Modules over a Ringed Space

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1 Modules over a Ringed Space Daniel Murfet October 5, 2006 In these notes we collect some useful facts about sheaves of modules on a ringed space that are either left as exercises in [Har77] or omitted completely. Most of these observations can be found in [Gro60], which is of course always the best reference. Nonetheless it may be useful for the reader to have an english reference when these points arise in our other notes on algebraic geometry. These notes are definitely not intended as an introduction to the subject. Contents 1 Modules over Ringed Spaces Stalks Inverse and Direct Image Sheafification Extension By Zero Generators Cogenerators Open Subsets Tensor products Ideals Locally Free Sheaves Exponential Tensor Products Sheaf Hom Coinverse Image Modules over Graded Ringed Spaces Quasi-Structures Stalks Tensor Products Properties of Inverse and Direct Image 43 1 Modules over Ringed Spaces Definition 1. Let X be a topological space and C the small category of open sets of X. In our Algebra in a Category notes we defined the following concepts: presheaf of rings,sheaf of rings, presheaf of commutative rings, sheaf of commutative rings. A ringed space is a pair (X, O X ) consisting of a topological space X and a commutative sheaf of rings O X. Note that if say that S is a sheaf of rings, we do not require S to be commutative. If R is a presheaf of rings on X, then we have the complete grothendieck abelian category (P (C); R)Mod of presheaves of left R-modules (LC,Corollary 3), which we denote by Mod(R) or Mod(X) for a ringed space (X, O X ). If S is a sheaf of rings on X, the complete grothendieck abelian category (Sh J (C); S )Mod of sheaves of left S -modules (LC,Corollary 4) is denoted Mod(S ) or Mod(X) for a ringed space (X, O X ). 1

2 Throughout this section (X, O X ) denotes a ringed space, and all modules are O X -modules. For any module F you can form the conglomerate of subobjects SubF. Recall that a subobject is a monomorphism φ : G F in Mod(X), whereas a submodule is a subobject for which φ U is the inclusion of a subset for all open U. Every subobject is equivalent to a submodule, and two submodules are equivalent iff. they are the same module with the same embedding in F, so there is a bijection between SubF and the set of submodules of F. You can form the intersection and union of any set of subobjects. Lemma 1. Let φ : F G be a morphism of sheaves of modules. Then the subsheaf Imφ is defined by the following condition: given an open set U and s G (U) we have s (Imφ)(U) if and only if for every point x U there is an open set V with x V U and s V Im(φ V ). Lemma 2. Let G be a subpresheaf of modules of a sheaf of modules F. Then the induced morphism of sheaves ag F is a monomorphism whose image G is defined by the following condition: given an open set U and s F (U) we have s G (U) if and only if for every point x U there is an open set V with x V U and s V G(V ). Note that the submodule G of F above is the smallest submodule of F containing G. Let {G i } i I be a nonempty family of submodules of a sheaf of modules F. Then the union i G i is the image of the induced morphism i G i F. Lemma 3. Let {G i } i I be a nonempty set of subsheaves of a sheaf of modules F. Then the subsheaf i G i is defined by the following condition: given an open set U and s F (U) we have s ( i G i)(u) if and only if for every point x U there exists an open neighborhood V with x V U and elements a i1,..., a in with a ik G ik (V ) such that s V = a i1 + + a in. Proof. Taking the union of subsheaves of modules is the same as taking the union of subsheaves of abelian groups, so the result follows from (SGR,Lemma 11). Lemma 4. Let {G i } i I be a direct family of subsheaves of a sheaf of modules F. Then for an open set U and s F (U) we have s ( i G i)(u) if and only if for every point x U there exists an open neighborhood V with x V U and k I such that s V G k (V ). If U is a quasi-compact open subset of X then any section s ( i G i)(u) belongs to G i (U) for some i I. Proof. Since the family is direct, this follows easily from the explicit description of the union given above. Remark 1. Since Mod(X) is grothendieck abelian, for any family of sheaves of modules {F i } i I the canonical morphism i F i F i is a monomorphism. In particular for every open set U X we have an inclusion Γ(U, F i ) Γ(U, F i ) i i which at first glance seems puzzling: the sheaf coproduct is not necessarily the presheaf (i.e. pointwise) coproduct, but nonetheless it consists of sequences. The difference is that Γ(U, i F i) can contain some genuinely infinite sequences that do not belong to the pointwise coproduct. In the case where X is quasi-noetherian and U quasi-compact there is sufficient finiteness about to stop any sequences escaping i Γ(U, F i) and becoming infinite (COS,Proposition 23), so the presheaf and sheaf coproduct agree. Since Mod(X) is a grothendieck abelian category it follows from Mitchell III 1.2 that if F i are a direct family of submodules of a module F then the induced morphism lim F i F is in fact a monomorphism, isomorphic as a subobject to the categorical union i F i. In the case where X is a noetherian space we can give a nice characterisation of this submodule. Lemma 5. Let {α i : F i F } i I be a nonempty family of submodules of a O X -module F. Then the intersection is the following submodule ( ) F i (U) = F i (U) i i 2

3 If X is a noetherian topological space and the F i are a direct family of submodules, then the union is the following submodule ( ) F i (U) = F i (U) i i Proof. Same proof as (SGR,Lemma 12). Lemma 6. Let F 1,..., F n be submodules of a O X -module F. Then for x X we have as O X,x -submodules of F x. (F 1 F n ) x = F 1,x F n,x Lemma 7. Let ψ : F G be a morphism of O X -modules and suppose H is a submodule of G. Then the inverse image ψ 1 H is the submodule defined by Proof. See the proof of (SGR,Lemma 14). (ψ 1 H )(U) = ψ 1 U (H (U)) Lemma 8. Let F be a O X -module. The following conditions on two submodules G, H of F are equivalent: (i) G precedes H as a subobject in the category O X Mod; (ii) G (U) H (U) for all open U; (iii) G x H x as submodules of F x for all x X. Proof. It is clear that (i) (ii) and (ii) (iii). If G x H x for all x X and s G (U) then we can find an open neighborhood x U x U and h x H (U x ) such that s Ux = h x for all x. The fact that H is a sheaf implies that the amalgamation s must belong to H (U), as required. Definition 2. Let (X, O X ) be a ringed space, F a sheaf of O X -modules and {s i } i I a nonempty set of sections s i F (U i ) over open sets U i X. The submodule of F generated by the set {s i } is the intersection of all submodules of F containing all the s i. This is the smallest submodule containing the s i, in the sense that it precedes any other such submodule. We say that F is generated by the set {s i } i I if the submodule generated by the s i is all of F. In particular, F is generated by global sections if there is a nonempty set of global sections {s i } i I which generate F. This property is stable under isomorphisms of modules and ringed spaces, and under restriction to open subsets. Proposition 9. Let (X, O X ) be a ringed space and F a O X -module. Given sections s i F (U i ) let G x F x be the O X,x -submodule generated by the set {germ x s i x U i }. For U X let G (U) = {s F (U) germ x s G x for all x U} Then G is the submodule generated by the set {s i }. Proof. It is clear that G (U) is a O X (U)-submodule of F (U), that these modules are closed under the restriction of F, and the resulting presheaf of O X -modules is actually a sheaf. So G is in fact a submodule of F, and clearly s i G (U i ) for all i. If H is any other submodule of F with s i H (U i ) for all i, we claim that G precedes H as a subobject of F. Since it is clear from the definition that G x = G x H x for all x X this follows from the previous Lemma. Remark 2. It follows that if (X, O X ) is a ringed space, F a O X -module and {s i } i I global sections, the s i generate F if and only if germ x s i generate F x as a O X,x -module for every x X. 3

