On Gluing Data, Finite Ringed Spaces and schemes

ON GLUING D A T A, FINITE RINGED SP A CES AND SCHEMES R. Fioresi ⋆ , A. Simonetti ⋆ , F. Zanc hetta ⋆ ⋆ F aBiT, Universit` a di Bologna Via Piero Gob etti 87, 40129 Bologna, Italy Abstract. F rom descent theory to higher geometry , the idea of gluing has been embedded in many elegan t and powerful techniques, proving instrumen- tal for the solution of man y problems. In this paper, we in tro duce a framew ork that allows to link important geometric ob jects, such as diﬀerentiable mani- folds or schemes, to certain ﬁnite ringed spaces arising from sheav es on 2 dimensional semisimplicial sets, thus op ening the door to their applications in ﬁelds such as discrete diﬀerential geometry . 1. Introduction Discrete geometry has seen renewed in terest ov er the last few years due to ad- v ances in machine learning and deep learning. Graph theory and metho ds from simplicial homotop y theory hav e b een fundamental for the dev elopment of geo- metric deep learning (GDL) [ BB2017 , GB2016 ] and top ological deep learning, [ Zam2024 ], which extend deep learning to data with geometric or top ological struc- ture. Indeed, algorithms lik e graph neural netw orks (GNNs) or simplicial neural net works ([ Spi2009 , BF2021 ]), central in this area, are gaining traction and are b ecoming more widely used in applications. In addition, sheav es attached to these ob jects are receiving particular atten tion as w ell: sheaf neural net works (SNNs), no w b ecoming an attractive ﬁeld of research, rely on the theory of cellular shea ves [ HG2019 , Cu2014 ]. On the other side, algebraic and diﬀerential geometry very often describ e their main ob jects of interest b y using the same combinatorics that discrete geometry is using. F or example a diﬀerentiable manifold or scheme is often b etter understo o d as the ˇ Cec h nerv e of a co vering of it: this simple y et crucial idea has b een ev entually formalized in descen t theory , that pla yed a key role in the theory of sc hemes, of stac ks and in deriv ed geometry . In this context, simplicial homotopy theory and category theory provide the background language for this notion to b e used. These theories, v ery elegant and sophisticated, required the introduction of many tec hnical to ols that allow ed the exp erts in the area to identify and prov e man y of the results that are now at the heart of geometry [ Ha1977 , Ja2001 , Lu2009 ]. Ho w ev er, these to ols are not easily accessible to practitioners in neighbouring ﬁelds that do not hav e exp ert knowledge of these tec hniques. F or example, simplicial ob jects asso ciated to manifolds or sc hemes are rarely p erceived as ’data attac hed to a discrete ob ject’, therefore ready to used also in applied contexts, and their complexity is very often high. As a consequence, a full transfer of the ideas and metho ds developed in ﬁelds suc h as diﬀerential geometry , higher geometry or algebraic geometry to discrete geometry has b een limited and many discrete geometric ob jects are often built 1 2 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES in analogy with their non-discrete counterparts and not from their non-discrete coun terparts (or viceversa) in a more conceptual wa y . Recen t research in the ﬁeld of algebraic geometry has started to address exactly this issue. In some recen t work, Salas and their collab orators (see [ S2017a ] and the references therein) study in more detail the notion of ﬁnite ringed spaces, i.e. ringed spaces having as underlying top ological space a ﬁnite top ological space. They ﬁnd a suitable sub category of ﬁnite ringed spaces that is equiv alen t, after lo calizing at certain weak equiv alences, to the category of ordinary schemes; the relations b etw een certain ﬁnite ringed spaces and diﬀerentiable manifolds are also considered, see [ CM2019 ] and the references therein. These works op en the do or to the understanding of highly structured ob jects, such as schemes, via simpler, in some sense discrete, ob jects. Indeed, b ecause of the equiv alence b etw een ﬁnite top ological spaces and partially ordered sets [ Al2011 ], one readily notices that the datum of a ﬁnite ringed space eﬀectively amounts to the datum of a poset and a sheaf of rings ov er it: at this p oint, the distance from the theory of cellular shea ves (that is the theory of sheav es, usually of vector spaces, on p osets coming from regular cell complexes [ HG2019 ]), recen tly introduced and used in discrete geometry and mac hine learning, b ecomes small. The fact that from a sheaf of rings ov er a ﬁnite top ological space w e can build a scheme do es not come as a surprise as, under certain assumptions, one migh t think to use the information enco ded into discrete ringed spaces to glue together some aﬃne schemes, corresp onding to the p oin ts of the ringed space. The informa- tion contained in a ﬁnite topological space, or equiv alently a partially ordered set, ho wev er, is not as structured or as readily informative as the one coming from a semisimplicial set or a cell complex. These, ho w ever, arise naturally if w e remind ourselv es of the notion of gluing datum (see [ GW2010 ]), that we can somehow think as a truncated Cech nerve in dimension 2. Indeed, consider a scheme (or, more generally , a ringed space) X , together with a co ver of op en aﬃne subsch emes { U i ⊆ X } i ∈ I . A “gluing datum” consisting of the U i ’s, their intersections and iden- tities coming from co cycle conditions then arises. Gluing this datum amounts to “glue” the U i ’s along their intersections in such a wa y that the appropriate co cycle conditions, inv olving the transition maps, are satisﬁed and this process gives bac k a space isomorphic to X . Observ e that we can build a 2 dimensional ﬁnite semisim- plicial set out of X taking the U i ’s as vertices, their intersections as edges and their triple intersections as 2-simplices. W e can then think of the pro cess of gluing geometric ob jects as ab ov e as the pro cess of gluing suitable data asso ciated to this 2 dimensional semisimplicial set. Note that, to this end, no higher dimensional simplices are inv olved to reconstruct X . Let us fo cus now on the task of mo del- ing sc hemes. Using the reasoning before as an heuristic argumen t, w e identiﬁed a category , that we will denote as C 2 Sch , having as ob jects pairs consisting of a 2 di- mensional ﬁnite semisimplicial set and a presheaf of rings on its face-incidence p oset satisfying certain assumptions. A category having ob jects of this type is indeed de- sirable as in many parts of discrete geometry or in applications simple geometric ob jects such as graphs often provide enough geometric structure for the theory to be dev elop ed or for the exp eriments to b e p erformed. Moreov er sheaf-based approac hes to quan tization in noncommutativ e geometry [ AFL W2021 , Ma2013 , DM1994 ] can b eneﬁt b y such categorical characterization of algebraic geometric ob jects. ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 3 Our constructions resembles some others app earing in simplicial descent theory [ Go2012 , Ja2001 ], but w e w ere not able to ﬁnd in the literature a category enco ding the notion of gluing data as extra data attac hed to a low dimensional semisimplicial set explicitly sp elled out and whose relation with ﬁnite ringed spaces is explored (usually simplicial ob jects in suitable categories are considered and ﬁnite ringed spaces are rarely used). In addition, our use of the language of semisimplicial sets distinguishes our approach from the one of [ S2017a ] and the one pursued in cellular sheaf theory: we link our ob jects to explicit 2 dimensional semisimplicial sets instead of p osets that could p ossibly hav e higher complexity (or that can ha v e non-geometric origin). It is worth noticing that Ladk ani in [ SP2006 ] studied the categories of sheav es ov er ﬁnite p osets, while Liu in [ SQ2013 ] studied stacks and torsors o ver quivers. Their w ork, how ever, is only partially related with ours. W e shall now describ e the conten ts of the pap er, highlighting our main results. In Section 2 w e review standard material on semisimplicial sets, with a particular fo cus on graphs, and w e establish our notation, see [ GJ1999 , SP2009 , EW2018 ] as references. In Section 3 we recast the theory of gluing data as the theory of certain functors of ringed spaces ov er the category of simplices of 2 dimensional semisimplicial sets. W e study the prop erties of the resulting ob jects. Intuitiv ely , we see gluing data as “shea ves of ringed spaces ov er a graph”. In Section 4 we recall some constructions of [ S2017a ] and w e introduce and study the category C 2 Sch . Then, we identify a class W of “weak equiv alences” in this category that can b e promoted to what we call “schematic right multiplicativ e system”, a w eaker notion of the ordinary righ t m ultiplicative system (see 4.27 and the discussion therein). It then b ecomes possible to lo calize in a weak wa y , see Section 4 for the details, the category C 2 Sch at the class W obtaining a category C 2 Sch [ W − 1 ]. Notice that our localization pro cedure is slightly diﬀerent from the ordinary one, since the natural candidate W is not a right m ultiplicativ e system. W e then obtain our main result. Theorem 1.1 ( 4.34 ) . The c ate gory C 2 Sch [ W − 1 ] is e quivalent to the c ate gory of quasi- c omp act and semi-sep ar ate d schemes. W e also compare our constructions with the ones of Salas [ S2017a ] and we p oint out to some possible future dev elopments in the context of diﬀerential geome- try while also making a few remarks useful to gain further insights on our work [ FSZ2026 ]. Ac kno wledgmen ts. The ﬁrst author thanks Barbara F antec hi for helpful dis- cussions. The ﬁrst and last authors thank F rancesco V accarino for helpful discus- sions and his in terest in our w ork. This researc h w as supp orted b y GNSA GA- Indam, INFN Gast Initiative, PNRR MNESYS, PNRR National Cen ter for HPC, Big Data and Quantum Computing CUP J33C22001170001, PNNR SIMQuSEC CUP J13C22000680006. This work w as also supported by Horizon Europ e EU pro jects MSCA-SE CaLIGOLA, Pro ject ID: 101086123, MSCA-DN CaLiF orNIA, Pro ject ID: 101119552, COST Action CaLIST A CA21109. Notation. W e shall record some of the notations we employ throughout the pap er here for the con venience of the reader. 