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Annex. Theory of Sheaves. Notion of Coherence

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Annex. Theory of Sheaves. Notion of Coherence Annex. Theory of Sheaves. Notion of Coherence In this Annex we collect general facts about sheaves to that extent as is necessary for the purposes of this book. We mention [TAG], [FAC] and [TF] as standard references related to the material here. Parts of this Annex overlap with Chapter A.O of [TSS]. We introduce the notion of a sheaf in the way we learned it in HIRZEBRUCH'S lectures held in 1954 at MUnster. Thus for us sheaves are primarily covering spaces (espaces etales). Needless to say that in many concrete cases we construct sheaves by means of presheaves. "Tantot Ie point de vue des faisceaux est plus commode, tantot c'est Ie point de vue des espaces etales" [CAR], p.825. Main emphasis is put on the notion of a coherent sheaf. The yoga of such sheaves is developed as in [FAC]; of utmost importance for Complex Analysis is the Extension Principle 4.3. §O. Sheaves We introduce the notion of a sheaf of sets and discuss basic properties. By X, Y we denote arbitrary topological spaces. 1. Sheaves and Morphisms. A pair (g, n) conslstmg of a topological space [/ and a local homeomorphism n: [/ -+ X from [/ into X is called a sheaf (of sets) on X. Instead of (g, n) we mostly just write Y. It follows that the projection n is open and that every stalk ~:=n-l(x), XEX, of [/ is a discrete and closed subset of Y. If ([/', n') and (g, n) are sheaves on X a continuous map ({J: [/' -+[/ is called a (sheaf) morphism if no ({J = n'. Then ({J maps stalks into corresponding stalks and we have induced stalk maps ({Jx: g; -+~, XEX. Since n', n are local homeomorphisms each morphism ({J: [/' -+[/ clearly is a local homeo­ morphism and in particular an open map. If ({J: [/' -+ [/ and "': [/ -+ [/" are morphisms then '" 0 ({J: [/' -+ [/" is likewise a morphism. Since the identity map id: [/ -+[/ is a morphism the sheaves on X form a category. The simplest sheaves are topological product spaces X x M together with the projection X x M -+ X, where M denotes an arbitrary set with discrete topology; such sheaves are called constant. 224 A. Theory of Sheaves. Notion of Coherence 2. Restrictions, Subsheaves and Sums of Sheaves. If Y is a topological subspace of X each sheaf (9; n) on X induces the sheaf (n- 1(y), nln-l(Y)) on Y where n-l(y) carries the relative topology. This sheaf is called the restriction of !/ to Yand is denoted by !/I Y or Yr. A subset :T of a sheaf 9; equipped with the relative topology, is called a subsheaf of !/ whenever (g; nl:T) is a sheaf on X. Obviously :T is a subsheaf of !/ if and only if :T is an open subset of Y'. For each subsheaf :T of !/ the injection :T --+!/ is a morphism. If (Y, n), (!/: n') are sheaves on X, we provide the fiber product !/EB!/':={(P,p')E!/X!/': n(p)=n'(p)} = U(Y~xg;) xeX with the relative topology in !/ x !/'. Then the map !/ EB!/' --+ X, (p, p')l-+n(p) is a local homeomorphism, hence !/EB!/' is a sheaf over X. This sheaf is called the direct sum or the WHITNEY sum of !/ and !/'. Analogously one defines the WHITNEY sum 9;. EB ~ EB ... EB g; of finitely many sheaves. We write !/P for the p-fold WHITNEY sum of !/ with itself. 3. Sections. Hausdorff Sheaves. A continuous map s: Y --+!/ from a subspace Y of X into a sheaf !/ on X is called a section over Y in !/ if nos=idy. It is customary to write Sx for the value of sat XEY and to call SxE~ the germ of s at x. The set of all sections over Yin !/ will be denoted by !/(Y). Sections over open sets are local homeomorphisms. The family {s(U): U open in X, SE!/(U)} forms a basis for the topology of 9; this especially implies: Each point pE!/ is the germ Sx of a section SE!/(U) over an open neighborhood U of x:=n(p). Each sheaf morphism q>: !/' --+!/ induces for each subspace Y of X the map !/'(y)--+!/(Y), s'--+q>0s'. It can be easily shown: A map q>: !/' --+!/ is a sheaf morphism if, for each point p' E!/', there exists an open set U c X and a section s' E!/'(U) with p' ES'(U) so that the map q> 0 s': U --+!/ is a section in !/ over U. A sheaf !/ is called a HAUSDORFF sheaf, if the sheaf space !