4 Lemma 10. Let (X, O X ) be a ringed space and ϕ : F G a morphism of O X -modules. The image of ϕ is the submodule of G generated by the elements {ϕ U (s) s F (U)}. Proof. Earlier we defined the submodule Imϕ of G by t (Imϕ)(U) iff. every x U has an open neighborhood x V U such that t V Im(ϕ V ), which is iff. germ x t Im(ϕ x ). For x X the O X,x -submodule G x G x corresponding to the collection {ϕ U (s) s F (U)} is precisely Im(ϕ x ), so the result is clear. Definition 3. Let (X, O X ) be a ringed space. A sheaf of O X -modules F is globally finitely presented if there are integers p, q > 0 and an exact sequence O p X Oq X F 0 We say that F is locally finitely presented if every point x X has an open neighborhood x U X such that F U is a globally finitely presented sheaf of O X U -modules. Clearly a finitely free sheaf is globally finitely presented, and a locally finitely free sheaf is locally finitely presented. Proposition 11. Let (X, O X ) be a ringed space. There is a nonempty set I of locally finitely presented sheaves of modules on X which is representative. That is, every locally finitely presented sheaf of modules is isomorphic to some element of I. Proof. The general idea is that up to isomorphism all the globally finitely presented sheaves can be obtained by taking p, q > 0 and the canonical cokernel of all morphisms O p X Oq X. To get all the locally finitely presented sheaves you get all the globally finitely presented sheaves over open covers and all the possible ways of glueing these sheaves. Let us now get into the specifics. Let C be the set of all nonempty open covers of X. For each open set U X we define the following union set F U = Hom OX U (O X p U, O X q U ) p,q>0 For every pair of open sets U, V X we define the following union set G U,V = Hom OX U V (Coker(f) U V, Coker(g) U V ) (f,g) F U F V A glueing datum D consists of the following data: (i) a nonempty open cover U of X (ii) for each open set U U an element f U of F U (iii) for each pair of open sets U, V U an isomorphism ϕ U,V : Coker(f U ) U V Coker(f V ) U V with the property that the sheaves Coker(f U ) and isomorphisms ϕ U,V satisfy the hypothesis of (GS,Proposition 1). Using the sets F U, G U,V it is clear how to formalise this in such a way that we have a set G of all glueing datums. We can glue any glueing datum D to produce a locally finitely presented sheaf of modules F D on X. This defines a set of locally finitely presented sheaves {F D } D G. Now let G be any locally finitely presented sheaf of modules on X and let U be a nonempty open cover with G U globally finitely presented for every U U. That is, we can find f U F U together with a canonical isomorphism ψ U : Coker(f U ) = G U for each U U. For each pair of open sets U, V U let ϕ U,V be ψ 1 V U V ψ U U V. This clearly defines a glueing datum D, and we have F D = G by construction, which completes the proof. Lemma 12. Let X be a topological space, and let Z be the sheafification of the constant presheaf of rings U Z. Then there is a canonical isomorphism Mod(Z) = Ab(X). Proof. Let P be the presheaf of rings defined by P ( ) = { } and P (U) = Z for nonempty open U. The restriction maps are the obvious ones, and we denote by Z the sheaf of rings associated to this presheaf. We claim that the canonical forgetful functor F : Mod(Z) Ab(X) is an isomorphism. Let F be a sheaf of abelian groups on X. Then F becomes a sheaf of P -modules in the obvious way, and therefore also a sheaf of Z-modules. This defines a functor G : Ab(X) Mod(Z). It is clear that F G = 1, and it is not difficult to check that GF = 1, which completes the proof. 4

5 1.1 Stalks Throughout this section let (X, O X ) be a fixed ringed space. All sheaves of modules will be over X. For any point x X we have the additive stalk functor ( ) x : Mod(X) O X,x Mod Given an O X,x -module M we define a sheaf of modules by { M x U Γ(U, Sky x (M)) = 0 otherwise where for x U the abelian group M = Γ(U, Sky x (M)) becomes an O X (U)-module via the ring morphism O X (U) O X,x. One defines Sky x on morphisms in the obvious way, so that we have an additive functor Sky x ( ) : O X,x Mod Mod(X) For a sheaf of modules F and O X,x -module M we have natural morphisms η : F Sky x (F x ), η U (m) = (U, m) ε : (Sky x M) x M, ε(u, m) = m In fact these are the unit and counit of an adjunction ( ) x Sky x. It is clear that ε is a natural equivalence, from which we deduce that Sky x is fully faithful (AC,Proposition 21). It is also distinct on objects, so Sky x gives a full embedding of O X,x Mod into Mod(X). Given a sheaf of modules F we define an abelian group F x = Hom OX (Sky x (O X,x ), F ) which becomes a O X,x -module with action (r s) V (t) = s V (rt) for open sets V containing x. A morphism of sheaves of modules ψ : F G determines a morphism of O X,x -modules ψ x defined by ψ x (s) = ψ s. This defines an additive functor ( ) x : Mod(X) O X,x Mod For an O X,x -module M, any element m M corresponds to a morphism of modules m : O X,x M and therefore to a morphism of sheaves Sky x (m) Sky x (M) x. Since Sky x ( ) is fully faithful, this is a natural isomorphism of O X,x -modules η : M Sky x (M) x η(m) = Sky x (m) In fact this is the unit of an adjunction Sky x ( ) ( ) x. To see this, suppose we are given a sheaf of modules G and morphism of O X,x -modules α : M G x. Then the following morphism of sheaves of modules φ : Sky x (M) G φ V (m) = α(m) V (1) is unique with the property that φ x η = α. This establishes the desired adjunction, so we have a triple of adjoints ( ) x Sky x ( ) ( ) x which in particular means that Sky x ( ) is exact and preserves all limits and colimits. Remark 3. You can t expect the stalk functor ( ) x to preserve products in general, because a sequence in i F i,x could contain germs over a strictly descending sequence of open sets. Then it makes sense that no open section of i F i (which is a sequence of sections over a fixed open set) can induce all of these germs at once. 5