4 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES • W e denote as (∆ + ) ∆ the category ha ving as ob jects the partially ordered sets [ n ] = { 0 → · · · → n } , n ∈ N , and arrows the (injective) order pre- serving maps b etw een them. F or any n ∈ N e denote as (∆ n, + ) ∆ n its full sub category of ob jects having cardinality smaller or equal than n . • W e will denote with Sets and V ect K the categories of sets and of ﬁnitely generated vector spaces ov er a base ﬁeld K resp ectively . W e will denote with T op the category of top ological spaces and with RngSp cs, LRngSp cs the categories of ringed spaces and lo cally ringed spaces resp ectively . • Giv en a set X a preorder on X is a binary relation that is reﬂexive and transitiv e. If this relation is also an tisymmetric, then we hav e a partial order: in this case we will say that X is a p oset. • Giv en a category C w e will denote as C op its dual (or opp osite) category . Finally , given t w o categories A and B we shall denote as F un( A , B ) or as B A the category of the co v arian t functors A → B and as Pre( A , B ) := F un( A op , B ) the category of contra v arian t functors A → B . W e refer to the ob jects of Pre( A , B ) as pr eshe aves . If the category B is the category Sets or it is clear from the con text, we shall denote Pre( C , B ) simply as Pre( C ). 2. Semisimplicial sets, graphs and shea ves In this section, we ﬁrst recall what a (semi)simplicial set is and then we in tro duce some basic notation ab out graphs in this setting. Finally , w e recall some basic notions ab out sheav es on a base and the Alexandrov top ology asso ciated with a preordered set. 2.1. The categories of (semi)simplicial sets and digraphs. Deﬁnition 2.1. W e deﬁne the category of (semi)simplicial sets to b e the category sSets := Pre(∆ , Sets) (ssSets := Pre(∆ + , Sets)). F or a giv en (semi)simplicial set X , for ev ery n ∈ N , w e shall denote the set X ([ n ]) as X n and call its elemen ts simplic es . Replacing the category of sets with an arbitrary category C gives us the notion of a (semi)simplicial obje ct in C . W e deﬁne (semi)c osimplicial sets as F un(∆ , Sets) (F un(∆ + , Sets)) and n -dimensional (semi)simplicial sets and (semi)cosimplicial sets as Pre(∆ n , Sets), (Pre(∆ n, + , Sets)), F un(∆ n , Sets), (F un(∆ n, + , Sets)) resp ectively . W e sa y that a (co)s emisimplicial set X has dimension n if X n  = ∅ and X m = ∅ for m > n . Remark 2.2. By deﬁnition, our n -dimensional simplicial sets are not simplicial sets, while in literature a simplicial set is usually said to ha ve dimension n if it is isomorphic to its n -sk eleton [ GJ1999 ]. How ever, the inclusion functor ∆ n ⊆ ∆ induces a truncation function tr n : sSets → sSets n ha ving as a fully faithful left adjoin t sk n : sSets n → sSets the n -skeleton functor. F or a given simplicial set X , sk n ◦ tr n ( X ) is the usual n -sk eleton (see also [ Lu2018 , T ag 04ZY ]). This recov ers the link b et ween our deﬁnition and the most common deﬁnition. Deﬁnition 2.3. W e deﬁne the category of dir e cte d gr aphs to b e Pre(∆ 1 , + , Sets) and w e denote it diGraphs. Replacing the category of sets with an arbitrary category C gives us the notion of dir e cte d gr aphs in C . Giv en a digraph G , we will call the sets G 0 and G 1 (or V G and E G ) the set of vertic es and e dges resp ectiv ely . W e can represen t a digraph using the following ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 5 diagram, where the maps h G and t G are called head and tail. E G h G / / t G / / V G Note that this deﬁnition of directed graph allows graphs with m ultiple edges with same head and tail (sometimes called dir e cte d multi-gr aphs ) and self-lo ops. W e denote with diGraphs ≤ 1 the full sub category of diGraphs whose ob jects hav e at most one edge connecting each pair of vertices. In addition, note that for a giv en G ∈ diGraphs b oth V G and E G can b e inﬁnite sets. Remark 2.4. Semisimplicial sets and simplicial sets are closely related. The latter category is the most frequently used in the context of simplicial homotop y theory b ecause of the nice properties of the geometric realisation [ Ma1992 , GJ1999 ]. The former app eared more frequently in early mo dern algebraic geometry , for example in [ SGA4 ]. F or the purpose of this work, semisimplicial ob jects provide a more natural con text b ecause of their easier com binatorics. The reader must b e assured, ho wev er, that they are not an artiﬁcial category: there exists a geometric realisation for semisimplicial ob jects (the “fat realisation”) and the geometric realisation of a simplicial set X is alwa ys homotopy equiv alent to the geometric realisation of the semisimplicial set obtained from X precomp osing with the inclusion ∆ op + ⊆ ∆ op (see [ EW2018 , Section 2] and the discussion therein). Moreo ver in dimension lo wer or equal to one the categories of simplicial sets and semisimplicial sets are equiv alent, [ Lu2018 , Prop osition 001N ], thus directed graphs may b e viewed as simplicial sets. As semisimplicial sets are simply presheav es of sets, w e hav e the following [ KS2006 ]. Prop osition 2.5. The c ate gory of semisimplicial sets has al l limits and c olimits and they ar e c ompute d p ointwise. In particular, ﬁbre pro ducts of semisimplicial sets exist. W e deﬁne the category of (semi)simplices of X , Γ( X ), for a given (semi)simplicial set X , see [ Ho2007 ]. W e recall brieﬂy its construction. F or a given (semi)simplicial set X , b y Y oneda’s Lemma, the maps ∆ n → X (resp. ∆ n + → X ) are in bijection with the elemen ts of X ([ n ]) = X n , where ∆ n = Hom ∆ ( − , [ n ]) (resp. ∆ n + = Hom ∆ + ( − , [ n ])). So we take such maps as the ob jects for Γ( X ) as n v aries. T o deﬁne the morphisms of Γ( X ), we sp ecify when w e hav e an arrow b et ween t wo ob jects in Γ( X ) that is, a ∈ X n and b ∈ X m . W e say w e hav e suc h an arrow, whenever we can write a commutativ e diagram: (1) ∆ n / / a ! ! ∆ m b } } X resp. ∆ n + / / a ∆ m + b } } X where again w e employ Y oneda’s lemma. Deﬁnition 2.6. F or a given (semi)simplicial set X , we deﬁne Γ( X ) the c ate gory of (semi)simplic es of X as the category with ob jects the maps ∆ n → X (resp. ∆ n + → X ) and morphisms the diagrams in ( 1 ). Notice that a morphism f : X → Y of (semi)simplicial sets induces a functor b et w een their resp ective categories of simplices that w e will denote as Γ( f ). 6 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES Observ ation 2.7. F or a semisimplicial set X , the ob jects and the arrows of Γ( X ) can b e used to deﬁne a p oset P X where the elements are the simplices of X and x ≤ y if and only if Hom Γ( X ) ( x, y ) is non empty . Remark 2.8. Notice that for a semisimplicial set X there is a non-canonical iso- morphism betw een the category Γ( X ) and the category asso ciated with the p oset P X ha ving as ob jects the elements of P X and arro ws x ≤ y . Recall that an undirected graph G is a pair ( V , E ) of v ertices and edges, where the edges are deﬁned as a multiset of unordered pairs of vertices and morphisms are deﬁned accordingly (see [ Di2006 , GR2001 ] for the case V and E ﬁnite sets, though here we are more general). Let Graphs denote their category . W e call the category of simple undirected graphs, i.e. undirected graphs having at most one edge connecting eac h pair of vertices, as Graphs ≤ 1 . Observ ation 2.9. In Observ ation 2.7 we hav e given a wa y to asso ciate a p oset to a semisimplicial set, and therefore to a directed graph. W e can do the same with undirected graphs as follows. Let G = ( V , E ) ∈ Graphs b e an undirected graph. Deﬁne the p oset P G asso ciated to it having as underlying set V ∪ E and where x ≤ y if and only if x is a vertex of the edge y or x = y . 2.2. Shea v es on preordered sets. A top ological space is called ﬁnite if it consists of a ﬁnite num b er of points. F or a giv en ﬁnite space X and a p oint p ∈ X , w e deﬁne U p := smallest op en subset of X containing p C p := p = smallest closed subset of X containing p Giv en a ﬁnite top ological space X , we can deﬁne the structure of a (ﬁnite) preorder on X b y setting p ≤ q if and only if U p ⊇ U q (equiv alently , p ∈ q ). Conv ersely , giv en a (not necessarily ﬁnite) preorder ≤ on a set P , we can see P as a top ological space, whose top ology is generated b y the base consisting of the follo wing open sets: U p := { q ∈ P | q ≥ p } p ∈ P This top ology is called the Alexandr ov top olo gy [ Al2011 ] (see also [ S2017a ]) and we denote as B P the base w e used to generate it. Notation 2.10. Whenev er necessary , to av oid confusion, w e shall use a diﬀerent notation for a pr e or der P (i.e. a preordered set) and the top ological space A ( P ) asso ciated with it, as ab ov e. Similarly , for a ﬁnite top ological space X w e will write P ( X ) for the preorder asso ciated with it as ab ov e. W e note explicitly that if X is a ﬁnite top ological space and P ( X ) is the asso ciated preorder, then X = A ( P ( X )). W e denote by Op en( X ) the category of the op en sets of a top ological space X . Recall that sheav es on X are deﬁned as preshea v es on Op en( X ) suc h that the iden tity and the gluability axioms hold (see for example [ V a2025 ] Deﬁnition 2.2.6). W e hav e the following result, see [ S2017a ] for a clear review of these statements. Theorem 2.11. The fol lowing statements hold: 1) The ab ove c onstruction deﬁnes an e quivalenc e of c ate gories b etwe en ﬁnite top olo gic al sp ac es FT op and ﬁnite pr e or der e d sets PreSets given by the func- tors: P : FT op − → PreSets , A : PreSets − → FT op ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 7 2) A ﬁnite top olo gic al sp ac e X is T 0 (i.e. diﬀer ent p oints have diﬀer en t clo- sur es) if and only if the pr e or der r elation ≤ induc e d by the top olo gy is an- tisymmetric i.e. X is a p oset. Remark 2.12. F or a given ﬁnite top ological space X , the irreducible op en and closed sets are U p and C p , p ∈ X . The same holds true if X is a top ological space of the form A ( P ) for some (not necessarily ﬁnite) preorder P . Observ ation 2.13. If X is a semisimplicial set, w e obtain the p oset P X as in Obs. 2.7 , hence we can consider sheav es on A ( P X ). It is well known (see for example [ V a2025 ] 2.5.1) that sheav es on a top ological space are uniquely determined by a sheaf on a base for the underlying top ological space. In our case, the base B P X of A ( P X ) consists of irreducible op en subsets of A ( P X ). As a consequence preshea ves on the base B P X are automatically sheav es on a the base B P X . In addition if we consider the full subcategory of Open( A ( P X )) having as ob jects the elemen ts of B P X , we see that it is isomorphic to P X . Putting all together, we get the follo wing prop osition. Prop osition 2.14. L et b e X a semisimplicial set or a ﬁnite top olo gic al sp ac e. Consider its p oset P X . Then the c ate gory of she aves on A ( P X ) is e quivalent to the c ate gory Pre( P X ) . The previous prop osition holds not only for the case of sheav es of sets but also for shea ves taking v alues in other categories, such as ab elian categories, etc. Example 2.15. Let G = ( V G , E G ) be in Graphs. The top ology of the space A ( P G ) is then generated b y the following base of op en sets: • U v = { e ∈ E G | v ≤ e } ∪ { v } , that is the op en star of v , for each vertex v ∈ V , • U e = { e } , i.e. the edge e , without its vertices, for each e ∈ E . F rom now until the end of the pap er we shall mak e the blanket assumption that all the semisimplicial sets w e consider are ﬁnite unless stated otherwise. 