an immersion. 
(b) If X, Y are local ringed spaces over a local ringed spaces S, then the fibre product (X ×_S Y, p, q) of X and Y over S in the site of ringed spaces is the fibre product as local ringed spaces. 
(c) Any immersion in RSp (resp. LSp, resp. GSp) is universally bicontinuous. (See 5.2.) 

Proof. (a) is a consequence of (4.2.12.b and d). 
(b) also follows from (4.2.12.d). 
(c) comes from (a) and the fact that immersions are bicontinuous morphisms. 

Proposition 4.3.6. Suppose f: X d Y is a surjective immersion of local ringed spaces and Y is a geometric spaces. Then X is a geometric space and f is an isomorphism. 

Proof. Since f is a homeomorphism, we may identify X with Y. We only need to prove that for any x Î X, the induced map f_x^#: O_Y,x ® O_X,x is an isomorphism. Since f is an immersion, f_x^# is surjective, it remains to prove that f_x^# is injective. Suppose s is a regular section of an open neighborhood f(x) such that f_x^#(s) is zero at x. Shrinking U if necessary we may assume that f_x^#(s) = 0 over U. Thus for any y Î U, s_y is in the kernel of f_x^#, hence a non-unit. Since Y is a geometrically reduced local ringed space, this means that s = 0. Thus f_x^# is injective. 

Corollary 4.3.7. Suppose f: X ® Y is an open immersion of local ringed spaces and Y is a geometric space. Then X is a geometric space and f is an open effective morphism. 

Proof. Let U = f(X) and let i: U ® Y be the effective inclusion. Then U is a geometric space. The induced morphism X ® U is an isomorphism by (4.3.6). 

Proposition 4.3.8. A morphism f: X ® Y in the site of geometric spaces is an immersion if and only if it is effective.  

Proof. One direction has been noticed (4.3.3.d). Suppose f is an immersion. Write f(X) = U. The inclusion morphism U_red ® Y is effective above U in the category of geometric spaces. Since by assumption X is a geometric space, f factors through a unique surjective immersion X ® U_red, hence an isomorphism by (4.3.6). This proves that f is effective in the category of geometric spaces. 

Recall that a regular monomorphism in a category C is a monomorphism which is the kernel (i.e., equalizer) of some pair of morphisms (see (5.1) for the properties of regular monomorphisms). 

Proposition 4.3.9. Any regular monomorphism in RSp, LSp, GSp, RSet, LSet, GSet, RPot, LPot, ASch, GASch is an immersion (hence is universally bicontinuous). 

Proof. Suppose g: S ® X is a section of a morphism f: X ® S in any of these sites. Then we have fg = 1_X, which implies that g is bicontinuous, and for each s Î S, the composition (f_g(s)g_s)^# = g_s^#f_g(s)^#: O_S,s ® O_X,g(s) ® O_S,s is the identity map of O_S,s. Hence g_s^# is surjective for any s Î S. Thus any section g is an immersion. In particular, any diagonal morphism of a morphism is an immersion (5.1.3). Since any regular monomorphism is the result of making a base extension of a diagonal morphism (see (5.1.10c), and immersions are stable under base extensions, it follows that any regular monomorphism is an immersion. Since an immersion is universally bicontinuous, any regular monomorphism in these sites are universally bicontinuous. 

Combining (4.3.8) and (4.3.9) we obtain the following 

Corollary 4.3.10. Any regular monomorphism in GSp is effective. 

Proposition 4.3.11. Suppose f: X ® y is a monomorphism in LSp where y is a geometric point. Then X = {x} is a geometric point and f is an isomorphism.  

Proof. Suppose y = spot k for a field k. Since LSp is separable, f is injective. Thus X is a local ringed point x. We make the following observations: 
(a) Since f: x ® y is monomorphism, the canonical projections p, q: x ×_y x ® x are isomorphisms and p = q. Thus w = x ×_y x is a local ringed point.  
(b) If f: x ® y is not an isomorphism, then the canonical maps i, j: O_x ® O_x Ä_k O_x given by a ® a Ä 1 and a ® 1 Ä a respectively are different since for any a Î O_x - k the elements a Ä 1 and 1 Ä a are linearly independent over k.  
(c) Assume O_x has only one prime ideal. Then x = spot O_x, w = spot O_x Ä_k O_x, and p^# = i, q^# = j. Applying (a) and (b) we see that f is an isomorphism. 
(d) Suppose O_x has more than one prime ideals. Moduling a minimal prime ideal p if necessary we may assume that O_x is an integral domain (the composition spot O_x/p ® spot O_x ® spot k is a monomorphism since the morphism spot O_x ® spot k and the immersion spot O_x/p ® spot O_x are monomorphisms). If k is algebraically closed, then O_x Ä_k O_x is an integral domain, and because O_w is a localization of O_x Ä_k O_x, we have O_w Ê O_x Ä_k O_x. Since O_x is not a filed, f is not an isomorphism, then by (b) the maps i, j: O_x ® O_x Ä_k O_x are different. Composing with the inclusion map O_x Ä_k O_x ® O_w we obtain different maps p^#, q^#: O_x ® O_w, contradicting to (a). This proves that O_x can not have infinite prime ideals under the assumption that k is algebraically closed. We have thus proved the proposition for algebraically closed k. 
(e) It remains to consider the case that k is not algebraically closed. By (c) we may assume x is not a geometric point. Suppose y' is a geometric point over y whose field is an algebraic closure of k. Then x_y' = x ×_y y' ® y' is a monomorphism and the local ring of x_y' is faithful flat over O_x since y' ® y is so. This implies that x_y' is not a geometric point, which is impossible according to the above (c) and (d). This finishes the proof.  

Corollary 4.3.12. Suppose f: X ® Y is a monomorphism of local ringed spaces. For any y Î Y, let Spec k_y = spot k_y ® Y be the canonical immersion sending spot k_y to y, where k_y is the residue field of the local ring O_y of y. Then the projection X_y = X ×_Y Spec k_y ® Spec k_y is an isomorphism.  

Proof. Since f is a monomorphism, the projection X_y ® Spec k_y is a monomorphism, thus we may apply (4.3.11).