6 Remark 4. It is clear that for any sheaf of modules F the morphism of O X,x -modules η x : F x Sky x (F x ) x is an isomorphism. Suppose now that x is a closed point and that Supp(F ) = {x}. Then the stalk of Sky x (F x ) is zero at every point y x, so it is clear that η is an isomorphism of sheaves of modules. So if x is a closed point and F a sheaf of modules, F is a skyscraper sheaf at x if and only if Supp(F ) {x}. Lemma 13. Let x X be a point and I an injective O X,x -module. Then Sky x (I) is an injective object in Mod(X). Proof. The functor Sky x ( ) has an exact left adjoint ( ) x (AC,Proposition 25). and therefore preserves injectives Lemma 14. Let x X be a point, F a sheaf of modules on X and M an O X,x -module. Then there is a canonical morphism of sheaves of modules natural in F and M λ : F OX Sky x (M) Sky x (F x OX,x M) a b (U, a) b with the property that λ x is an isomorphism. If x is a closed point then λ is an isomorphism. Proof. Tensoring the counit ε : (Sky x M) x M with F x and composing with the canonical isomorphism yields an isomorphism of O X,x -modules (F OX Sky x (M)) x F x OX,x M By the adjunction this corresponds to a morphism λ with the required properties. This result should be compared with Lemma 80. Lemma 15. Let x X be a point and M, N two O X,x -modules. There is a canonical isomorphism of sheaves of modules natural in M, N Sky x Hom OX,x (M, N) H om OX (Sky x M, Sky x N) α Sky x (α) U Lemma 16. Let x X be a point, M an O X,x -module and F a complex of sheaves of modules. There is a canonical isomorphism of sheaves of modules natural in F, M Sky x Hom OX,x (F x, M) H om OX (F, Sky x M) Proof. We compose the isomorphism of Lemma 15 with the unit of the adjunction between Sky x ( ) and ( ) x to obtain a canonical morphism of sheaves of modules Sky x Hom OX,x (F x, M) = H om OX (Sky x (F x ), Sky x M) H om OX (F, Sky x M) which is clearly natural in both variables, and is easily checked to be an isomorphism. This result should be compared with Corollary Inverse and Direct Image Lemma 17. Let f : (X, O X ) (Y, O Y ) be a morphism of ringed spaces and F a sheaf of modules on Y. Then for open U X, s f F (U) and x U there exists open V with x V U such that s V = s s n where each s j has the form with T f(v ), c F (T ) and b O X (V ). s j = [T, c] b Lemma 18. Let f : X Y be a morphism of ringed spaces and φ : F G a morphism of sheaves of modules on Y. For open U X, T f(u) and c F (T ), b O X (U) we have (f φ) U ([T, c] b) = [T, φ T (c)] b 6

7 Theorem 19. Let f : X Y be a morphism of ringed spaces. Then there is an adjunction f f. For a sheaf of modules F on X the unit is η : F f f F η U (s) = [U, s] 1 If φ : G f F is a morphism of sheaves of modules on Y, the adjoint partner is for open V f(u) and t G (V ). ψ : f G F ψ U ([V, t] r) = r φ V (t) U Remark 5. Let f : (X, O X ) (Y, O Y ) be a morphism of ringed spaces and G a sheaf of modules on Y. For x X the O X,x -module (f G ) x is also a (f 1 O Y ) x -module as f G is the tensor product of f 1 O Y -modules f 1 G f 1 O Y O X. The ring isomorphism (f 1 O Y ) x = OY,f(x) therefore defines a O Y,f(x) -module structure on (f G ) x. It is straightforward to check this agrees with the structure obtained from the canonical ring morphism O Y,f(x) O X,x. The canonical isomorphism of abelian groups (f 1 G ) x = Gf(x) maps the action of (f 1 O Y ) x to the action of O Y,f(x). Proposition 20. Let f : (X, O X ) (Y, O Y ) be a morphism of ringed spaces and G a sheaf of modules on Y. Then there is a canonical isomorphism of O X,x -modules natural in G τ : (f G ) x G f(x) OY,f(x) O X,x germ x ([V, s] b) germ f(x) s germ x b Proof. Identifying the rings O Y,f(x) and (f 1 O Y ) x using the canonical isomorphism, we have an isomorphism of O Y,f(x) -modules (f G ) x = (f 1 G f 1 O Y O X ) x = (f 1 G ) x (f 1 O Y ) x O X,x = (f 1 G ) x OY,f(x) O X,x = G f(x) OY,f(x) O X,x It is straightforward to check that this is an isomorphism of O X,x -modules natural in G. Lemma 21. Let f : (X, O X ) (Y, O Y ) and g : (Y, O Y ) (Z, O Z ) be morphisms of ringed spaces. Then as functors between the categories of sheaves of modules (gf) = g f. Also 1 = 1 and therefore if f is an isomorphism of ringed spaces, f : Mod(X) Mod(Y ) is an isomorphism of categories. Remark 6. Let (X, O X ) be a ringed space and U X be an open subset with inclusion i : U X. If F is a sheaf of O X modules then the isomorphism of sheaves of abelian groups F U i 1 F is actually an isomorphism of sheaves of O X U -modules. The module structure on i 1 F comes from the natural i 1 O X -module structure together with the isomorphism O X U i 1 O X. Proposition 22. Let f : (X, O X ) (Y, O Y ) and g : (Y, O Y ) (Z, O Z ) be morphisms of ringed spaces and let F be a sheaf of O Z -modules. There is a canonical isomorphism of sheaves of (gf) 1 O Z modules natural in F ψ : (gf) 1 F f 1 g 1 F Proof. The sheaf g 1 F is naturally a g 1 O Z -module, so f 1 g 1 F is naturally a f 1 g 1 O Z - module. We use the isomorphism (gf) 1 O Z f 1 g 1 O Z to make f 1 g 1 F into a (gf) 1 O Z - module. 7