3. Finite ringed sp aces, gluing da t a and semisimplicial sets Gluing data as in [ GW2010 ], [ SP ] allo w us to deﬁne ob jects lik e manifolds, sc hemes or morphisms b et ween them b y “gluing” some lo cal data satisfying certain compatibilit y conditions. The aim of this section is to study this notion, via the semisimplicial set language of the previous section, in a wa y that can b e useful for applications. 3.1. Gluing data. W e recall the deﬁnition of gluing datum of ringed spaces, (see 3.5 of [ GW2010 ], or 6.33 of [ SP ]). A ringed space is a pair ( X , O X ) consisting of a top ological space X and of a sheaf of rings O X on it; a gluing datum reco vers a ringed space through a collection of ringed spaces and compatibilit y conditions. Deﬁnition 3.1. A gluing datum of ringe d sp ac es consists of: • a collection of ringed spaces ( U i , O U i ) i ∈ I ; • a collection of op en subspaces ( U ij ⊆ U i ) i,j ∈ I , U ii = U i , for all i ∈ I . 8 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES • a collection of isomorphisms of ringed spaces ( φ j i : U ij → U j i ) i,j ∈ I suc h that: φ ki = φ kj ◦ φ j i on U ij ∩ U ik , (co cycle condition) W e will denote such datum as (( U i ) i ∈ I , ( U ij ) i,j ∈ I , ( φ ij ) i,j ∈ I ). W e may replace the sheaf of rings with a sheaf of ab elian groups, sets, v ector spaces etc. and obtain similar notions of gluing datum, for which the constructions w e give b elow will hold. As customary , we ma y sometimes denote both the top ological space and the ringed space with the same letter, whenever there is no danger of confusion. Re- placing ringed spaces and morphisms betw een them with topological spaces and op en embeddings (resp. sc hemes and op en embeddings and so on) giv es rise to the notion of gluing datum of top ological spaces (resp. sc hemes, etc.). T o ﬁx the ideas, w e will examine the case of gluing data in the category of ringed spaces RngSp cs, but many of the deﬁnitions, lemmas and prop ositions we will prov e, hold also re- placing RngSp cs with other categories (like top ological spaces T op, schemes Sch, etc.). F rom no w on if we hav e t wo op en embeddings f : U  → X , g : V  → X w e shall use the in tuitive notation U ∩ V instead of the more precise one U × X V . Deﬁnition 3.2. A ringed spaced X is obtained by gluing a gluing datum (( U i ) i ∈ I , ( U ij ) i,j ∈ I , ( φ ij ) i,j ∈ I ), if there exists an open co v er of X of ringed subspaces V i , such that V i ∼ = U i and such isomorphisms restrict suitably to give V i ∩ V j ∼ = U ij and the co cycle conditions. Giv en a gluing datum, we can alwa ys ﬁnd a ringed space obtained by gluing it (see [ GW2010 ] or [ SP ] for example). The gluing construction essentially amounts to taking a colimit i.e. direct limit of a certain diagram in the category of ringed spaces: w e shall no w explore this statement, so that we ﬁt it in to our discussion. This is a generally known fact, how ev er we prefer to recast it so that it b ecomes useful in the study of ﬁnite ringed spaces (as in [ S2017a ]). W e b egin with the notion of gluing cub e and w e consider, for the moment, just the problem of gluing three ringed spaces. Deﬁnition 3.3. W e deﬁne the category Ξ, as the index category consisting of 7 ob jects X 1 , X 2 , X 3 , X 12 , X 13 , X 23 , X 123 and arrows as in Fig. 3.1 , identit y arrows not depicted. A functor F : Ξ → RngSp cs is called gluing cub e , if all the morphisms in Ξ are sent to op en embeddings and the 3 commutativ e squares of diagram in Fig. 3.1 are sent to cartesian squares. Giv en a gluing cub e F , we can glue the ringed spaces F ( X 1 ), F ( X 2 ) and F ( X 3 ) along the “intersections” (ﬁb ered pro ducts) F ( X ij ) in such a wa y that a co cycle condition is satisﬁed, obtaining a ringed space Y , that visually o ccupies the “missing v ertex” in the gluing cub e diagram 3.1 . This pro cess amounts to tak e the colimit of the gluing cube F . The gluing construction yields natural open embeddings F ( X i )  → Y (see [ SP ] 6.33). The next observ ation clariﬁes these statemen ts. Observ ation 3.4. A gluing cube F : Ξ → RngSp cs deﬁnes a gluing datum GD( F ) = (( U i ) , ( U ij ), ( φ ij )) i,j ∈{ 1 , 2 , 3 } and viceversa. T o construct a gluing datum from F , w e set U i = U ii := F ( X i ) for i = 1 , 2 , 3. W e then deﬁne the U ij as follows. W e kno w that the map F ( α ij ) : F ( X ij ) → F ( X i ) is an op en embedding, so it factors through an op en embedding e ij : ( U ij , O F ( X i ) | U ij ) → F ( X i ) where U ij is an op en ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 9 X 123 pr 13   pr 23 / / pr 12 ( ( X 23 α 32   α 23 ' ' X 12 α 12   α 21 / / X 2 X 13 α 13 ( ( α 31 / / X 3 X 1 F ( X 123 ) F (pr 13 )   F (pr 23 ) / / F (pr 12 ) ) ) F ( X 23 ) F ( α 32 )   F ( α 23 ) ) ) F ( X 12 ) F ( α 12 )   F ( α 21 ) / / F ( X 2 ) F ( X 13 ) F ( α 13 ) ) ) F ( α 31 ) / / F ( X 3 ) F ( X 1 ) Figure 1. The category Ξ and a gluing cub e subset of F ( X i ). This deﬁnes the desired U ij . In addition, we hav e isomorphisms ψ ij : F ( X ij ) ∼ = − → U ij suc h that e ij ◦ ψ ij = F ( α ij ). Note here that if i > j , strictly sp eaking, we do not hav e a X ij : in this case by F ( X ij ) w e mean F ( X j i ). W e set φ j i := ψ j i ◦ ψ − ij 1 : U ij ∼ = − → U j i . One can chec k that this deﬁnes a gluing datum, that w e denote GD( F ) := (( U i ) , ( U ij ) , ( φ ij )) i,j ∈{ 1 , 2 , 3 } . Con versely , if we start from a gluing datum inv olving only three spaces, W = (( U i ) , ( U ij ) , ( φ ij )) i,j ∈{ 1 , 2 , 3 } , we can deﬁne a gluing cube F W as follo ws. W e set F W ( X i ) = U i , F W ( X ij ) := U ij and F W ( α ij ) to b e the inclusion U ij ⊆ U i for i, j = 1 , 2 , 3, i < j . If i > j we deﬁne F W ( α ij ) to b e the comp osition of φ j i and the canonical inclusion U ij ⊆ U i . Finally , we deﬁne F W ( X 123 ) and F W (pr ij ) by taking the pullbac ks of the maps F W ( α ij ) in RngSp cs and using the co cycle conditions. If the gluing datum W comes from a gluing cub e F , one can then easily chec k that this pro cedure giv es back the gluing cub e F , that is F GD( F ) ∼ = F . Conv ersely , starting from a gluing datum W as ab o ve, if we construct a gluing cube and then a gluing datum from it, w e obtain W back. 3.2. Gluing data for 2-dimensional semisimplicial sets. W e no w consider gluing data in volving more than 3 spaces. Recall that ∆ 2 is isomorphic to the follo wing semisimplicial set, that we shall call T : T 2 = { A } δ 0 / / δ 1 / / δ 2 / / T 1 = { e 12 , e 23 , e 31 } δ ′ 0 / / δ ′ 1 / / T 0 = { v 1 , v 2 , v 3 } As usual T i denotes T ([ i ]) and δ i , δ ′ i are the usual face maps satisfying the face iden tities [ GJ1999 ]. The semisimplicial set T can b e represented as the following 10 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES diagram: • v 3 A v 1 • e 12 / / e 13 A A • v 2 e 23 ] ] W e ha ve that the opp osite category of the category of simplices Γ( T ), whose ob jects are the elements in T 0 , T 1 , T 2 (see Deﬁnition 2.6 ) is isomorphic to Ξ and so is the opp osite of the category arising by the poset P T (see Obs. 2.7 ). As a consequence, a gluing cube can b e seen as a functor (Γ( T )) op ∼ = ( P T ) op → RngSp cs. Deﬁnition 3.5. X ∈ ssSets is called r e gular if for every n -simplex a ∈ X n , the smallest sub-semisimplicial set con taining a is isomorphic to ∆ n [ FP1990 ]. Let G be a ﬁnite graph in diGraphs ≤ 1 or Graphs ≤ 1 together with a total ordering of its v ertices. Recall that the notion of k -cliques of G is deﬁned in b oth these categories: a k -clique is a subset of k vertices such that each tw o distinct v ertices are adjac ent , i.e. linked by an edge if G is undirected or linked by an edge in eac h direction if G is directed. Deﬁnition 3.6. Let G b e either an ob ject of diGraphs ≤ 1 or Graphs ≤ 1 . Given a total ordering on its vertices, w e deﬁne the semisimplicial set G • as the regular semisimplicial set of dimension 2 consisting of 1, 2 and 3-cliques. Explicitly: • G 0 , G 1 , G 2 are the sets of 1-cliques (vertices), 2-cliques and 3-cliques re- sp ectiv ely . • F ace maps are deﬁned by ordering the complex of 1, 2 and 3-cliques using the giv en total ordering of the vertices of G . Diﬀeren t choices of the total ordering on the vertices give rise to isomorphic G • . This follo ws from the fact that all total orderings on ﬁnite sets are isomorphic. Observ ation 3.7. The simplices of a semisimplicial set G • are in bijection with sub-semisimplicial sets isomorphic to T := T 2 / / / / / / T 1 ⇒ T 0 , E := • ⇒ • and P := • and are represented respectively b y the diagrams: • v 3 A v 1 • e 12 / / e 13 A A • v 2 e 23 ] ] v i • e ij − → • v j • v i The opp osite categories of their categories of simplices Γ( T ), Γ( E ) and Γ( P ) are isomorphic resp ectively to the categories Ξ, Λ and ∗ , deﬁned b y the diagrams b elow: A   / / ( ( e 23   ' ' e 12   / / v 2 e 13 ( ( / / v 3 v 1 v i • α i ← − e ij • α j − → • v j • v i ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 11 As usual, the iden tity morphisms are not depicted in these diagrams. Deﬁnition 3.8. W e deﬁne a gluing we dge and a gluing p oint to b e functors Λ → RngSp cs and ∗ → RngSp cs resp ectively , such that all the morphisms in Λ and ∗ are sen t to op en embeddings. Remark 3.9. Gluing wedges and gluing poin ts giv e rise to gluing data (( U i ) , ( U ij ) , ( φ ij )) i,j ∈{ 1 , 2 } and (( U 1 ) , ( U 11 ) , ( φ 11 )) resp ectiv