GENERALIZED RINGED SPACES AND SCHEMES DAVID I. SPIVAK November 29, 2011 Contents 1. Introduction 1 2. Categorical Preliminaries 1 2.1. Basic category theory 1 2.2. Natural transformation diagrams 3 2.3. Correspondences 3 2.4. Downarrow categories 5 3. Sieves and Covering Sieves 6 3.1. First definitions 6 3.2. Covering Sieves 8 3.3. (C; CovC)-spaces 13 3.4. Sheaves 13 4. R-ringed Categories and R-ringed Spaces 14 4.1. R-ringed Categories 14 4.2. Examples 15 4.3. R-ringed spaces 16 5. The Structure theorem 17 6. Simplifying assumptions 17 6.1. Op(X) 17 6.2. C1-rings 17 7. Deleted stuff 17 7.1. 1 17 7.2. 2 21 7.3. 3 21 7.4. 4 23 7.5. 5 24 1. Introduction 2. Categorical Preliminaries 2.1. Basic category theory. For n 2 N, let [n] denote the \subdivided interval" category with n + 1 objects f0; 1; : : : ; ng and exactly one morphism i ! j for every i ≤ j ≤ n. 1 2 DAVID I. SPIVAK Let C be a category. A diagram in C is a functor X : I !C, where I is a small category (such a functor is also called an I-shaped diagram in C). The I- shaped diagrams in C form the objects of a category, whose morphisms are natural transformations. If i 2 I is an object, we sometimes denote X(i) by Xi. There is an isomorphism of categories C =∼ C[0]; that is we may identify objects in C with functors [0] !C, and morphisms with natural transformations between them. Any functor F : X ! Y naturally induces a functor F I : XI ! Y I , which we may denote simply by F : XI ! Y I . Let F : C!D be a functor. By an object in F we mean a functor [0] !D which factors through F . More generally, we refer to a diagram I !D which factors through F as an I-shaped diagram in F . If C = D and F is the identity, then diagrams in F are simply diagrams in C. Let Pre(C) denote the category of contravariant functors from C to Sets. It is called the category of presheaves on C. There is a natural functor r : C! Pre(C) given by r(C) = Hom(−;C) called the representation functor, or the Yoneda imbedding. Let C and D be categories. An adjunction from C to D is a triple (L; R; φ), where L: C!D and R: D!C are functors, and φ is a natural isomorphism of functors φ: HomD(L(−); −) ! HomC(−;R(−)): That is, for any C 2 C and D 2 D, there is a natural isomorphism Hom(LC; D) =∼ Hom(C; RD): The functor L is called a left adjoint and the functor R is called a right adjoint. If φ(β[) = β], then we refer to a pair of morphisms (β[ : LC ! D; β] : C ! RD) as a φ-partnership or (L; R)-partnership, and we refer to β[ and β] as the left and right partners, respectively . We denote such a φ-partnership by β : C ! D. We say that φ is the adjunction isomorphism of the adjoint pair L C o /D R: We sometimes suppress mention of φ when it is inconvenient. If F : A!B and X : A!C are functors, depicted in the diagram X A / C ? F λF X B then the left Kan extension of X along F (if it exists) is denoted λF X : B!C: A natural transformation ηF (X): X ! λF X ◦ F is part of the data of a left Kan extension, but is typically suppressed. By the term Basic Category Theory or BCT we will mean anything that can be proven using the following facts (all of which can be found in [?]). Lemma 2.1. Let C and D be categories, and let I be a small category. GENERALIZED RINGED SPACES AND SCHEMES 3 (1) Suppose that Y : I !C is a functor which has a limit Y C. Then for any object X 2 C, one has a natural isomorphism ∼ HomC(X; Y C) = lim(HomSets(X; Y )): (2) Suppose that X : I !C is a functor which has a colimit XB. Then for any object Y 2 C, one has a natural isomorphism ∼ HomC(XB;Y ) = lim(HomSets(X; Y )): (3) Suppose that L C o /D R is a pair of adjoint functors. Then L commutes with colimits and R com- mutes with limits. Explicitly, if a diagram X : I !C has a colimit XB, then the diagram LX : I !D has a colimit (LX)B, and there is a natural isomorphism ∼= (LX)B −! L(XB); (and similarly for R and limits). L (4) If C o /D is a pair of adjoint functors and I is a small category, then R L I I C o /D are also adjoint. R (5) The Yoneda imbedding r : C! Pre(C) is fully faithful. Moreover, for any presheaf F on C and object C 2 C, we have an isomorphism of sets ∼ HomPre(C)(rC; F ) = F (C): 2.2. Natural transformation diagrams. Let A; B; C, and D be categories, A −!w y B −! D and A −!x C −!z D pairs of composable functors, and β : zx ! yw a natural transformation of functors. We can express these, data in a natural transformation square: w AAB/ B β x ;C y Cz / D: This is an abbreviation of the diagram w AAB/ B yw β x ;C y zx + Cz / D: 4 DAVID I. SPIVAK A natural transformation square in which yw = zx and β is the identity trans- formation is called a commutative square (of categories), and is written w AAB/ B x y Cz / D: Similarly, there are natural transformation diagrams of any given shape; for exam- ple, see Lemma 2.8. There is one other special case worth mentioning. Suppose that y : B ! D has a right adjoint y0 : D ! B. Then if A = C and x: A ! C is the identity, then there is a natural isomorphism ∼ 0 HomDA (yw; z) = HomBA (w; y z): Any natural transformation β[ : yw ! z has a right partner β] : w ! y0z, and these two transformations represent the same data. We represent this data by the diagram w AAB/ B O id β y y0 ADA z / D 2.3. Correspondences. Definition 2.2. Let C and D be categories. A correspondence F from C to D, written F : C < D is a functor F ] : D! Pre(C). The correspondence F is called full (resp. faithful) if F ] is full (resp. faithful). The correspondence F can also be regarded as a functor F [ : Cop × D ! Sets via the Cartesian adjunction. Remark 2.3. There are several reasons for the notation F : C < D to denote a correspondence F : Cop ×D ! Sets. First, it is good to write C before D, since it is the contravariant variable. Second, correspondences generalize functors by saying that the correspondence underlying a functor F : D!C is [−;F (−)]C : C < D: Thus the direction on the tip on the arrow is preserved by the less-than symbol. We will sometimes denote a correspondence F : C < D by some variant of the notation [−; −]. There is often a way to do so which makes F clear. See Example 2.4. Example 2.4. Let G: D!C be a functor. It naturally induces two correspondences, op op [−;G(−)]C : C × D ! Sets and [G(−); −]C : D × C ! Sets; defined for C 2 C and D 2 D by [C; G(D)]C : = HomC(C; GD) and [G(D);C]C : = HomC(GD; C); respectively. In particular, the identity functor idC : C!C gives op the correspondence [−; −]C = HomC : C × C ! Sets. More generally, functors G: D!C and H : E!C induce correspondences [G(−);H(−)]C and [H(−);G(−)]C in the obvious way. GENERALIZED RINGED SPACES AND SCHEMES 5 As seen in Example 2.4, a functor G: D!C induces two correspondences. In op this work, we emphasize the correspondence [−;G(−)]C : C × D ! Sets. Definition 2.5. Let G: D!C be a functor. The correspondence associated to G is the composition of G with the Yoneda imbedding, D −!CG −!r Pre(C); and is denoted rG. Note that correspondences can be composed. If F : D < E and G : C < D are correspondences, we can define a correspondence F ◦ G : C < E as follows. Consider F as a functor E! Pre(D) and consider G as a functor D! Pre(C). The left Kan extension of G along the Yoneda imbedding is a functor λrG : Pre(D) ! Pre(C), and we define F ◦ G := F ◦ λrG: Let C and D be categories, and let F; G : C < D be two correspondences between them. A natural transformation of correspondences a: F!G is simply a natural transformation of left partners (F [ )G[): Cop × D ! Sets or equivalently of right partners (F ] )G]): D! Pre(C): Note that both of these ares equivalent to a correspondence a:(C × [1]) < D. Lemma 2.6. Let F : C < D be a correspondence. It determines an adjunction λr F Pre(D)o /Pre(C) ρr F of presheaf categories in which λrF(rD) = F(−;D) for any D 2 D. ] Proof. Let F = F : D! Pre(C), L = λrF and R = ρrF. For a presheaf P 2 Pre(D), one defines L(P ) = colimrX!P F (X). For a presheaf Q 2 Pre(C), one defines R(Q) = HomPre(C)(F (−);Q).