8 We claim the isomorphism of sheaves of abelian groups (gf) 1 F f 1 g 1 F is actually an isomorphism of sheaves of (gf) 1 O Z -modules. First we set up our notation. Let Z, Z be the presheaves on X which sheafify to give (gf) 1 F and f 1 g 1 F respectively and let R, R be the presheaves of rings sheafifying to give (gf) 1 O Z and f 1 g 1 O Z respectively. Let φ : Z Z and ψ : Z Z be the canonical morphisms of presheaves. For x X and a Z x and b R x it is not difficult to check that φ x(b a) = ψ x(b) φ x(a) Let φ : (gf) 1 F f 1 g 1 F and ψ : (gf) 1 O Z f 1 g 1 O Z be the sheafifications of φ, ψ respectively. If U X is open and m (gf) 1 F (U) and r (gf) 1 O Z (U) then φ U (r m)(x) = φ x((r m)(x)) = φ x(r(x) m(x)) = ψ x(r(x)) φ x(m(x)) = (ψ U (r) φ U (m)) (x) This shows that φ : (gf) 1 F f 1 g 1 F is an isomorphism of (gf) 1 O Z -modules, as required. Proposition 23. Let f : X Y and g : Y Z be morphisms of ringed spaces. There is a canonical isomorphism of sheaves of modules on X natural in F α : (gf) F f g F which is unique making the following diagram commute F η g g F (1) η (gf) (gf) F (gf) α g η (gf) f g F Proof. We have the following diagram of adjoint pairs (gf) Mod(X) f Mod(Y ) f g Mod(Z) g (gf) Let η f, η g, η gf be the units for the adjunctions f f, g g and (gf) (gf). By composition of adjoints, the functor f g is left adjoint to g f = (gf) with unit ν = g η f g η g. We deduce a canonical natural equivalence α : (gf) f g which is unique making the necessary diagram commute for each sheaf of modules F on Z. Remark 7. With the notation of Proposition 23 one checks that for a sheaf of modules F on X the following diagram commutes f g g f F f ε f F f f F α 1 ε (gf) F (gf) (gf) F ε F 8

9 Remark 8. With the above notation, let F be a sheaf of modules on Z, U X and T (gf)(u) open sets and c F (T ), b O X (U) sections. Then we have α : (gf) F f g F α U ([T, c] b) = [g 1 T, [T, c] 1] b In checking this one can reduce to the case b = 1 and U = (gf) 1 T, which follows from commutativity of (1). Also for open U X, V f(u), T g(v ) and r O X (U), s O Y (V ) and c F (T ) we have α 1 U ([V, [T, c] s] r) = [T, c] (rf # V (s) U ) Checking this for s = 1 is trivial. We can reduce to this case in the following way [V, [T, c] s] r = ([V, s] [V, [T, c] 1]) r = [V, [T, c] 1] (rf # V (s) U ) since the tensor product is over f 1 O Y (U). 1.3 Sheafification Throughout this section (X, O X ) is a ringed space and a : Mod(X) Mod(X) the canonical sheafification functor. Lemma 24. Let U X be an open subset. Then the following diagram of functors commutes up to canonical natural equivalence Mod(X) a Mod(X) U Mod(U) a U Mod(U) Proof. The analogous result for presheaves of sets, groups or rings is easily checked (see p.7 of our Section 2.1 notes). Let F be a presheaf of modules on X, and let φ x : (F U ) x F x be the canonical isomorphism of modules (compatible with the ring isomorphism (O X U ) x = OX,x ) for x U. Then we define η : a(f U ) (af ) U η W (s)(x) = φ x (s(x)) This is a morphism of sheaves of O X -modules, which is clearly an isomorphism. Naturality in F is also easily checked. Lemma 25. Let f : (X, O X ) (Y, O Y ) be an isomorphism of ringed spaces. Then the following diagram of functors commutes up to canonical natural equivalence Mod(X) a Mod(X) f Mod(Y ) a f Mod(Y ) Proof. We checked this for abelian groups in Section 2.5 p.33. The natural equivalence given there is easily checked to give a natural equivalence of functors in our present situation. 9

11 a morphism of ψ : H G X by ψ V (s) = ϕ f(v ) (ṡ). This is the unique morphism of sheaves of O X -modules making the following diagram commute ϕ f! H G f! ψ ε f! (G X ) which shows that f! is left adjoint to ( ) X with counit ε. Remark 9. Let (X, O X ) be a ringed space and U X an open subset with inclusion j : U X. There is a canonical isomorphism of sheaves of modules η : H (j! H ) U for any sheaf of modules H on U. It gives rise to an isomorphism of O X,x -modules H x (j! H ) x for any x U. Clearly (j! H ) x = 0 for any x / U. Corollary 28. Let f : (X, O X ) (Y, O Y ) be an open immersion of ringed spaces. Then the functor ( ) X : Mod(Y ) Mod(X) preserves injectives. Proof. This follows from the fact that ( ) X has an exact left adjoint (AC,Proposition 25). There is an intuitive description of the sections of the extension by zero as those sections of the original sheaf which fade out in an appropriate way, and can therefore be extended by zero to a larger open set. Lemma 29. Let (X, O X ) be a ringed space and U an open subset with inclusion j : U X. For any sheaf of modules F on U there is a canonical monomorphism of sheaves of modules natural in F ψ : j! F j F whose image is the submodule G defined by Γ(V, G ) = {s Γ(V U, F ) for every x V \ U there is an open set x W V such that s W U = 0} Proof. Let F E be the presheaf of modules on X defined by Γ(V, F E ) = Γ(V, F ) for V U and zero otherwise. This is a subpresheaf of j F and the induced morphism of sheaves ψ : j! F j F has an image G defined by the condition of Lemma 2. Examining this condition one checks that Γ(V, G ) Γ(V, j F ) consists of the sections satisfying the given condition. Remark 10. With the notation of Lemma 29 let G be a sheaf of modules on X and set F = G U. Let V X be open and s Γ(V, j! (G U )) a section identified with a section of Γ(V U, G ) with the fading out property. For every x V \ U choose an open set x W x V with s Wx U = 0. Then these sets together with V U cover V, and s together with all the zero sections amalgamates to give a section t Γ(V, G ). This is the extension of s by zero. Of course the counit ε : j! (G U ) G already knows all about this: it is precisely the map which extends every section by zero. In other words, ε V (s) = t. This is the observation that the following diagram commutes ε j! (G U ) G ψ j (G U ) since ε x is zero for x / U, so the image under ε V of any section must have zero germs at every point of x V \ U. Definition 8. Let (X, O X ) be a ringed space, U X an open subset and G a sheaf of modules on X. Let V X be open and suppose s Γ(V U, G ) is a section with the property that for every x V \ U there is an open set x W V such that s W U = 0. Then there is a unique section s V Γ(V, G ) with the property that (s V ) V U = s and germ x (s V ) = 0 for all x V \ U. We call this the extension by zero of s. Clearly ε V (s) = s V. 11