ely . W e extend this remark to obtain a result in Prop 3.11 , key for our subsequent treatmen t, which generalizes Obs. 3.4 , leaving all the details to the reader. Deﬁnition 3.10. Given a graph G w e sa y that a functor F : Γ( G • ) op → RngSp cs is a gluing functor for G if: 1. F or all T , E , P ⊆ G • , F | Γ( T ) op , F | Γ( E ) op , F | Γ( P ) op are resp ectively a gluing cub e, w edge or p oint, where F | Γ( T ) op : Γ( T ) op → RngSp cs , F | Γ( E ) op : Γ( E ) op → RngSp cs , F | Γ( P ) op : Γ( P ) op → RngSp cs 2. F or all pairs of gluing wedges F ( a ) ← F ( b ) → F ( c ) ← F ( d ) → F ( e ), if a and e are not linked b y an edge, the pullbac k of ringed spaces F ( b ) × F ( c ) F ( d ) is the empty ringed space. Let G b e a ﬁnite graph belonging to either the category diGraphs ≤ 1 or Graphs ≤ 1 , together with a total ordering of its v ertices and let Γ( G • ) b e the category of simplices of G • . Prop osition 3.11. L et F b e a gluing functor for a ﬁnite gr aph G as ab ove. F deﬁnes a gluing datum GD( F ) = (( U i ) , ( U ij ) , ( φ ij )) i,j ∈{ 1 ,...,n } and colim F is iso- morphic to the ringe d sp ac e obtaine d by gluing GD( F ) . Conversely, given a gluing datum W = (( U i ) , ( U ij ) , ( φ ij )) i,j ∈{ 1 ,...,n } , we c an de- ﬁne a 2-dimensional semisimplicial set G • whose 1-skeleton is isomorphic to a gr aph G ∈ diGraphs having at most one e dge joining e ach p air of vertic es and a functor F W : Γ( G • ) op → RngSp cs , such that colim F W is isomorphic to the ringe d sp ac e obtaine d by gluing W . Remark 3.12. Consider a gluing functor F for a ﬁnite graph G . If tw o vertices u , v of G are not linked by any edge, then the ringed spaces F ( a ) and F ( b ) are disjoin t op en sub ringed spaces of colim F . Observ ation 3.13. In the previous prop osition, the choice of the total ordering on the vertices of G is immaterial: tw o distinct total orderings on G give rise to an automorphism of G • inducing on the colimits of the tw o resulting gluing functors an isomorphism. 3.3. Gluing Ringed Spaces. Given a ﬁnite graph G , w e now sho w how the infor- mation enco ded in gluing functor F : Γ( G • ) op → RngSp cs for G eﬀectively reco v ers a ringed space. W e hav e a fully faithful inclusion functor T op ⊆ RngSpcs and forgetful functors LRngSp cs → RngSp cs → T op. As these forgetful functors ha ve righ t adjoints, they preserve colimits. In particular, whenever w e glue a gluing datum of (lo cally) ringed spaces, the underlying top ological space is obtained as if we were gluing a gluing datum of top ological spaces. The same also applies if we replace the sheav es of rings with shea ves of ab elian groups, etc. 12 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES Deﬁnition 3.14. Consider a ringed space ( S, O S ) and a ﬁnite cov ering by op en em b eddings U = { U i  → S } i =1 ,...,n of it. W e iden tify eac h U i with the subspace of S isomorphic to it. This co v ering gives rise to a gluing datum ( U i , U ij := U i ∩ U j , φ ij ) where the φ ij are isomorphisms. W e deﬁne the graph G ( U ) ∈ Graphs ≤ 1 b y taking as vertices the ordered set of U i ’s and placing an edge b et w een t w o v ertices v i and v j i  = j if and only if U ij  = ∅ . Notice that, since vertices of G ( U ) are naturally ordered we can immediately construct the semisimplicial set G ( U ) • according to Def. 3.6 . It has one 0-simplex for each U i , a 1-simplex for eac h non empty U i ∩ U j ( i < j ) and one 2-simplex for each triple intersection U i ∩ U j ∩ U k corresp onding to a 3-clique of G ( U ) (these triple intersections can b e empt y). Moreo v er, the 2-dimensional semisimplicial set G ( U ) • is regular. Remark 3.15. The identiﬁcation of each element of a co v ering U by op en em b ed- dings of a given ringed s pace S with the subspace of S isomorphic to it is merely a w ay to av oid c ho osing a cleav age as in [ Vi2005 ]. Please note that this c hoice do es not “remov e rep etitions” in the following sense. If we hav e t w o schemes in U that em b ed to the same op en subspace of S w e still hav e in G ( U ) 0 t wo distinct vertices corresp onding to these ringed spaces. Giv en S , U as ab o v e we can deﬁne a ﬁnite ringed space ( S U , O S U ) as in [ S2017a ]. Deﬁnition 3.16. Let ( S, O S ) b e a ringed space and U a ﬁnite cov ering b y open em- b eddings. F or each point s ∈ S , w e deﬁne U s := ∩ s ∈ U i U i . Consider the equiv alence relation: s ∼ s ′ if and only if U s = U s ′ W e deﬁne the ﬁnite ringed space ( S U , O S U ) asso ciated to S and the co ver U as S U := S/ ∼ and O S U := π ∗ O S , where π : S → S U is the contin uous pro jection morphism. Notice that π can b e promoted to a morphism of ringed spaces. Observ ation 3.17. Let π : S → S U and U as in Def. 3.16 . One can c heck that the image of U i ∩ U j =: U ij and U i ∩ U j ∩ U k =: U ij k under π is an op en subset for ev ery i, j, k . By looking at U i , U ij and U ij k as ringed spaces, we deﬁne V i := ( π ( U i ) , ( π | U i ) ∗ O S | U i ), V ij := ( π ( U ij ) , ( π | U ij ) ∗ O S | U ij ) and V ij k := ( π ( U ij k ) , ( π | U ij k ) ∗ O S | U ij k ). These are op en sub-ringed spaces of S U : the V i form a co ver of S U b y op en em b eddings and V ij , V ij k are the double and the triple inter- sections of them so that they naturally deﬁne a gluing datum that returns S U after gluing. W e will denote as RngSp cs Fin the category of ringed spaces ha ving as underlying top ological space a ﬁnite top ological space. F or a given ringed space S and a given op en cov ering U of it as ab ov e, using Prop osition 3.11 and Observ ation 3.17 we can deﬁne: • F U can : Γ( G ( U ) • ) op → RngSp cs, sending the ob jects of Γ( G ( U )) op to U i , U ij , U ij k . • F U ﬁn : Γ( G ( U ) • ) op → RngSpcs Fin , sending the ob jects of Γ( G ( U )) op to V i , V ij , V ij k . Lemma 3.18. L et the notation b e as ab ove. Then colim F U ﬁn ∼ = S U colim F U can ∼ = S. ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 13 In addition we have a natur al tr ansformation of functors ˜ π : F U can → F U ﬁn induc e d by π that, under the isomorphisms colim F U ﬁn ∼ = S U and colim F U can ∼ = S r e c onstructs the morphism π : S → S U on the c olimits. Pr o of. W e leav e to the reader to c hec k the compatibility conditions. □ W e end this section by constructing a ringed space similar, but diﬀerent from ( S U , O S U ) in Def. 3.16 , that will b e imp ortant later. Observ ation 3.19. If ( S, O S ) is a ringed space and U = { U i } is a ﬁnite co v ering of it by op en embeddings, Γ( G ( U ) • ) is a category isomorphic to the category as- so ciated with the poset P G ( U ) • (see Observ ation 2.7 , Remark 2.8 ). Unra v eling the deﬁnitions, we see that this p oset has one element p i for each U i , an element p ij for eac h U ij ( i < j , U i ∩ U j  = ∅ ) and an element p ij k for each U ij k corresp onding to a 3-clique of G ( U ). W e can consider the top ological space A ( P G ( U ) • ) asso ciated to the p oset P G ( U ) • using the Alexandro v top ology (see 2.10 ). One can chec k, by the v ery deﬁnition of the Alexandrov top ology , that P G ( U ) • , viewed as a category , is isomorphic to the dual category of irreducible op en subsets of A ( P G ( U ) • ). This iso- morphism asso ciates to eac h p i , p ij , p ij k the irreducible op en subsets U p i , U p ij and U p ij k of A ( P G ( U ) • ) (see Section 2.2 for the notation). Hence, a functor from P G ( U ) • to Rings will yield a sheaf on the top ological space A ( P G ( U ) • ) (see Prop osition 2.14 ). Hence, the functor F 2 U : P G ( U ) • ∼ = Γ( G ( U ) • ) → Rings , p i 7→ O S ( U i ) , p ij 7→ O S ( U ij ) , p ij k 7→ O S ( U ij k ) is a sheaf of rings on the base { U p i , U p ij , U p ij k } of A ( P G ( U ) • ). In other words, F 2 U is deﬁned b y comp osing the functor F U can with the global section functor. Deﬁnition 3.20. Let S be a ringed space and let U b e a ﬁnite co vering of it by op en em b eddings. W e deﬁne the ringed spac e ( S 2 U , O S 2 U ): • S 2 U is the top ological space A ( P G ( U ) • ). • O S 2 U is the sheaf of rings on S 2 U obtained from the sheaf F 2 U of Observ ation 3.19 on a base of A ( P G ( U ) • ). W e will say also in this case that ( S 2 U , O S 2 U ) is the ringed space asso ciated to S and the co ver U . Remark 3.21. Given a ringed space S and a ﬁnite cov ering by op en em beddings U , in general the topological spaces S 2 U and S U are not homeomorphic and conse- quen tly the tw o resulting ringed spaces according to Deﬁnitions 3.16 and 3.20 are not isomorphic. Indeed, consider a cov ering U consisting of S itself and of a given non empty op en subspace U ⊂ S . Then S U is a ﬁnite top ological space consisting of t wo p oints while S 2 U is a ﬁnite top ological space consisting of 3 p oin ts. Remark 3.22. Giv en a cov ering U of S as ab ov e one can deﬁne the sub-semisimplicial set i : e G ( U ) •  → G ( U ) • obtained by G ( U ) • b y removing in G ( U ) 2 the elements p ij k asso ciated with triple intersections U ij k that are empty . F rom i we get a map Γ( i ) : Γ( e G ( U ) • ) → Γ( G ( U ) • ) and therefore, as P G ( U ) • ∼ = Γ( G ( U ) • ) and P e G ( U ) • ∼ = Γ( e G ( U ) • ), a functor: e F 2 U := F 2 U ◦ Γ( i ) , e F 2 U : P G ( U ) • ∼ = Γ( e G ( U ) • ) → Rings F rom i we also get a top ological space e S 2 U , a contin uous inclusion f : e S 2 U  → S 2 U , a sheaf O e S 2 U on e S 2 U obtained from e F 2 U and a functor e F U can = F U can ◦ Γ( i ) op . As for every 14 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES sheaf F on a top ological space we ha v e that F ( ∅ ) is equal to the zero ring, one can c heck that b y the v ery deﬁnition of O e S 2 U w e ha ve an isomorphism ϕ : O S 2 U ∼ = − → f ∗ O e S 2 U giving rise to a morphism of ringed spaces ( f , ϕ ) : ( e S 2 U , O e S 2 U ) → ( S 2 U , O S 2 U ) suc h that ϕ is a sheaf isomorphism. This morphism of ringed space is not an isomorphism but its latter prop ert y , together with the fact that the colimits of e F U can and F U can ◦ Γ( i ) op can be c heck ed to be b oth isomorphic to S might prompt us to think that ( e S 2 U , O e S 2 U ) and ( S 2 U , O S 2 U ) are weakly equiv alent in some sense to b e deﬁned. This heuristic argumen t motiv ates part of the reasoning found in Section 4.2 . 4. Finite ringed sp aces, schemes and semisimplicial sets In this section we sp ecialize some of the previous constructions to the case of sc hemes and manifolds. W e also introduce a conv enient category to study gluing data of schemes leveraging the language of semisimplicial set that we will cal C 2 Sch . 