/ is HAUS­ DORFF. If S, t are sections in any sheaf !/ over an open set U the set {XE U: SX = tJ always is open in U. If in addition !/ is a HAUSDORFF sheaf this set is also closed in U. This shows: If two sections s, tE U of a HAUSDORFF sheaf !/ have the same germ at one point XE U, then these sections coincide in the connected component of x in U. § 1. Construction of Sheaves from Presheaves 225 Thus for HAUSDORFF sheaves a kind of "principle of analytic con­ tinuation" holds. The standard example is the sheaf of germs of hoI om or­ phic functions on a domain in <en (cf. 1.1.1). § 1. Construction of Sheaves from Presheaves Sheaves are frequently constructed from data which are called presheaves. In this section we will describe carefully this very important device. 1. Presheaves. Suppose that to every open set U in X there is associated some set S(U), we demand that S(0) is the one point set of maps of the empty set into itself. We further postulate that for every pair of open sets V, U in X with VC U we have a "restriction map" p~: S(U)-l-S(V) satisfying pg=idu and p:;'op~=p~, whenever We V c U, where W is open too. Then the family S: = {S (U), p~} is called a presheaf on X. Thus a presheaf on X is a contravariant functor from the category of open subsets of X (with inclusions as morphisms) to the category of sets (with set theoretic maps as morphisms). If S' = {S' (U), p;P} is another presheaf on X a presheaf map <P: S' -l- S is a family {<Pu} of maps <Pu:S'(U)-l-S(U) such that for all pairs of open sets U, V with VC U we have <PvO p~u = p~o<Pu. Clearly the presheaves on X form a category. Every sheaf fI' on X gives rise to its canonical presheaf (fI'(U), r~), where r~(s):=slV for sEfI'(U) denotes the canonical restriction. Every sheaf mor­ phism fI"-l-fI' induces in an obvious manner a presheaf map between the canonical presheaves, it is an easy exercise to show that in this way we obtain a covariant functor of the category of sheaves into the category of presheaves. 2. The Sheaf Associated to a Presheaf. Every presheaf S = {S( U), p~} on X gives rise in a natural way to a sheaf fI' which is constructed as follows: for each point XEX the subsystem (S(U), p~, XE U) is directed with respect to inclusion. Thus the direct limit ~:=lim S(U) and the maps p~: S(U)-l-~ -­xeU are well-defined. Each element SES(U) thus determines an element sx: = p~ s of ~ called the germ of s at x. Every point of ~ is a germ. If V, V' are open neighborhoods of x two elements SES(V), s' ES(V') determine the same germ at x, i.e. Sx = s~, if and only if there is an open neighborhood We V II V' of x such that p:;' s = p!; s'. 226 A. Theory of Sheaves. Notion of Coherence Let Y be the union U ~ and let n: Y --> X map points of ~ to x. The xeX topology of Y is defined by means of a basis. To each element SES(V) we attach the map su: V --> Y, XI->Sx given by its germs. The family {su(V): V open in X, SES(V)} forms the wanted basis for the topology on Y. Then n is a local homeomorphism and hence Y is a sheaf on X. We call Y the sheaf associated to the presheaf S; all maps su: V -->Y are sections in Y. Let c]J: S'-->S be a presheaf map where S'={S'(V),p~u} and c]J={c]Ju}. Then c]J induces a sheaf morphism rp: Y' --> Y of the associated sheaves in the following way: For PE~' choose V and SES'(V) with p~u(s)=p and put rp(p): = p~ c]Ju(s). This definition is independent of the choice of V and s, the map rp: Y' --> Y is a sheaf morphism. A direct verification shows that we constructed a covariant functor from the category of presheaves into the category of sheaves. Remark: In applications the data S(V) of a presheaf are sometimes given only for a subfamily {V} of open sets in X, which is a base of the topology of X. To any such "general pre sheaf" one can associate in exactly the same way as above a sheaf. In this book we encounter this situation twice: in 1.4.7 the definition of higher dimensional direct image sheaves is given by using only those open sets (in Y) which are STEIN spaces; in 10.3.1 the definition of sheaves of polydisc modules is performed by using only those open sets ~ E which are polydiscs.
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