12 1.5 Generators Throughout this section (X, O X ) is a fixed ringed space. For any open subset U X with inclusion i : U X we have the functor i! ( ) : Mod(U) Mod(X) of Definition 7. We denote by O U the sheaf of modules i! (O X U ). For open sets U V there is a canonical monomorphism of sheaves of modules ρ U,V : O U O V. Clearly ρ U,U = 1 and ρ V,T ρ U,V = ρ U,T. We have O = 0 and the module OU for U = X is isomorphic to the structure sheaf O X. The adjunction i! ( ) U means that we can compute explicitly morphism sets of the form Hom(O U, F ). Proposition 30. For any open subset U X and O X -module F there is a canonical isomorphism of abelian groups Hom OX (O U, F ) Γ(U, F ) which is natural in U and F in the sense that for an inclusion U V and a morphism F G the following two diagrams commute Hom(O V, F ) Γ(V, F ) Hom(O U, F ) Γ(U, F ) Hom(O U, F ) Γ(U, F ) Hom(O U, G ) Γ(U, G ) Proof. Given U X let F U be the presheaf sheafifying to give O U. For s F (U) we define F U F by sending 1 F U (V ) to s V F (V ). Conversely a morphism φ : F U F is mapped to φ U (1). This gives an isomorphism of abelian groups Hom(F U, F ) = F (U) which is natural in U and F. Using the adjunction between inclusion and sheafification, it is straightforward to finish the proof. We note that the isomorphism Hom(O U, F ) F (U) sends φ to φ U (1) where 1 is the section mapping every x U to 1 O X,x. Remark 11. With the notation of Proposition 30 let O U F correspond to a section m Γ(U, F ). Then it is easy to check that the composite O X,x = OU,x F x sends the identity to germ x m. Corollary 31. Let {U i } i I be a basis of open sets for X. Then the sheaves {O Ui i I} form a generating family for Mod(X). Proof. The previous result shows that if φ : F G is a morphism of O X -modules and s F (U) corresponds to α : O U F, then the composite φα corresponds to φ U (s) G (U). So it is clear that if φ is nonzero, there is i I and a morphism O Ui F such that O U F G is nonzero. Lemma 32. For any open set U X we have Supp(O U ) U with equality if (X, O X ) is a locally ringed space. Proof. This is immediate from Remark 9. Lemma 33. Let f : (X, O X ) (Y, O Y ) be a morphism of ringed spaces and V Y an open subset. There is a canonical isomorphism of sheaves of modules natural in V α : f O V O f 1 V Proof. The identity 1 Γ(V, f O f 1 V ) corresponds to a morphism O V f O f 1 V and by adjointness to a morphism f O V O f 1 V. To check this is an isomorphism it suffices to check on stalks, which follows immediately from Proposition 20. By naturality in V we mean that for open sets W V of Y the following diagram commutes f O W O f 1 W f O V O f 1 V which is easily checked. 12

13 Next we introduce the analogue of the sheaf O U for closed subsets. Let Z X be a closed subset with open complement U and let j : Z X be the inclusion. Then O Z = j (j 1 O X ) is a sheaf of rings on X and there is a canonical morphism of sheaves of rings ω : O X O Z which makes the latter sheaf into a sheaf of O X -modules. From (SGR,Lemma 26) we deduce a canonical short exact sequence of O X -modules 0 O U O X O Z 0 (2) and in particular whenever Z Q are closed subsets of X we have an epimorphism of O X - modules O Q O Z compatible with the projections from O X. By passing to stalks one checks that O Z,x = 0 for z / Z, and ω x : O X,x O Z,x is an isomorphism for x Z. Proposition 34. For any sheaf of O X -modules F there is a canonical isomorphism of abelian groups Hom OX (O Z, F ) = Γ Z (X, F ) natural in F and Z. Proof. Recall that Γ Z (X, F ) denotes the submodule consisting of sections s Γ(X, F ) satisfying s U = 0, where U = X \ Z. Given a morphism of O X -modules O Z F we can compose with the projection O X O Z to obtain a morphism O X F which corresponds to an element s Γ(X, F ). We claim that this is the desired isomorphism. To see this, apply Hom OX (, F ) to the exact sequence (2) to obtain an exact sequence 0 Hom OX (O Z, F ) Γ(X, F ) Γ(U, F ) as required. Naturality in F is obvious. By naturality in Z we just mean that given closed sets Z Q and morphism O Z F corresponding to s Γ Z (X, F ) Γ Q (X, F ), the composite O Q O Z F also corresponds to s. Lemma 35. Let f : (X, O X ) (Y, O Y ) be a morphism of ringed spaces and Z Y a closed subset. There is a canonical isomorphism of sheaves of modules natural in Z β : f O Z O f 1 Z Proof. The global section 1 Γ(X, O f 1 Z) is also a global section of f O f 1 Z and it corresponds by the isomorphism of Proposition 34 to a morphism of sheaves of modules O Z f O f 1 Z, and therefore by the adjunction to a morphism β : f O Z O f 1 Z. One checks this is an isomorphism by passing to stalks. By naturality in Z we mean that for closed sets Z Q of Y the following diagram commutes f O Q O f 1 Q which is easily checked. f O Z O f 1 Z 1.6 Cogenerators Throughout this section (X, O X ) is a fixed ringed space. For each x X let Λ x be an injective cogenerator of O X,x Mod and let λ x = Sky x (Λ x ) be the skyscraper sheaf. By adjointness of ( ) x and Sky x ( ) for any sheaf of modules F we have a canonical isomorphism of abelian groups natural in both variables Hom OX (F, λ x ) Hom OX,x (F x, Λ x ) By Lemma 13 the sheaves λ x are injective, and we claim they form a family of injective cogenerators for Mod(X). In particular F is zero if and only if Hom OX (F, λ x ) = 0 for every x X. Lemma 36. The sheaves {λ x x X} form a cogenerating family for Mod(X). Proof. Suppose we are given a nonzero morphism of sheaves of modules φ : F G. Then φ x is nonzero for some x X and therefore admits a morphism a : G x Λ x with aφ x 0. By adjointness this corresponds to a morphism α : G λ x with αφ 0, completing the proof. 13