4.1. Finite ringed spaces and schemes. In this section we study more in detail the gluing constructions introduced in the previous section for the case of schemes. F or the main deﬁnitions as scheme, Sp ec, etc. we refer the reader to [ Ha1977 ] Ch. 2. Our next deﬁnition is directly inspired by the theory of schematic spaces (see [ S2017a ] and refs therein and also Sec. 4.6 for the diﬀerentiable category). Consider a ﬁnite ringed space ( X , O X ), i.e. a ringed space having as underlying top ological space a ﬁnite top ological space. There is a natural preorder on X as explained in Section 2.2 , Theorem 2.11 . Deﬁnition 4.1. W e sa y that a ﬁnite ringed space ( X , O X ) is p ar aschematic if for all p ≥ q , p, q ∈ X , (recall that p ≥ q if and only if U q ⊂ U p ), the scheme morphism Sp ec( O X ( U q )) → Sp ec( O X ( U p )) is an op en embedding. W e denote the category of parasc hematic spaces as PSch. Recall that a sc heme is called semi-separated if its diagonal morphism is an aﬃne morphism or, equiv alen tly , if the in tersection of any tw o aﬃne subschemes of it is an aﬃne sc heme. Recall also that a sc heme is called quasi-compact if admits a ﬁnite cov er of aﬃne op en subschemes. Notable categories of schemes enjoy these prop erties, for example all quasi-pro jective schemes (ov er a regular ring, say) or more generally all divisorial schemes ha v e this prop erty . W e denote the category of quasi-compact and semi-separated schemes with Sc h. Unless otherwise sp eciﬁed, w e assume schemes to b e in this category . W e no w make an observ ation regarding the ﬁnite ringed space ( S U , O S U ) as in tro duced in Def. 3.16 , for the case in which S is a scheme. Observ ation 4.2. Let S b e a scheme together with a ﬁnite cov ering by aﬃne open subsc hemes U . As b efore, w e can view the preorder P ( S U ) as a category and it is isomorphic to the dual category of irreducible op en subsets in S U . Then, the functor: P ( S U ) − → Rings , p − → O S U ( U p ) is a (pre)sheaf on a base of S U , where U p denotes the smallest open set in S U con taining p (see Section 2.2 ). Moreo ver its extension to the whole S U is O S U . W e can rep eat the construction in Obs. 4.2 replacing S U b y a paraschematic space X , thus obtaining a presheaf: P ( X ) − → Rings , p − → O X ( U p ) ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 15 extending to the sheaf O X on X . This leads us to the following deﬁnition [ S2017a ]. Deﬁnition 4.3. Let ( X , O X ) b e a parasc hematic space. W e deﬁne the functor: F X : ( P ( X )) op → RngSp cs , p 7→ Sp ec O X ( U p ) This allo ws us to deﬁne the functor: Sp ec S : PSc h − → RngSp cs , X 7→ colim F X where PSc h denotes the category of paraschematic spaces. W e state a result found in [ S2017a ], though in a diﬀerent language with slightly diﬀeren t assumptions (quasi-compact and quasi-separated schemes). The pro of is a simple c heck. Prop osition 4.4. L et S b e a scheme and let U b e a ﬁnite c overing of S by aﬃne schemes. Then S U is p ar aschematic and Spec S ( S U ) ∼ = S in the c ate gory of schemes. W e now turn to examine the ringed space S 2 U (see Deﬁnition 3.20 ), whic h is more in teresting to our purp oses. Prop osition 4.5. L et S b e a scheme and let U b e a ﬁnite c overing of S by aﬃne op en subschemes. Then S 2 U is p ar aschematic. In addition, Sp ec S ( S 2 U ) ∼ = S in the c ate gory Sch . Pr o of. First of all, w e chec k that it is parasc hematic. As S is semi-separated, this follo w from the v ery deﬁnition of S 2 U as w e ha v e that for ev ery p ≤ q the ring morphisms O S 2 U ( U p ) → O S 2 U ( U q ) corresp onds, via the antiequiv alence b etw een aﬃne schemes and rings, to an op en em b edding. Using this equiv alence, we see that the functor F 2 U : P G ( U ) • → Rings (see Deﬁnition 3.20 ) inducing the functor F X : P op G ( U ) • → Sch (see Deﬁnition 4.3 ) is isomorphic to F U can : Γ( G ( U ) • ) op → RngSpcs. Using Lemma 3.18 , this concludes the pro of. □ Remark 4.6. By Prop ositions 4.4 , 4.5 , starting with a scheme S and a ﬁnite aﬃne cov er U of it, we can build tw o distinct paraschematic spaces that w e called S U and S 2 U . The former is a sc hematic space as in [ S2017a ] and has some go o d cohomological prop erties (see [ CM2019 ]). The latter might not b e schematic (see Remark 4.7 ), but its underlying ﬁnite top ological space arises from the poset of simplices of a regular 2 dimensional semisimplicial set, while this is not alwa ys the case for S U . Despite of these diﬀerences, in b oth cases (see [ S2017a ] and Prop osition 4.5 ) w e hav e that Sp ec S ( S U ) ∼ = Sp ec S ( S 2 U ) ∼ = S . In our next remark we p oint out that the parasc hematic space S 2 U w e obtained, while b eing parasc hematic, a priori might not b e in general sc hematic in the sense of [ S2017a ]. Remark 4.7. In [ S2017a ] a paraschematic space ( X , O X ) is said to b e schematic if for any p, q ∈ X , any p ′ ≥ p and an y i ≥ 0 the natural morphism H i ( U p ∩ U q , O X ) ⊗ O X ( U p ) O X ( U p ′ ) → H i ( U p ′ ∩ U p , O X ) is an isomorphism (see [ S2017a ] 4.1 and 4.2). A paraschematic space S 2 U as ab ov e is not alw ays schematic. Indeed, assume that in the co v ering U there are four distinct aﬃne schemes U 1 , U 2 , U 3 and U 4 suc h that U 1 ∩ U 2 =: Sp ec( A ), Sp ec( B ) := U 1 ∩ U 2 ∩ U 3 , Spec( C ) := U 1 ∩ U 2 ∩ U 4 and 16 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES that Sp ec( D ) := U 1 ∩ U 2 ∩ U 3 ∩ U 4 is not empty . The corresp onding simplices in G ( U ) • can b e depicted as follo ws. c • a • v / / e 13 @ @   • b e 23 ^ ^ d • @ @ where a, b, c, d corresp ond to U 1 , U 2 , U 3 , U 4 resp ectiv ely and v corresp onds to U 1 ∩ U 2 . W e denote as p, q the 2-simplices corresp onding to the upper and low er 3-clique in the ﬁgure. In this case, w e hav e that U a , U b , U c , U d , U v , U p and U q seen as op en subspaces of S 2 U are aﬃne ringed spaces in the sense of Salas [ S2017a , Deﬁnition 3.10] by 4.11 in [ S2017a ] (note they are acyclic b ecause of 2.13 of [ S2017a ]). No w, if S 2 U w ere schematic, then U v w ould b e aﬃne and schematic [ S2017a ] and in particular it should b e semiseparated [ S2017a , Prop osition 4.9]. How ever, O S 2 U ( U p ∩ U q ) ∼ = 1 is the trivial ring as U p ∩ U q = ∅ but O S 2 U ( U p ) ⊗ O S 2 U ( U v ) O S 2 U ( U q ) ∼ = B ⊗ A C ∼ = D con tradicting 4.12 (2) in [ S2017a ]. Therefore, S 2 U is not sc hematic. W e end this section with a remark regarding quasi-coheren t mo dules on (para) sc hematic spaces. Remark 4.8. Let S b e a scheme and qCoh( S ) the category of quasi-coherent mo dules on S [ Ha1977 , Chapter 2]. F or a given ﬁnite ringed space W , we also hav e the notion of quasi-coherent mo dule on it and w e denote by qCoh( W ) the category of quasi-coherent mo dules on W . Then, by faithfully ﬂat descent [ Vi2005 ], we ha ve an equiv alence of categories: qCoh( S ) ∼ = qCoh( S 2 U , O S 2 U ) W e leav e to the reader all the chec ks in v olved. 4.2. The category C 2 Sch . Let b e S b e a sc heme, U a ﬁnite cov ering by aﬃne op en subsc hemes of S . Recall we assume all sc hemes to b e quasi-compact and semi-separated. Moreov er, all graphs we consider are ﬁnite and b elong to either diGraphs ≤ 1 or Graphs ≤ 1. W e w ant to understand some key properties of the ﬁnite ringed space S 2 U as deﬁned in 3.20 , leading us ﬁrst to deﬁne the cate gory C 2 Sch , its lo calization and then to our main result of Sec. 4 Theorem 4.34 . The cov ering U determines the 2-dimensional semisimplicial set G ( U ) • , which has 1-sk eleton isomorphic to a digraph having at most one edge connecting each pair of distinct vertices (vertices corresp ond to op en sets in U , while edges to intersections of tw o op en sets). Recall that the ﬁnite ringed space S 2 U allo ws us to deﬁne tw o functors (see Sec. 3.3 ): F 2 U : P G ( U ) • ∼ = Γ( G ( U ) • ) → Rings , p i 7→ O S ( U i ) , p ij 7→ O S ( U ij ) , p ij k 7→ O S ( U ij k ) F U can : Γ( G ( U ) • ) op → RngSp cs , p i 7→ U i , p ij 7→ U ij , p ij k 7→ U ij k iden tifying with a small abuse of notation the ob jects of Γ( G ( U )) op with those in P G ( U ) • ∼ = Γ( G ( U ) • ) (see also Obs. 2.9 and 3.19 ). ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 17 Let T b e a 2-sub-semisimplicial set of G ( U ) • as in Defs. 3.3 , 3.14 . Notice that Γ( T ) op = Ξ, so that, by Prop. 3.11 ( F U can ) | Γ( T ) op is a gluing cub e, hence it corresp onds to a gluing datum (Defs. 3.1 , 3.2 , Sec. 3.1 ). Using the anti-equiv alence b et ween aﬃne schemes and rings and the fact that U consists of aﬃne schemes, we see that ( F U can ) | Γ( T ) op b eing a gluing cub e is equiv alent to F 2 U preserving pushouts, i.e. ﬁb ered copro ducts. Our purp ose is to deﬁne a category C 2 Sch , having ob jects constructed using the semisimplicial sets w e ha ve in tro duced so far, that we can localize at a certain class of morphisms to get a category equiv alent to the category of schemes. Heuristically , if S is a sc heme and U a ﬁnite aﬃne cov ering, we iden tify S with the ob ject ( G ( U ) • , F 2 U ) of C 2 Sch . Ho wev er, to deﬁne C 2 Sch and the equiv alence of category , we need a preliminary deﬁnition follo wed by a discussion. Deﬁnition 4.9. Let b e G be a graph and F : Γ( G • ) op → RngSp cs a gluing functor (Def. 3.10 ). W e deﬁne the the de gener ate exp ansion G • of G • to b e the 2-dimensional semisimplicial set obtained by adding to G • exactly one self loop for each v ertex of G and all the p ossible t wo dimensional simplices u → u → u , u → u → v , u → v → v and v → v → v suc h that u, v ∈ G 0 and u → v is an elemen t of G 1 . Notice that we are neither adding extra v ertices, nor edges, except lo ops. • u • u • v • v • • • u u v • • • u u u • • • u v v • • u v • • • v v v Figure 2. All the simplices of the degenerate expansion G • of G • : u → v W e call the added simplices de gener ate simplic es of G • . W e also deﬁne the the de gener ate exp ansion F of F to b e the functor F : Γ( G • ) op → RngSp cs as follows. 