14 1.7 Open Subsets Proposition 37. Let (X, O X ) be a ringed space and U X an open subset with inclusion i : (U, O X U ) (X, O X ). Then U = i and so we have a diagram of adjoints i Mod(U) Mod(X) i U = i U The functor U is exact and preserves all limits and colimits. The functor i is fully faithful and injective on objects. Proof. We know from our Section 2.5 notes that i is left adjoint to i and i = U, so it follows that U is also left adjoint to i. This shows that U preserves epimorphisms and all colimits. It is not hard to check that for a morphism ψ : F G of sheaves of modules on X, (Kerψ) V = Kerψ V for any open set V, so U also preserves kernels and is therefore exact. To prove U preserves all limits, we need only show it preserves products. But this is trivial since products in Mod(X) are taken pointwise. The claims about i are easily checked. Note that the unit of the adjunction U i is the map F i (F U ) given by restriction F (V ) F (U V ) and the counit (i G ) U G is the identity. It follows from Proposition 37 that i gives an isomorphism of Mod(U) with a full abelian subcategory of Mod(X), so we can identify Mod(U) with this subcategory. It also follows from Proposition 37 that Mod(U) is a Giraud subcategory of Mod(X). The reflection of a sheaf of modules F is the sheaf i (F U ). Lemma 38. Let (X, O X ) be a ringed space and {U i } i I a nonempty open cover. Suppose we have a sequence of sheaves of modules on X F F F (3) This sequence is exact if and only if F Ui F Ui F Ui is an exact sequence of sheaves of modules on U i for every i I. Proof. The condition is clearly necessary, so assume the sequences on the U i are all exact. We know the sequence (3) is exact iff. it is exact as a sequence of sheaves of abelian groups, which by (H, Ex. 1.2) is iff. the sequence of abelian groups F x F x F x is exact for every x X. But if x U i we have a commutative diagram of abelian groups i F x F x F x (F Ui ) x (F Ui ) x (F Ui ) x By assumption the bottom row is exact, hence so is the top row, which completes the proof. Proposition 39. Let (X, O X ) be a ringed space and {U α } α Λ a nonempty open cover. Let D be a diagram of sheaves of modules on X. Then a cone {H, ρ i : H D i } i D is a limit if and only if {H Uα, ρ i Uα : H Uα D i Uα } i D is a limit for the restricted diagram of sheaves of modules on U α for every α Λ. Proof. The categories Mod(X), Mod(U α ) are all complete and cocomplete, so let µ i : Y D i be a limit for D in Mod(X). There is an induced morphism η : H Y with µ i η = ρ i, and the morphisms ρ i are a limit iff. η is an isomorphism. But this is iff. η Uα is an isomorphism for all α, which is iff. the morphisms ρ i Uα are a limit for all α. This completes the proof. Proposition 40. Let (X, O X ) be a ringed space and {U α } α Λ a nonempty open cover. Let D be a diagram of sheaves of modules on X. Then a cocone {H, ρ i : D i H} i D is a colimit if and only if {H Uα, ρ i Uα : D i Uα H Uα } i D is a colimit for the restricted diagram of sheaves of modules on U α for every α Λ. 14

15 1.8 Tensor products Definition 9. Let (X, O X ) be a ringed space and F 1,..., F n sheaves of O X -modules for n 1, and G a sheaf of abelian groups. A morphism of sheaves of sets f : F 1 F n G is multilinear if for every open set U X the function f U : F 1 (U) F n (U) G (U) is multilinear, in the sense of (TES,Definition 2). If G is a sheaf of O X -modules and each f U is a multilinear form then we say that f is a multilinear form. The canonical morphism of sheaves of sets γ : F 1 F n F 1 F n (a 1,..., a n ) a 1 a n is clearly a multilinear form. If n = 1 then a multilinear map is just a morphism of sheaves of abelian groups F 1 G, and a multilinear form is a morphism of sheaves of modules. Proposition 41. Let (X, O X ) be a ringed space and F 1,..., F n, G sheaves of O X -modules for n 1. If f : F 1 F n G is a multilinear form then there is a unique morphism of sheaves of modules θ : F 1 F n G making the following diagram commute γ F 1 F n F 1 F n f θ G Proof. For n = 2 this follows from the definition of the tensor product. For n > 2 fix an open set U X and a F 1 (U). Define a multilinear form f : F 2 U F n U G U f V (a 2,..., a n ) = f V (a V, a 2,..., a n ) By the inductive hypothesis this induces a morphism of sheaves of modules θ a F n ) U G U. Define a multilinear form : (F 2 f : F 1 (F 2 F n ) G f U (a, t) = (θ a) U (t) The induced morphism of sheaves of modules θ : F 1 F n G is the one we require. Remark 12. Let (X, O X ) be a ringed space and F 1,..., F n sheaves of O X for n 1. Define a presheaf of O X -modules by P (U) = F 1 (U) F n (U) using the multiple tensor product of (TES,Definition 3). This sheafifies to give a sheaf Q of O X -modules together with a multilinear form F 1 F n Q, which one checks has the universal property described in Proposition 41. We deduce a canonical isomorphism of sheaves of modules Q F 1 (F 2 ( (F n 1 F n ) ) (4) defined by a 1 a n a 1 (a 2 ). If we define another presheaf of O X -modules by P (U) = F 1 (U) (F 2 (U) ( (F n 1 (U) F n (U)) ) using the usual tensor product repeatedly, then there is by (TES,Lemma 2) a canonical isomorphism of presheaves P = P and therefore a canonical isomorphism of the sheafification of P with the sheaves of (4). These isomorphisms are all natural in every variable. 15

16 Definition 10. Let φ : O O be a morphism of sheaves of commutative rings on a topological space X. Restriction of scalars defines an additive functor F : Mod(O ) Mod(O). If F is a sheaf of O-modules then define the following presheaf of O -modules P (U) = F (U) O(U) O (U) Let F O O denote the sheaf of O -modules associated to P. If φ : F F is a morphism of sheaves of O-modules then define a morphism of presheaves of O -modules φ : P P φ U : F (U) O(U) O (U) F (U) O(U) O (U) φ U = φ U 1 Let φ O O : F O O F O O be the morphism of sheaves of O -modules associated to φ. This defines an additive functor O O : Mod(O) Mod(O ). Proposition 42. Let φ : O O be a morphism of sheaves of commutative rings on a topological space X. Then we have a diagram of adjoints Mod(O) O O Mod(O ) O O F F Proof. Let F be a sheaf of O-modules and define a morphism of sheaves of O-modules natural in F η : F F O O η U (s) = s 1 So to complete the proof we need only show that the pair (F O O, η) is a reflection of F along F. If α : F G is another morphism of sheaves of O-modules, where G is a sheaf of O -modules, then for each open U define a O X (U)-bilinear map ε : F (U) O (U) G (U) (m, r) r α U (m) This induces a morphism of sheaves of O -modules τ : F O O G which is unique such that τη = α, as required. 1.9 Ideals Definition 11. Let (X, O X ) be a ringed space. A sheaf of ideals on X is a submodule J of O X. If J is a sheaf of ideals on X and F is a sheaf of modules on X, then J F denotes the submodule of F given by the image of the following morphism J F O X F = F It follows from the next result and Lemma 2 that J F is the submodule of F given by sheafifying the presheaf U J (U)F (U). So J F is the smallest submodule of F containing J (U)F (U) for every open U. In particular if J 1, J 2 are two sheaves of ideals, then we have the product sheaf of ideals J 1 J 2. If J is a sheaf of ideals then J n denotes the n-fold product for n 1. It is clear that if U X is open then (J F ) U = J U F U. Fix a sheaf of ideals J. If φ : F G is a morphism of sheaves of modules then there are induced morphisms of sheaves of modules J φ : J F J G and F /J F G /J G unique 16

17 making the following diagram commute with exact rows 0 J F F F /J F 0 J φ φ 0 J G G G /J G 0 In this way we define additive functors J : Mod(X) Mod(X) and ( )/J ( ) : Mod(X) Mod(X). Note that if J, K, L are sheaves of ideals then J K L if and only if for every open U X we have J (U)K (U) L (U). This follows from the fact that J K is the sheafification of U J (U)K (U) and Lemma 2. Lemma 43. Let J be a sheaf of ideals on X, F a sheaf of modules. For open U X and s F (U) we have s (J F )(U) if and only if for every x U there is an open neighborhood x V U such that s V = s s n where each s i is of the form for some r J (V ) and m F (V ). Proof. This follows immediately from Lemma 1. s i = r m Lemma 44. Let J be a sheaf of ideals on X and let n 1. For open U X and s F (U) we have s J n (U) if and only if for every x U there is an open neighborhood x V U such that s V belongs to the ideal J (V ) n of O X (V ). Lemma 45. Let F be a sheaf of modules on X. Then we have the following properties of ideal product (i) If J, K are sheaves of ideals then J K = K J J K. (ii) If J 1, J 2 are sheaves of ideals then J 1 (J 2 F ) = (J 1 J 2 )F. In particular the product of sheaves of ideals is associative. (iii) If J is a sheaf of ideals then (J F ) x = J x F x. (iv) O X F = F. (v) If J, K, L are sheaves of ideals with J K then J L K L. In particular if J K then J n K n for all n 1. Proof. (i) and (ii) follow from Lemma 43. For (iii) use the fact that the stalk functor Mod(X) O X,x Mod is exact to see that (J F ) x is the image of the following morphism which is clearly J x F x. (iv) and (v) are trivial. J x F x O X,x F x = Fx Definition 12. Let f : X Y be a morphism of ringed spaces and let J be a sheaf of ideals on Y. The morphism of sheaves of modules J O Y f O X induces a morphism of sheaves of modules f J O X. Let f 1 J O X denote the sheaf of ideals on X given by the image of this morphism. We use the notation J O X if no confusion seems likely to result, and call J O X the inverse image ideal sheaf. Lemma 46. Let f : X Y be a morphism of ringed spaces, J a sheaf of ideals on Y, and let U X be open. If s O X (U) then s (J O X )(U) if and only if for every point x U there are open neighborhoods x V U and f(v ) W such that s V = s s n where each s i is of the form s i = r f # W (t) V for some r O X (V ) and t J (W ). 17

18 Proof. Let ψ : f J O X be the morphism of sheaves of modules in the Definition. For open V X, W f(v ) and r O X (V ), t J (W ) we have ψ V ([W, t] r) = r f # W (t) V by Theorem 19. Suppose s (J O X )(U) and let x U be given. Then there is an open neighborhood x V U such that s V = ψ V (m) for some m (f J )(V ) (by (SGR,Lemma 9)). By making V sufficiently small we can assume m is of the form m = m m n with each m i of the form [W, t] r for W f(v ), t J (W ), r O X (V ). We can assume W is constant for all i, and therefore s V = ψ V (m) has the desired form. The converse is obvious. Corollary 47. Let f : X Y be a morphism of ringed spaces, J a sheaf of ideals on Y. Then J O X is the smallest submodule of O X containing all the sections f # W (t) for open W Y and t J (W ). Proof. This follows immediately from Lemma 46. Lemma 48. Let f : X Y be a morphism of ringed spaces, V Y open and set U = f 1 V. If J is a sheaf of ideals on Y then (J O X ) U = J V O X U as submodules of O X U. Proof. To be a little more precise, let g : U V be induced by f. We claim that (f 1 J O X ) U = g 1 (J V ) O X U This follows immediately from the definition and Proposition 112. Lemma 49. If h : X Y is an isomorphism of ringed spaces with inverse f, then there is a bijection between the sets of submodules of O Y and O X Sub(O Y ) Sub(O X ) J J O X For open U X and s O X (U) we have s (J O X )(U) if and only if f # U (s) J (h(u)). In particular we have (J O X ) O Y = J. Proof. Let J be a sheaf of ideals on Y. Then J O X is the image of the morphism h J O X, which is the top row in the following commutative diagram h J h ε O Y O X (f # ) 1 f J f O Y Where ε is the adjoint partner of h # : O Y h O X. To check commutativity, use Proposition 107. Therefore J O X is the image of f J f O Y = OX, which shows that s (J O X )(U) iff. f # U (s) J (h(u)). Then it is clear that (J O X) O Y = J and therefore we have a bijection, whose inverse is the map Sub(O X ) Sub(O Y ) induced by the isomorphism f : Y X. Lemma 50. Let f : X Y and g : Y Z be morphisms of ringed spaces and J a sheaf of ideals on Z. Then J O X = (J O Y ) O X. Proof. One inclusion is obvious from Corollary 47, and the other follows easily from Lemma 46. Lemma 51. Suppose we have a commutative diagram of ringed spaces X h X f Y g 18

19 If J is a sheaf of ideals on Y then the canonical bijection between submodules of O X and O X identifies J O X and J O X. In particular h (J O X ) = J O X. Now suppose we have a commutative diagram of ringed spaces Y g X h f Y If J is an ideal sheaf on Y which corresponds to the ideal sheaf K on Y then J O X = K O X. Proof. Immediate from Lemma 50. Proposition 52. Let f : X Y be an isomorphism of ringed spaces, J a sheaf of ideals on X and F a sheaf of modules on X. Then f (J F ) = (J O Y )f F J n O Y = (J O Y ) n Proof. The first claim follows immediately from Lemma 49 and Lemma 43, while the second uses Lemma 44. Definition 13. Let F, G be submodules of a sheaf of O X -modules H. Then we define a sheaf of ideals (F : G ) on X by (F : G )(U) = {r O X (U) r V G (V ) F (V ) for all open V U} It is clear that (F : G ) = O X if and only if G F, and (F : G )G F. If U X is an open subset then (F : G ) U = (F U : G U ). Lemma 53. Let f : X Y be an isomorphism of ringed spaces. If F, G are O X -submodules of a sheaf of O X -modules H then (F : G ) O Y = (f F : f G ). Lemma 54. Let (X, O X ) be a ringed space and J a sheaf of ideals on X. canonical isomorphism of sheaves of modules on X natural in F Then there is a ρ : F OX O X /J F /J F a (b + Γ(U, J )) ba + Γ(U, J F ) Proof. By definition J F is the image of the morphism F J F O X = F. By tensoring F with the exact sequence 0 J O X O X /J 0 we obtain a commutative diagram with exact rows F J F O X F O X /J 0 J F F F /J F 0 where ρ is the unique morphism making the right hand square commute. isomorphism, and naturality in F is easily checked. This is clearly an 1.10 Locally Free Sheaves Definition 14. Let (X, O X ) be a ringed space. A sheaf of O X -modules F is free if it is a coproduct of copies of O X. A sheaf of modules F is locally free if every point x X has an open neighborhood U with F U a free O X U -module. We say that F is invertible if every point x X has an open neighborhood U with F U = OX U. We say that F is locally finitely free if every point x X has an open neighborhood U with F U a coproduct of a finite set of copies of O X. 19

20 Remark 13. It is treacherous to talk about the rank of a free module over the zero ring, and it is similarly treacherous for free sheaves of modules. We say a ringed space (X, O X ) is nontrivial if X is nonempty and O X,x 0 for every x X. A nonempty locally ringed space is nontrivial. Definition 15. Let (X, O X ) be a nontrivial ringed space. If F is a free sheaf of O X -modules then for x X the O X,x -module F x is free and the ranks rank OX,x F x are all equal to the same element n {0, 1, 2,..., }, called the rank of F and written rank OX F. Clearly if F can be written as a coproduct of n copies of O X then rank OX F = n. If F is locally free then for x X the O X,x -module F x is free and we call rank OX,x F x the rank of F at x. If U is an open subset with F U free then rank OX U F U = rank OX,x F x for every x U. For n {0, 1, 2,..., } we say that F is locally free of rank n if it is locally free and the rank of F at every point x X is n. Equivalently, every point x X has an open neighborhood U with F U free of rank n. A sheaf of modules F is locally free of rank 0 if and only if F = 0. We say that a sheaf of modules F is locally free of finite rank if it is locally free of rank n for some finite n 0. Remark 14. Note that the concept of an invertible sheaf is defined for all ringed spaces, including those that are empty or have trivial local rings. But in the case where (X, O X ) is nontrivial, a sheaf of O X -modules F is invertible iff. it is locally free of rank 1. Lemma 55. Let (X, O X ) be a nontrivial ringed space and U X open. If F is a free module of rank n then F U is a free O X U -module of the same rank. If F is locally free of rank n then F U is also locally free of the same rank. Proof. The morphism of ringed spaces O X U O X induces a pair of adjoint functors between the module categories, with restriction being left adjoint to direct image. Hence restriction preserves coproducts, and thus preserves free modules and their rank. If F is locally free of rank n and x U let x V X be such that F V is free of rank n. Then (F U ) U V = (F V ) U V which is free of rank n, completing the proof. Lemma 56. Suppose (X, O X ) is a nontrivial ringed space and let F, G be O X -modules. Then (i) If F, G are free of finite ranks m, n then F G is free of rank mn. (ii) If F, G are locally free of finite ranks m, n then F G is locally free of rank mn. (iii) If F, G are locally free of finite ranks m, n then F G is locally free of rank m + n. (iv) For any ringed space, the tensor product of invertible sheaves is invertible. Proof. Tensoring with a O X -module is an additive functor, hence preserves finite products and coproducts since O X Mod is abelian. So (i) is obvious. For (ii) let x X be given and find open neighborhoods U, V of x such that F U is free of rank m and G V is free of rank n. Since restriction preserves coproducts the same can be said of F U V and G U V respectively, so by (i) the tensor product F G restricts to F U V G U V which is a free module of rank mn, as required. (iii) and (iv) are trivial. Lemma 57. Let (X, O X ) be a ringed space and f : L M an epimorphism of invertible sheaves on X. Then f is an isomorphism. Proof. We reduce immediately to proving the following result of algebra: if f : M N is a surjective morphism of free modules of rank 1 over a commutative ring R, then f is an isomorphism. Let m be a basis for M and n a basis for N, and suppose f(m) = α n for α R. Then since f is surjective, there is k R with f(k m) = n and therefore n = kα n and 1 = kα. Therefore α is a unit. If f(l m) = 0 then lα = 0 and therefore l = 0, so f is injective and therefore an isomorphism, as required. 20

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