1) F | Γ( G • ) op = F 2) Let u, v ∈ G 0 and u → v ∈ G 1 . W e set F ( u → u ) := F ( u ) , F ( u → u → u ) := F ( u ) , F ( u → u → v ) := F ( u → v ) , F ( u → v → v ) := F ( u → v ) 3) F or each morphism u → ( u → u ) in Γ( G • ) its image F ( u → u ) = F ( u ) → F ( u ) = F ( u ) is deﬁned to b e the identit y , for each morphism ( u → v ) → ( u → u → v ) and ( u → v ) → ( u → v → v ), F ( u → u → v ) → F ( u → v ) and 18 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES F ( u → v → v ) → F ( v → v ) are deﬁned to b e the identit y and for eac h morphism ( u → u ) → ( u → u → v ) and ( u → u ) → ( v → u → u ), F ( u → u → v ) → F ( u → u ) and F ( v → u → u ) → F ( u → u ) are deﬁned to b e the op en em b eddings F ( u → v )  → F ( u ) and F ( v → u )  → F ( u ) resp ectively . Observ ation 4.10. In the notation of Deﬁnition 4.9 one can chec k that the colimit of the functors F and F are isomorphic. W e are ready to deﬁne the category C 2 Sch that we shall lo calize at a certain class of morphisms and compare to the one of schemes. Deﬁnition 4.11. W e deﬁne the category C 2 Sch as follo ws. Obje cts: the ob jects of C 2 Sch are pairs A = ( A , F A : Γ( A ) → Rings) where A is a 2-dimensional ﬁnite semisimplicial set isomorphic to the degenerate expansion G • of a given semisimplicial set G • , G a graph, and the functor Sp ec ◦ F A factors through the degenerate expansion of a gluing functor Γ( G • ) op → RngSp cs. Ar r ows: the morphisms of C 2 Sch are pairs ψ = ( f , ε ) : A = ( A , F A ) → B = ( B , F B ), where f : A → B is a morphism b et ween 2-dimensional ﬁnite semisimplicial sets and ε : F B ◦ Γ( f ) → F A is a natural transformation. Observ ation 4.12. W e notice some k ey facts regarding the category C 2 Sch . (1) Giv en an ob ject ( A , F A ) in C 2 Sch , the functor F A : Γ( A ) → Rings gives a sheaf on a base for the top ological space A ( P A ). Hence we can asso ciate to ( A , F A ) the ringed space ( A ( P A ) , O A ), where O A is obtained by from F by noticing that F A is a sheaf of rings on a base. This space is also parasc hematic, as one can readily see. (2) Giv en tw o ob jects A = ( A , F A ), B = ( B , F B ) in C 2 Sch , a morphism ψ = ( f , ε ) : A → B gives rise to a map betw een the corresponding parasc hematic spaces. Indeed f induces a contin uous map | f | : A ( P A ) → A ( P B ). F rom the natural transformation ε we get a sheaf morphism ¯ ε : | f | − 1 O B → O A and so, b y adjunction, a morphism ε ♭ : O B → | f | ∗ O A , see [ GW2010 ] page 55. Giv en an ob ject A = ( A , F A ) ∈ C 2 Sch , by applying the functor Sp ec S to ( A ( P A ) , O A ), obtained as in Obs. 4.12 , we get a scheme. Henc e, we can give the following deﬁ- nition. Deﬁnition 4.13. Let the notation b e as ab ov e. W e deﬁne the functor: Sp ec C : C 2 Sch → Sch , Spec C ( A , F ) := Sp ec S ( A ( P A ) , O A ) , the deﬁnition on the morphisms b eing clear. W e conclude this section with a key deﬁnition and an observ ation we will need in the sequel. Deﬁnition 4.14. Let S b e a scheme and U a ﬁnite cov ering by op en aﬃne sub- sc hemes of S . Deﬁne the functor F 2 U : P G ( U ) • ∼ = Γ( G ( U ) • ) → Rings , via F U can , the degenerate expansion of F U can . Deﬁne also: ( U ) 2 sch := ( G ( U ) • , F 2 U ) Observ ation 4.15. W e ha v e that ( U ) 2 sch is an ob ject of C 2 Sch and that Sp ec C (( U ) 2 sch ) ∼ = S by Observ ation 4.10 . ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 19 4.3. The category C 2 Sch and w eak equiv alences. W e wan t to localize the cat- egory C 2 Sch at a class of morphisms, con taining all isomorphisms and closed under comp osition, that we call, with a small abuse of terminology , a set of we ak e quiva- lenc es . Deﬁnition 4.16. Let us deﬁne the set of W in C 2 Sch as the set of morphisms ψ = ( f , ε ) such that the map ε ♭ (Obs. 4.12 ) is an isomorphism and Sp ec C ( ψ ) is aﬃne. As one can readily chec k W is a set of morphisms containing all isomorphisms and closed under comp osition, hence W is a set of weak equiv alences. W e shall refer to its elemen ts as we ak e quivalenc es . W e now record in the following prop osition some useful facts concerning the set of w eak equiv alences W . Prop osition 4.17. The fol lowing ar e true. 1) The functor Sp ec C sends we ak e quivalenc es to isomorphisms of schemes. 2) If A = ( A , F A ) ∈ C 2 Sch and Spec C ( A , F A ) =: S , we have that U A := { Sp ec( F A ( p )) } p ∈A 0 is an aﬃne op en c over of S and we have a c anoni- c al we ak e quivalenc e i A = ( f , ε ) : ( U A ) 2 sch  → A such that f is an inclusion of 2-dimensional semisimplicial sets and ε is the identity. 3) Using the notation of 2), let V = { V j } j ∈ J b e a ﬁnite c overing by op en aﬃne subschemes of S that is a r eﬁnement of U A . Assume that for al l p ∈ A 0 ther e exist a subset J p ⊆ J such that { V j } j ∈ J p is a c overing of Sp ec( F A ( p )) ⊆ S and the J p ar e a p artition of J . Then ther e exists a c anonic al we ak e quivalenc e ( V ) 2 sch → ( U A ) 2 sch that is an inclusion on the underlying 2-dimensional semisimplicial sets. Pr o of. 1) Consider a weak equiv alence ψ = ( f , ε ) : A = ( A , F A ) → B = ( B , F B ). Denote Spec C ( A ) =: X , Sp ec C ( B ) =: Y and consider the parasc hematic spaces ( A ( P A ) , O A ) and ( A ( P B ) , O B ) asso ciated to A and B as in Observ ation 4.12 (2). W e wan t to prov e that Sp ec C ( ψ ) : X → Y is an isomorphism. As isomorphisms are lo cal on target (see [ GW2010 ] App endix C for this notion), it suﬃces to show that for all v ∈ B 0 the morphism Spec C ( ψ ) − 1 (Sp ec( O B ( U v )) → Sp ec( O B ( U v )) is an isomorphism. By deﬁnition of w eak equiv alence, w e kno w that Sp ec C ( ψ ) is aﬃne. As a consequence Sp ec C ( ψ ) − 1 (Sp ec( O B ( U v )) is an op en aﬃne subscheme of X so Sp ec C ( ψ ) − 1 (Sp ec( O B ( U v )) → Spec( O B ( U v )) is a morphism of aﬃne schemes and therefore to chec k that it is an isomorphism it suﬃces to show that its induced map on global sections is a ring isomorphism. First of all, Sp ec C ( ψ ) − 1 (Sp ec( O B ( U v )) ∼ = Sp ec( O A ( | f | − 1 ( U v ))). Moreov er, as ψ is a weak equiv alence, ε ♭ is an isomorphism. Therefore, w e get the required isomorphism. W e shall now prov e 2). By deﬁnition, A ∼ = G • for some graph G and Sp ec ◦ F A factors through the degenerate expansion of a gluing functor As a consequence U A is a cov ering by op en aﬃne subschemes of Sp ec C ( A ). Here a commen t should b e made: strictly sp eaking, a cov ering by open embeddings of a scheme is a set and therefore, as it could happ en that the op en embeddings Sp ec( F A ( p ))  → S and Sp ec( F A ( q ))  → S are equal (so in particular Sp ec( F A ( p )) = Sp ec( F A ( q ))), we end up with a cov ering of S that a priori do es not hav e an element for eac h p ∈ A 0 . This is not substan tial as in this case we can for example replace one of the sc hemes in volv ed with another one isomorphic to it. 20 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES Consider no w the semisimplicial set G ( U A ) • . By construction we hav e a ϕ : G ( U A ) 0 ∼ = − → G 0 . Consider u, v ∈ G ( U A ) 0 and assume that ϕ ( u ) , ϕ ( v ) ∈ G 0 are connected b y and edge e ∈ G 1 suc h that F A ( e ) is not trivial. Then, as Spec ◦ F A factors through the degenerate expansion of a gluing functor we ha ve that u and v are connected by an edge as w ell. Con versely , if F A ( e ) is trivial or ϕ ( u ) and ϕ ( v ) are disconnected then u and v are disconnected. It follows that ϕ can b e extended to an inclusion of graphs G ( U A ) ⊆ G and therefore G ( U A ) • and G ( U A ) • are sub semisimplicial sets of G • and G • resp ectiv ely . Deﬁne f to b e the inclusion G ( U A ) • ⊆ G • just obtained. F or a giv en p ∈ G ( U A ) • w e deﬁne the ring mor- phism ε ( p ) : ( F A ◦ Γ( f ))( p ) → O S ( U p ) to b e the one induced by the isomorphism Sp ec( F A ( p )) and the op en subscheme U p corresp onding to it in S (note that this deﬁnition do es not require choices). As ( f , ε ) is trivially a w eak equiv alence this completes the pro of of 2). W e shall now pro v e 3) b y building explicitly a canonical weak equiv alence ( r , ε r ) : ( V ) 2 sch → ( U A ) 2 sch . T o ease the notation, let us denote as { U i ⊆ S } i ∈ I = A 0 the cov- ering U A and for each i ∈ I let us denote as { V ij ⊆ U i } j ∈ J i the sub cov er of U i as in the h ypothesis. The fact that V is a reﬁnement of U A together with the fact that eac h V ij is a subscheme of a particular U i allo ws us to deﬁne a graph morphism ˜ r : G ( V ) → G ( U A ) in the obvious w a y: ˜ r ( V ij ) := U i and ˜ r ( V ij ∩ V lk ) = U i ∩ U l (here w e are denoting with the in tersections their corresp onding edges if the considered in tersections are not empty). This map clearly extends to a map r : G ( V ) • → G ( U A ) • . Finally , we construct the required ε r considering the inclusions of aﬃne sc hemes V ij ⊆ U i , V ij ∩ V lk ⊆ U i ∩ U l and V ij ∩ V lk ∩ V mn ⊆ U i ∩ U l ∩ U m . □ Deﬁnition 4.18. Let S b e a scheme and U = { U k } k ∈ K b e a ﬁnite co v er of it b y op en but p ossibly not aﬃne subschemes. W e say that a ﬁnite reﬁnement V = { V j } j ∈ J of U b y aﬃne sc hemes is a c omplete aﬃne sub c over of U if for all k ∈ K there exists a subset J k ⊆ J such that { V j } j ∈ J k is a co vering of Sp ec( U k ) ⊆ S and the J k are a partition of J . Deﬁnition 4.19. Let S b e a scheme and V = { V j ⊆ S } j ∈ J , U = { U i ⊆ S } i ∈ I t wo ﬁnite op en aﬃne co verings of it. W e deﬁne the co v ering V × S U of S as V × S U := { U i × S V j ⊆ S } ( i,j ) ∈ I × J Observ ation 4.20. Notice that, as S is semiseparated, the co vering V × S U is a complete aﬃne subcov er of b oth U and V . Therefore, mimic king the proof of Prop osition 4.17 3) w e can deﬁne weak equiv alences p V : ( V × S U ) 2 sch → ( V ) 2 sch , p U : ( V × S U ) 2 sch → ( U ) 2 sch . 4.4. Sc hematic Lo calization of C 2 Sch . W e would like to lo calize the category C 2 Sch at weak equiv alences. One of the classical wa ys to do this is to pro ve that the given class of w eak equiv alences is either a righ t or a left m ultiplicative system. This is not how we will pro ceed, but, since we mo dify such construction, let us brieﬂy recall the deﬁnition of righ t multiplicativ e system, see [ SP , T ag 04VC ]. Deﬁnition 4.21. Let C b e a category . A set of arrows S of C is called a right multiplic ative system if it has the following prop erties: RMS1) The identit y of every ob ject of C is in S and the comp osition of tw o com- p osable elemen ts of S is in S . ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 21 RMS2) F or every solid diagram X / / t   Y s   Z / / W with s ∈ S , there exists an X (not unique in general), so that the diagram can b e completed to a comm utative dotted square with t ∈ S . RMS3) F or ev ery pair of morphisms f , g : X − → Y and s ∈ S with source Y such that s ◦ f = s ◦ g there exists a t ∈ S with target X , such that f ◦ t = g ◦ t . When the set of w eak equiv alences is a right (or left) multiplicativ e system, the usual calculus of fractions can b e used to construct an explicit mo del of the desired lo calization (see [ SP , T ag 04VB ] or the very nice exp osition in [ Bor1994 ]). In our case, how ever, the set of weak equiv alences W fails to b e a multiplica- tiv e system as we see in the next example, showing an explicit counterexample to prop ert y RMS3. Example 4.22. Let b e R X a comm utative ring and let X := Sp ec( R X ). Consider ob jects A = ( A , F A ) and B = ( B , F B ) where A and B are the degenerate expansions of G • and H • where G is the graph consisting of a single vertex u and H is a graph a → b consisting of t wo vertices joined by a single edge. F A and F B are deﬁned to b e the constant functors with v alue R X on the category of simplices of A and B resp ectively . Note that Sp ec C ( A ) and Sp ec C ( B ) are b oth isomorphic to X . Consider the tw o morphisms ψ a = ( f a , ε a ) , ψ b = ( f b , ε b ) : A → B having f a and f b deﬁned extending the unique inclusions sending u to a and b resp ectively and where ε a and ε b are obtained using the iden tit y R X → R X . Consider the map π : ( f π , ε π ) : B → A , where f π is the semisimplicial set map collapsing the tw o distinct vertices of B to the single vertex of u and ε π is deﬁned using the iden tity of R X . π is a w eak equiv alence and π ◦ ψ a = π ◦ ψ b but it is not p ossible to ﬁnd a semisimplicial set Q and a morphisms f s : Q → G ( U ) • suc h that f a ◦ f s = f b ◦ f s , so we get that that RMS3 is not satisﬁed b y the collection of w eak equiv alences. How ever, one can chec k that Sp ec C ( ψ a ◦ π ) = Sp ec C ( ψ b ◦ π ). As Sp ec C ( π ◦ ψ a ) = Sp ec C ( π ◦ ψ b ), we then observe that in our example a mo diﬁed v ersion of property RMS3 holds at the lev el of the scheme morphisms induced b y morphisms in C 2 Sch . This suggests a new deﬁnition of right m ultiplicativ e system, with a w eaker form of RMS3, that as we shall see, allows for lo calization. The previous example motiv ates the following deﬁnitions. Deﬁnition 4.23. W e say that tw o morphisms ψ , ψ ′ : A → B in C 2 Sch are schematic e qual if Sp ec C ( ψ ) = Sp ec C ( ψ ′ ). W e say that a diagram in C 2 Sch is schematic c om- mutative if all the morphisms app earing in it, having the same source and target, are sc hematic equal. By deﬁnition, a commutativ e diagram in C 2 Sch is also sc hematic comm utative. W e will need some useful prop erties of schematic equal morphisms. W e start with a tec hnical lemma. Lemma 4.24. Every morphism ψ = ( f , ε ) : A = ( A , F A ) → B = ( B , F B ) in C 2 Sch r estricts to a morphism ψ ′ = ( f ′ , ε ′ ) : ( U A ) 2 sch → ( U B ) 2 sch such that ψ ◦ i A = i B ◦ ψ ′ . 22 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES Pr o of. Note that if for a one dimensional simplex e ∈ A we hav e that F A ( e ) is not trivial (i.e. e b elongs to the subsemisimplicial set G ( U A ) • ⊆ A ), then F B ( f ( e )) must b e not trivial as well (i.e. f ( e ) b elongs to the subsemisimplicial set G ( U B ) • ⊆ B ), as there can b e no ring homomorphisms from the trivial ring to a non trivial ring. The same holds true for 2 dimensional simplices. As a consequence, f restricts to a map G ( U A ) • → G ( U B ) • and, as a consequence, so is ε . □ Remark 4.25. Explicitly note that if ψ = ( f , ε f ) , ψ ′ = ( g , ε g ) : A ⇒ B are sc hematic equal then there exists an isomorphism ϕ : | f | ∗ O A → | g | ∗ O A suc h that ε ♭ g = ϕ ◦ ε ♭ f . In addition, if A and B are of the form ( U ) 2 sch and ( V ) 2 sch for tw o given op en aﬃne co verings of tw o schemes X and Y we can take ϕ to b e the identit y . W e no w pro v e an important technical prop osition that can be interpreted as the fact that our weak equiv alences satisfy a mo diﬁed version of RMS2 and RMS3 where all the equalities required on the morphisms b y these t w o conditions are relaxed to sc hematic equalities. Lemma 4.26. The fol lowing holds true: 1) L et b e f : A → B a we ak e quivalenc e and let b e g : C → B a morphism of C 2 Sch . Then ther e exist Q ∈ C 2 Sch , a we ak e quivalenc e f ′ : Q → C and a morphism g ′ : Q → A such that g ◦ f ′ and f ◦ g ′ ar e schematic e qual. In addition, if g is a we ak e quivalenc e we c an c onstruct g ′ in such a way that g ′ is a we ak e quivalenc e as wel l. 2) Given two maps f , g : A ⇒ B , if ther e exists a we ak e quivalenc e s : B → Q in C 2 Sch such that s ◦ f is schematic e qual to s ◦ g then ther e exists a we ak e quivalenc e t : Q → A such that f ◦ t and g ◦ t ar e schematic e qual. Pr o of. W e b egin with the pro of of 1). Denote Y := Sp ec C ( B ), X := Spec C ( A ), Z := Sp ec C ( C ). Because of Lemma 4.24 we can assume that A = ( U A ) 2 sch =: ( A , F A ), B = ( U B ) 2 sch =: ( B , F B ) and C = ( U C ) 2 sch =: ( C , F C ). As f is a w eak equiv a- lence, w e know that Spec C ( f ) is an isomorphism. Accordingly , Spec C ( f )( U A ) = { Sp ec C (Sp ec( F A ( q )) } q ∈A 0 is a co vering of Y by op en aﬃne subschemes. As a con- sequence, U g − 1 U A := { Sp ec C ( g ) − 1 (Sp ec C (Sp ec( F A ( q ))) } q ∈A 0 is an op en (possibly non aﬃne) cov ering of Z : let V aﬀ b e a complete aﬃne sub cov er of it. W e deﬁne Q := ( Q , F Q ) := ( U C × Z V aﬀ ) 2 sch and f ′ := p U C . By Def. 4.19 f ′ is a weak equiv- alence. W e now need to deﬁne g ′ =: ( ˜ g ′ , ε g ′ ). W e deﬁne ˜ g ′ as the comp osition of the canonical maps G ( U C × Z V aﬀ ) • → G ( V aﬀ ) • → G ( U g − 1 U A ) • ∼ = G ( U A ) • = A . W e deﬁne ε g ′ as follows. F or every a ∈ Γ( G ( U C × Z V aﬀ ) • ), w e deﬁne the map ε g ′ ( a ) : F A ( ˜ g ′ ( a )) → F Q ( a ) as the ring homomorphism given b y the global sections of the scheme morphism obtained comp osing the op en embedding Spec( F Q ( a )) → Sp ec C ( g ) − 1 (Sp ec C (Sp ec( F A ( ˜ g ′ ( a )))) (recall that V aﬀ is a complete aﬃne sub cov er of U g − 1 U A ) with the map Sp ec C ( g ) − 1 (Sp ec C (Sp ec( F A ( ˜ g ′ ( a )))) → Sp ec C (Sp ec( F A ( ˜ g ′ ( a ))) obtained b y taking the pullback of Sp ec C ( g ) along the morphism Sp ec( F A ( ˜ g ′ ( a )))  → X f − → Y One can chec k that g ′ is the required morphism and that it is a weak equiv alence if g is a w eak equiv alence. W e turn to the pro of of 2). Denote Y := Sp ec C ( B ), X := Sp ec C ( A ), Z := ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 23 Sp ec C ( Q ). Again, w e can assume that A = ( U A ) 2 sch =: ( A , F A ) and that B = ( U B ) 2 sch =: ( B , F B ). Consider the co vering by op en (p ossibly non aﬃne) subsc hemes U f := Sp ec C ( f ) − 1 ( U B ) and U g := Sp ec C ( g ) − 1 ( U B ) of X . As s is a weak equiv a- lence, Sp ec C ( s ) is an isomorphism. The h yp othesis implies Sp ec C ( s ) ◦ Sp ec C ( f ) = Sp ec C ( s ) ◦ Sp ec C ( g ) so that Sp ec C ( s ) = Sp ec C ( f ). As a consequence U f = U g : con- sider a complete aﬃne subcov er V of it. Deﬁne Q := ( U A × X V ) 2 sch and t := p U A . One can c heck that these are the desired ob ject and morphism. □ Deﬁnition 4.27. A set of arro ws S of C 2 Sch is called a schematic right multiplic ative system , if it has the follo wing prop erties: sRMS1) The identit y of every ob ject of C is in S and the comp osition of tw o com- p osable elemen ts of S is in S . sRMS2) F or every solid diagram X t ′ / / t   Y s   Z s ′ / / W with s ∈ S , there exists an X (not unique in general), so that the diagram can b e completed to a schematic commutativ e diagram with t ∈ S . If s ′ ∈ S then one can complete the diagram in such a wa y that t ′ ∈ S as well. sRMS3) F or ev ery pair of morphisms f , g : X − → Y and s ∈ S with source Y such that s ◦ f is sc hematically equal to s ◦ g there exists a t ∈ S with target X , suc h that f ◦ t is schematically equal to g ◦ t . By Lemma 4.26 we hav e immediately the follo wing result, which is the k ey for the lo calization of C 2 Sch at W . Prop osition 4.28. The set W of we ak e quivalenc es in C 2 Sch is a schematic right multiplic ative system. W e are now ready to deﬁne the localized category C 2 Sch [ W − 1 ] mimicking the usual construction used in the case of lo calization at right multiplicativ e system (see for example [ Bor1994 , 5.2.4] or [ S2017a ] for a construction closer to our case). Deﬁnition 4.29. W e deﬁne the schematic lo c alization C 2 Sch [ W − 1 ] of C 2 Sch at the sc hematic multiplicativ e system W to b e the the category deﬁned as follows: 1) The ob jects of C 2 Sch [ W − 1 ] coincide with the ob jects of C 2 Sch . 2) The morphisms from an ob ject A to an ob ject B are obtained considering the set of zig-zags A s ← − P f − → B , where s is a weak equiv alence and f is a morphism of C 2 Sch , and quotienting with the equiv alence relation given b y declaring tw o zig-zags A s ← − P f − → B , A s ′ ← − P ′ f ′ − → B to b e equiv alent if there exists a zig-zag P u ← − P ′′ v − → P ′ suc h that u and v are weak equiv alences, s ◦ u and s ′ ◦ v are schematic equal and f ◦ u and f ′ ◦ v are schematic equal (see diagram 2 ). 24 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES (2) P ′′ u ~ ~ v ! ! P s   f * * P ′ s ′ t t f ′ A B 3) The comp osition of tw o zig-zags A s ← − I f − → B and B t ← − J g − → C is deﬁned to b e any zig-zag A s ◦ r ← − − K g ◦ h − − → C where K r − → I is a weak equiv alence and K h − → J is a morphism such that f ◦ r and t ◦ h are schematic equal (see diagram 3 ). (3) K r   h   I s   f   J t   g   A B C Observ ation 4.30. W e explicitly note that the category C 2 Sch [ W − 1 ] is not a lo- calization of categories in the usual sense, i.e. a category satisfying a univ ersal prop ert y such as the one stated in [ Bor1994 , 5.2.1]. In our case, only the prop erties stated in Prop osition 4.32 hold. Prop osition 4.31. C 2 Sch [ W − 1 ] is wel l deﬁne d. Pr o of. The pro of follo ws arguing formally as in the case of the lo calization of a category at a righ t m ultiplicative collection of morphisms (see for example [ Bor1994 ] 5.2.4 and the diagrams therein) using Prop erties sRMS2 and sRMS3 in place of the prop erties RMS2 and RMS3 of a right multiplicativ e system (see Deﬁnitions 4.27 and 4.21 ). Note that we require both u and v in point 2) of the previous deﬁnition to b e weak equiv alences (as done also in [ S2017a ], for example), while when lo calizing at right calculi of fractions usually a weak er hypothesis is required (indeed, in the case of sRMS2, using the terminology of Prop osition 4.26 1), if g is a weak equiv alence it follows that w e can take g ′ to b e a weak equiv alence as w ell). □ 4.5. The equiv alence of categories b etw een C 2 Sch [ W − 1 ] and sc hemes. T o es- tablish an equiv alence of categories betw een C 2 Sch [ W − 1 ] and the category of sc hemes w e ﬁrst deﬁne a functor b etw een them. Prop osition 4.32. We have the fol lowing: 1) Ther e exists a functor I W : C 2 Sch → C 2 Sch [ W − 1 ] that is the identity on the obje cts and which sends a morphism f : A → B in C 2 Sch to the (class of ) the zig-zag A id A ← − − A f − → B . ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 25 2) The functor I W factors thr ough a functor Sp ec C W : C 2 Sch [ W − 1 ] → Sch such that Sp ec C W ◦ I W = Sp ec C . This functor is the identity on the obje cts and sends the (class of ) a zig-zag A s ← − P f − → B to Sp ec C ( f ) ◦ Sp ec C ( s ) − 1 : Sp ec C ( A ) → Sp ec C ( B ) . Pr o of. The pro of follows mutatis mutandis arguing as in [ SP , T ag 04VK ] (or as in [ Bor1994 , page 186]) using the fact that w eak equiv alences are sent to isomorphisms b y the functor Sp ec C . □ Observ ation 4.33. Let S b e a scheme and U a ﬁnite cov ering of it by aﬃne op en subsc hemes. Then one can chec k that Sp ec C W (( U ) 2 sch ) ∼ = S . Theorem 4.34. The functor Sp ec C W : C 2 Sch [ W − 1 ] → Sch is an e quivalenc e of c ate gories, wher e Sc h is the c ate gory of quasi-c omp act semi- sep ar ate d schemes. Pr o of. W e w an t to pro v e that Sp ec C W is essen tially surjectiv e, faithful and full. It is essen tially surjective b ecause of Observ ation 4.33 . W e prov e that the functor is faithful. T o do this, we need to show that if tw o zig- zags F := A s ← − I f − → B and G := A s ′ ← − J f − → B are sent to the same morphism by Sp ec C W , then they are equiv alen t. W e can assume that A = ( U A ) 2 sch , B = ( U B ) 2 sch , I = ( U I ) 2 sch and J = ( U J ) 2 sch . As Sp ec C ( s ) and Spec C ( s ′ ) are isomorphisms, w e ha ve that the cov ers U A I := Sp ec C ( s )( U I ) and U A J := Sp ec C ( s )( U J ) are cov erings by op en aﬃne subschemes of X := Sp ec C ( A ). W e explicitly note that G ( U A I ) • ∼ = G ( U I ) • and G ( U A J ) • ∼ = G ( U J ) • . As a consequence, using the global sections of the aﬃne sc heme isomorphisms Sp ec C ( F 2 U A ( s ′ ( p ))) ⊇ Sp ec C ( s ′ )(Sp ec C ( F 2 U J ( p ))) ∼ = − → Sp ec C ( F 2 U J ( p )) , p ∈ G ( U J ) • w e can deﬁne a map s ′ A : ( U A J ) 2 sch ∼ = − → J . Analogously , we can construct a map s A : ( U A I ) 2 sch ∼ = − → I . Consider now the cov ering V := U A × X U A J × X U A I and the pro- jections p I := p U A I , p J := p U A J and p A := p U A deﬁned in Def. 4.19 . Let Q := ( V ) 2 sch . Then the zig-zag I p I ← − Q p J − → J exhibits an equiv alence b etw een F and G . Indeed, using the very deﬁnition of the ob jects and the morphisms we constructed we can c heck that s ◦ p I and s ′ ◦ p J are schematic equal and that f ◦ p I and g ◦ p J are sc hematic equal. W e no w pro ve that Sp ec C W is full. Consider tw o ob jects A, B ∈ C 2 Sch [ W − 1 ] and an y map f : X := Sp ec C ( A ) → Sp ec C ( B ) := Y b etw een them. W e need to pro ve that there exists a zigzag F := A s ← − Q f ′ − → B such that Spec C W ( F ) = f . Because of Lemma 4.24 we can assume that A = ( U A ) 2 sch , B = ( U B ) 2 sch . Con- sider the cov ering f − 1 ( U B ) by open (p ossibly non aﬃne) subschemes of X . T ak e a complete aﬃne cov er Q of it, consider the co v ering V := U A × X Q (that is still a complete aﬃne subcov ering of f − 1 ( U B )) and deﬁne Q := ( V ) 2 sch , s := p U A : Q ≃ − → A . As V is an op en aﬃne sub cov er of f − 1 ( U B ), w e hav e an in- duced semisimplicial morphism ˜ f : G ( V ) • → G ( f − 1 ( U B )) • ∼ = G ( U B ) • . Using the fact that the functor Hom Sch ( − , Sp ec C ( B )) is a sheaf [ GW2010 , Prop osition 26 ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 3.5] we obtain, for all p ∈ Γ( G ( V ) • ) a scheme morphism b etw een aﬃne sc hemes X ⊇ Sp ec( F 2 V ( p )) → Sp ec( F 2 U B ( ˜ f ( p ))) ⊆ Y factoring through an elemen t of f − 1 ( U B ) and gluing to f . W e can use these morphisms, taking global sections, to deﬁne a map ε f : F 2 U B ◦ Γ( ˜ f ) → F 2 V . W e deﬁne f ′ := ( ˜ f , ε f ). The zig-zag A s ← − Q f ′ − → B is the one needed to complete the pro of as one can c hec k that Sp ec C ( f ′ ) = f ◦ Sp ec C ( s ). □ 4.6. Finite spaces and manifolds. In this section w e mak e some remarks re- garding the category of diﬀeren tiable manifolds. Let M be a diﬀerentiable manifold and let us consider, with a small abuse of notation, M also as the ringed space ( M , C ∞ M ), where C ∞ M is the sheaf of diﬀeren- tiable functions on M . The algebra of global sections C ∞ M ( M ) allows us to recov er the p oin ts of M via the Milnor exercise (see [ KM1993 , N2003 ]). More precisely , we sa y that a maximal ideal m ⊆ C ∞ M ( M ) is r e al if C ∞ M ( M ) / m ∼ = R . The set of real maximal ideals Sp ec m ( C ∞ M ( M )) is called the r e al sp e ctrum of C ∞ M ( M ) and we hav e the follo wing natural corresp ondence (Milnor’s exercise): (4) Hom( C ∞ M ( M ) , R ) ∼ = M resem bling the corresp ondence b et w een C -points of a complex aﬃne sc heme ( S, O S ) and the maximal sp ectrum of O S ( S ), the global sections of O S . The real sp ectrum of a diﬀeren tiable manifold can then b e given a topology , so that the correspondence ( 4 ) is an homeomorphism. W e hav e even more similarities b etw een the category of smo oth manifolds (that w e shall assume from now on to hav e a ﬁnite atlas) Man and aﬃne sc hemes (see [ NS2003 ]). In fact: • There is fully faithful contra v ariant functor Man → R -algebras, M 7→ C ∞ M . • W e can deﬁne a functor Sp ec m : R -algebras → RngSp cs such that Sp ec m ( A ) ∼ = M , for A ∼ = C ∞ M for some manifold M . Ho wev er, as p ointed out in [ NS2003 ], the category of smooth manifolds lac ks some desirable prop erties that the category of sc hemes has. Hence to obtain results similar to the ones of [ S2017a ] and their generalization in Section 4 one should then mo ve to the category of diﬀer entiable sp ac es as in [ NS2003 ]. Without getting into the tech nicalities of such construction is worth noticing that most of the treatmen t in Sec. 3 can b e extended with no diﬃculty to the case of smo oth manifolds and diﬀeren tiable spaces, in particular: • The concept of gluing data and gluing cub e in Sec. 3.1 . • Giv en a smo oth manifold S and an op en cov ering U , the construction of the t wo ringed spaces S U , S 2 U and the functor F 2 U in Sec. 3.3 . As for the category C 2 Man , the construction is more in volv ed, w e make some commen ts b elow. Remark 4.35. W e could deﬁne a category C 2 Man b y rep eating verbatim the con- struction of the category C 2 Sch and getting results analogue to the ones of Section 4 using diﬀerentiable algebras as in [ NS2003 , page 30] in place of Rings and the functor Spec m of [ NS2003 , page 44] in place of Sp ec. Giv en a manifold M admitting a ﬁnite atlas U , one could then rep eating the reasoning of Observ ation 4.33 to get an ob ject of C 2 Man . W e believe ob jects of the category C 2 Man to b e useful for appli- cations: w e leav e the study of this category , and more in general the diﬀerentiable manifolds and spaces case, to future work. Similarly , w e observ e that ev erything we ON GLUING DA T A, FINITE RINGED SP ACES AND SCHEMES 27 ha ve stated for diﬀerentiable manifold could b e extended to their sup er analogues [ CC2011 ]. W e end this section with an observ ation, which is imp ortan t for the study vector bundles on graphs in [ FSZ2026 ]. Observ ation 4.36. Giv en a diﬀere n tiable (or complex) manifold M , there exists a natural correspondence betw een isomorphism classes of C ∞ (or holomorphic) v ector bundles equipped with a ﬂat connection and isomorphism classes of lo cally constan t shea ves of real (complex) vector spaces on M (see for example [ V oi2002 ] Section 9.2). In particular, if M is a diﬀeren tiable manifold and ( V , ∇ ) is a vector bundle on it with a ﬂat connection, the desired locally constan t sheaf L V of real v ector bundles on M in this corresp ondence is ker( ∇ ) (interpreted as the lo cal syste m of the ﬂat sections of V , that is the ones annihilated by ∇ ) and we hav e V ∼ = L V ⊗ O M . Now, ﬁx a diﬀeren tiable manifold M , a vector bundle together with a ﬂat connection ( V , ∇ ) and assume that there exists a ﬁnite op en co ver U = { U i ⊆ M } of M trivializing V and such that L V obtained as before is isomorphic to a constant sheaf of vector spaces when restricted to eac h open submanifold of the cov er U , that is L V | U i ∼ = R n for all i and a giv en n ∈ N . Consider no w the space ( M 2 U , O M 2 U ) considered, for example, in Deﬁnition 3.20 . • V ∼ = L V ⊗ O M deﬁnes a lo cally free mo dule V M 2 U ⊗ O M 2 U where V M 2 U is a sheaf of R -v ector spaces on M 2 U . • V M 2 U is equiv alen t to the datum of a presheaf of vector spaces on the p oset P G ( U ) • where all the restriction morphisms are isomorphisms. The latter com binatorial datum amounts to a choice of an n -dimensional vector space F v for each vertex v of the graph G ( U ), of an isomorphism F v ∼ = F w for eac h edge joining t wo v ertices v and w of G ( U ) and some cocycle conditions coming from the combinatorics triple intersections of the elements of the co ver U (there are no such cocycle conditions if we ha v e a cov er with empty triple in tersections). As a consequence, our treatment of vector bundles on graphs in [ FSZ2026 ] inspired b y the work [ BY2024 ] app ears to b e, under suitable assumptions, the discrete anal- ogous of the study of v ector bundles with a ﬂat connection (see also [ DM1994 ]). 4.7. Homology and Cohomology. W e discuss the homology and cohomology of the t wo ﬁnite ringed spaces examined in Subsection 4.1 ( S U , O S U ) and ( S 2 U , O S 2 U ). Assume that X is a ﬁnite top ological space and that F is a sheaf of ab elian groups on it. Then giving a sheaf of ab elian groups on X amounts to giving a presheaf of ab elian groups on the p oset asso ciated to X by Prop osition 2.14 . Recall that this amoun ts to the datum of an ab elian group F ( U p ) for each p ∈ X together with morphisms F ( U p ) → F ( U p ) for each p ≤ q . F ollowing [ CM2019 , 1.6.1] we deﬁne shea ves C i F and cosheav es C i F on X as C i F ( U ) = Y ( x 0 <...

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment