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 § 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 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. 3. Canonical Presheaves. Let Y be the sheaf constructed from a presheaf S={S(V),p~}. For each element SES(V) we have the section su: V-->Y', XI->Sx of Y over V. In this way we get a map IU: S(V)-->Y(V). One has no trouble verifying that the family (Iu) is a presheaf map from S into the canon• ical presheaf (Y(V), rF) of Y, which gives rise to the identity sheaf morphism id: Y --> Y. In general lu is neither injective nor surjective. A presheaf S is called a canonical presheaf, if all maps lu are bijective. We have the following basic Proposition: The presheaf S=(S(V), p~) is canonical if and only if for all open sets V in X the following two conditions are satisfied: (Y1) If S, tES(V) are such that there exists an open covering {Val of V with pg.s=pg.tfor all r:t.., then s=t. (Y2) Given an open covering {Va} of V and elements saES(Va) satisfying pg:nupsa= pg~nupsp for all r:t.., {J, there always exists an element SES(V) such that pga s = Sa for all r:t... Proof: It is easy to show that these conditions are necessary. Conversely we first remark: If (Y1) holds then all maps IU are injective. Take s, tES(V) and assume lu(S)=lu(t). Then sx=tx for all XEV, therefore each point XEV § 1. Construction of Sheaves from Presheaves 227 has an open neighborhood UxcU with pgxs=pgJ Thus s=t by (9"1). - Next we show: If (9"1) and (9"2) hold then all maps IU are surjective. Take a section h in 9" over U. We can find an open covering {Ua} of U and elements S"ES(U,,) such that luJS,,) = hlUa for all ex. This implies: IU"n u/pg:n Up sa) = lu"nup(P~nupsp) for all ex, p. Since 'U"nUp is already shown to be injective we conclude pg:n Up s" =pg!nupsp for all ex, p. Thus by (9"2) there exists an element SES(U) such that pg"s=sa for all ex. Since 1(8)1 U,,= luJSa) = hi Ua, we conclude I(s)=h. 0 The conditions (9"1) and (9"2) were first stated in [FAC], p.200/201. In the literature a sheaf is often defined as a canonical presheaf, e.g. in [TF], p. 109. In view of the Proposition one usually identifies a canonical presheaf with the canonical presheaf of its associated sheaf 4. Image Sheaves. Sheaves are frequently constructed using the following device: Starting from a sheaf (resp. from sheaves) one passes to the canoni• cal presheaf (resp. presheaves), constructs a new presheaf from these data and goes back to the associated sheaf. This principle is used for example to define the notion of an image sheaf, which plays a key role in this book. Let f: X -+ Y be a continuous mapping from X to Y, let 9" be a sheaf on X. To every open set V c Y we associate the (possibly empty) set 9"(f-1(V)). If V' c V we have the restriction mapping p~,: 9"(f-1(V))-+9"(f-1(V')) for sections. Then it is clear that the family (9"(f- 1(V)), p~,) is a presheaf on Y. A simple verification yields that this is a canonical presheaf; the associated sheaf is denoted by f*(9") and is called the image sheaf of 9" with respect to f Due to the natural bijection 9"(f-1(V))~ f*(9")(V) we always identify f* (9")(V) with 9"(f- 1 (V)). Every germ tEf*(9")!(x)' XEX, is a germ of a section in 9" along the fiber f- 1(f(x)) and is represented by a section sE9"(f- 1(V)), where V is an open neighborhood of f(X)E Y. The section s determines a germ SxE~ which is independent of the choice of the representation and uniquely determined by t. Thus it is clear: For every point XEX there exists a natural germ map Ix: f*(9")f(x)-+~' If f* is a covariant functor from the category of sheaves on X into the category of sheaves on Y. We call f* the image functor. 228 A. Theory of Sheaves. Notion of Coherence If, along with f, there is given another continuous map g: Y --+Z from Y into a topological space Z then one has sheaves (g0f)*(9') and g*(f*(9'» on Z. Since g* (f* (9'» (W) = 9'(f- l(g-l(W» = 9'((gof)- l(W» = (go f)* (9') (W) for every open set Win Z we see: g*(f* (9'»= (go f)* (9'), furthermore one easily sees that g*(f*(cp»=(g0f)*(cp) for all sheaf mor• phisms cp: 9" --+ fI'. Let us emphasize that image sheaves already occurred in 1946 in LERAY's first note on sheaves (cf. Introduction of this book). § 2. Sheaves and Presheaves with Algebraic Structure In all applications the stalks ~ of a sheaf carry algebraic structures, which depend continuously on xeX. Most important are sheaves 9' of d-mod• ules, where d is a sheaf of rings on X. These notions along with basic properties are carefully explained in this section. We put our main emphasis on the category of d-modules, in function theoretic applications d always is a sheaf (!) of germs of holomorphic functions on a complex space which even is a sheaf of local 1. Sheaves of Groups, Rings and d-Modules. A sheaf 9' on X is called a sheaf of abelian groups, if each stalk ~, xeX, is an (additively written) abelian group such that the "subtraction map" 9' (f; 9' --+ !/', (p, q)1-+ P - q is continuous (recall from 0.2, that for each element (p, q)e9' (f; 9' we have p, qe~ with x:=n:(p)=n:(q), so that p-qe~ is well-defined). If 9' is a sheaf of abelian groups and Ox is the neutral (= zero) element in ~, then the map 0: X --+S, Xl-+Ox is a section in 9' over X. We call Oe9'(X) the zero section. The set Supp9':={xeX: ~=I=Ox} is called the support of 9'. For each open set U in X the set 9'(U) of sections in 9' over U is an abelian group: for s, te9'(U) one defines s-te9'(U) by (s-t)x:=sx-tx, xeU. § 2. Sheaves and Presheaves with Algebraic Structure 229 A sheaf .91 of abelian groups is called a sheaf of (commutative) rings, if, along with the additive group structure, there is a continuous "multipli• cation map" d(£Jd--+d, (a,b)Hab which provides every stalk d x with the structure of a commutative ring. We always assume that every stalk d x has a multiplicative identity element 1x depending continuously on x (i.e. XH 1x is a section in d); the case 1x=Ox is not excluded, obviously 1x=l=Ox for all xESuppd. Let .91 be a given sheaf of rings. A sheaf [I' of abelian groups is called a sheaf of modules over .91, or simply an d-sheaf or an d-module, if a sheaf morphism .91 (£J [I' --+ [I' is defined such that every stalk ~ becomes an d x- module. Obviously .91 is an d-module. As in the case of sheaves of groups the algebraic structure on a sheaf induces the same kind of structure on the sets of sections via point-wise definitions. Thus for each open set U in X the sets d(U) resp. [I'(U) are likewise rings resp. d(U)-modules if .91 is a sheaf of rings and [I' an .91- module. If ~, ... ,.Sj, are d-modules then their WHITNEY sum ~(£J ... (£J.Sj, is an d-module with operations defined component-wise. In particular for all natural numbers p;;:.: 1 the sheaf d P : = .91 (£J ... (£Jd with p summands is an d-module. 2. The Category of .91 -Modules. Quotient Sheaves. By .91 we denote a sheaf of rings given once and for all; in all considerations Yo [1", [I'" are .91- modules. A sheaf morphism cp:[I"--+[I' (cf. 0.1) is called an d-morphism, if each induced stalk map cpx: fI'; --+~ is an dx-module homomorphism, XEX. Then clearly we can say: The .91 -modules over X along with the .91 -morphisms form a category. A subsheaf:Y of [I' is called an d-submodule of Yo if every stalk !Y,. is an dx-submodule of ~. If we have an dx-submodule 1'" of ~ for every XEX, their union U Tx is an d-submodule of [I' if and only if this set is open in !I'. It follows immediately: Let [1'; [I'" be d-submodules of !I'. Then their sum [1" + [I'": = U(fI'; + fI';') XEX and their intersection [1" n [I'": = n(fI'; n fI';') are likewise d-submod- ules of [1'. XEX If cp: [1" --+ [I' is an .91 -morphism, the kernel and the image of cp, i.e. :ffe'tCP:= UKercpx and Ymcp:= U Imcpx' XEX XEX are d-submodules of [1" and [I' respectively. We call cp a monomorphism resp. an epimorphism resp. an isomorphism, if :ffe't cp = 0 resp. if Ym cp = [I' resp. if :ffe't cp = 0 and Ym cp =!I'. It follows that each .91 -isomorphism [1" --+ [I' is a homeomorphism of the underlying sheaf spaces. 230 A. Theory of Sheaves. Notion of Coherence A system of d -sheaves and d -morphisms 'C C{Ji-l ,c C{Ji ,c ···~Ji_l ~Ji-----+Ji+l ~ ... , iElL, is called an d-sequence. Such a sequence is called exact at 9';, if Jm({Ji_l =:fle't ({Ji' An d -sequence is said to be exact if it is exact at every 9';. An important role will be played by short exact d -sequences, i.e. exact d• sequences of the form o~Sf" -'4ff ~ff" ~O, O:=zero sheaf: = sheaf of O-modules. Every d -morphism ({J: ff' -4- ff gives rise to the short exact d -sequence o ~ ffe't ({J ---4 ff' ~ JrN ({J -4- O. For each d-submodule ff' of ff we consider the union fflff':= U~/~' of all dx-quotient modules, XEX. We define the map x: ff-4-fflff' stalkwise by the canonical quotient homomorphisms. We provide ff Iff' with the quotient topology: thus a set in ff Iff' is open if and only if its x• inverse in ff is open. Then the natural projection ff Iff' -4- X is a local homeomorphism onto X. Hence we see: ff Iff' is an d-module and x: ff -4-ff Iff' is an d-epimorphism with :fle't X= ff'. We call ff Iff' the quotient sheaf of ff by ff'. We have the short exact d-sequence O~ff' ---4ff .J:...."fflff' ~o. We have canonically induced exact ff(U)-sequences 0-4-ff'(U)-4-ff(U)~(fflff')(U) for all open sets U in X; however in general the map ff(U)-4-fflff'(U) is not surjective. Any d-morphism tjJ: ff-4-ff" gives rise to the short exact d-sequence O~Jj'N tjJ-4-ff" ~ff"jJm tjJ~O, the d-module C(}o:£elt tjJ: = ff"jJm tjJ is called the cokernel of tjJ. D d -submodules of d itself are called sheaves of ideals, or for short ideals in d. For every ideal Jed the product J. ff is defined stalkwise by J x ' ~, where the latter denotes the dx-submodule of ~ generated by all germs axsx' axEJx' SxE~. Since the set J·ff obviously is open in ff, the product J·ff is an d-submodule of ff. For any ideal J in d the quotient sheaf d jJ is a sheaf of rings on X. Every d jJ-module ff is, via the epimorphism d -4-S;{ jJ, an d-module. This device of "changing rings" is an important technique and will be discussed in detail in Section 4.3 of this Annex. 3. Presheaves with Algebraic Structures. A presheaf S =(S(U), rF) is called a presheaf of abelian groups if S(U) is always an abelian group and rF is always a group homomorphism. A presheaf of rings A =(A(U), rF) is defined analogously. The sheaf ff resp. d associated to such a presheaf is a sheaf of abelian groups resp. of rings: one just carries over the algebraic structure via the direct limit maps, the continuity of the operations is evident. §2. Sheaves and Pres heaves with Algebraic Structure 231 We denote by A a fixed presheaf of rings with associated sheaf .-;,1. A presheaf S is called a presheaf of A-modules, or simply an A-presheaf if every set S(U) is an A(U)-module and ifrf(as)=rf(a)rf(s) for all aEA(U), SES(U), V c U. The associated sheaf !I' is an .s1-sheaf. On the other hand for every d-sheaf!l' the canonical presheaf (!I'(U), rf) is an (. .w'(U), rf)-presheaf. A presheaf homomorphism iP: S' ~S, iP=(iPu), is a presheaf map where every iPu is a homomorphism of the underlying algebraic structure. The induced map 4. The Functor :Yt'<»n. Let !I",!I' be sli'-modules. The set Homx(!I",!I') of all d -morphisms !I" ~!I' obviously is an abelian group; it becomes an d(X)-module, if we define the product of aEd(X) with :Yt'omd(!I", !I') (U) = Homu(Y{;, 51;;), :Yt'omd(d, .9");:::; !I'. Warning: The d(U)-modules Homd(u) (!I"(U), !I'(U» cannot be used for the definition of the ~m-sheaf because, among other things, there are no restriction maps rf· 0 Each element of :Yt'omd(!I", !I')x is the germ Sx of a section sEHomu(Y{;,5I;;) in a suitable neighborhood U of x. Two such sections induce the same dx-homomorphism !I'; -> Y;,. Thus we have a canonical d~-homomorphism p: :Yt'm""zd(!I", !I')x ~ Homd )!!';, y;') which, in general, is not bijective. In 4.4 we shall give a sufficient condition for the bijectivity of p. A pair l/J: 5' ~ !I", X:.9" ~ 5 of d -morphisms evidently induces an d-morphism :Yt'£Y»I:<1(l/J, X): :Yt'N'/~<1(!I",!I') -+ :Yt'mnd(5', 5). Clearly :Yt'o.-mg; is a 232 A. Theory of Sheaves. Notion of Coherence functor in the category of d-modules which is contravariant in the first and covariant in the second argument. Moreover :ifo')Jz.r<1 is left exact in the sense that each exact sequence 0-+5'-+5 -+5" gives rise to an exact sequence 0-+ :ifom.r<1(Y, 5')-+:ifo')Jz.r<1(Y, 5)-+:ifooPJlJ;((Y, 5"). Furthermore we have canonical isomorphisms :ifoom.r<1(9", 9'E85) ~ :ifom.", (9", 9')E8:if£wFi.r<1(9", 5), .Yl'mJ'l.r<1(9" E85', 9') ~ :ifo~",(9", 9')E8:ifmn.r<1(5', 9'). We note, however, that another well known property of the usual functor Hom is not valid for :ifom.r<1: an exact d-sequence 9"-+9'-+9'''-+0 in general does not induce an exact d -sequence 0-+ :ifo.m.r<1(9''', 5) -+ :ifo~",(Y, 5) -+ :iffMH;od(9", 5), though all sequences 0 -+ Hom.odj9';',~) -+ Hom.r<1J~,~) -+ Hom.r<1j~, ~)are exact. 5. The Functor ®. Let 9",9' be d-modules. To any set U in X we attach the d(U)-module K(U):=9"(U)®9'(U), where the tensor product is taken with respect to d (U). If V is open and V c U, the restriction homo• morphisms 9"(U)-+9"(V), 9'(U)-+9'(V) define, by passing to their tensor product, a restriction homomorphism rr K(U)-+ K(V). It is easily seen that the family {K(U), rn is a canonical d-presheaf. The associated d-module is denoted by 9"®.r<19' or simply by 9"®9' and is called the tensor product of 9" and 9' (over d). There is a canonical d-isomorphism 9"®9' ~9'®9". Any two ~qf-homomorphisms q/: 9"-+5', § 3. Coherent Sheaves As was already said in the Introduction of this book, coherent sheaves play a fundamental role in many parts of Local and Global Complex Analysis. §3. Coherent Sheaves 233 In this section we introduce the general notion of a coherent sheaf; there is no need here to restrict ourselves to complex or 1. Sheaves of Finite Type. Finitely many sections S1' ... ,spE9"(U) define an du-homomorphism p 0": dlJ-+//f;, (a 1x' ... ,apx)1-+ LaiXsix , XEU. 1 We say that //f; is generated by the sections S1' ... , sp if 0" is surjective; this is true if and only if each element of each stalk ~, XE U, is a linear com• bination of the germs S1x' ... , spx with coefficients in d x' i.e. if and only if An d-sheaf 9" is called finitely generated or of finite type at XEX if there is an open neighborhood U of x such that //f; is generated by finitely many sections in 9" over U. This is equivalent to saying that there is an open neighborhood U of x and an integer p;;::: 1 and an exact du-sequence dP-+//f;-+O. An d-sheaf 9" is called of finite type on X if it is of finite type at all points XEX. If 9" is of finite type on X all stalks ~ are finite dx-modules, however being of finite type means much more for 9". Examples: 1) All sheaves s Lemma: Let 9" be of finite type at x EX; let s l' ... , s P be sections in a neighborhood U of x such that the germs S1x' ... ' spx generate ~ as an d x- module. Then there exists a neighborhood V c U of x such that s 1> ••• , s p generate Yy. 234 A. Theory of Sheaves. Notion of Coherence Proof: Since !/ is of fmite type at x there exists a neighborhood W of x and sections t 1, ... , tqe!/(W) generating !/w. Since Six' ... , spx generate ~ we can find sections fij in a neighborhood of x such that p tjx= I hjxS;x, 1 ::;'j::;,q. ;=1 p Therefore we may choose V c W in such a way that tjlV = I hjS;, l::;,j ::;'q. ;=1 Since t11 V, ... , tqlV generate 0% these equations imply that S1' ... , sp also generate 0%. D As a simple application of the Lemma we note: The support Supp!/ of any d-module !/ of finite type is closed in X. Proof: Clear since all zero stalks are generated by the zero germ. D Let !/' be of finite type at x, and let cp: !/' -+ !/, l/I: !/' -+ !/ be d - homomorphisms such that CPx = l/Ix· Then there exists an open neighborhood U of x such that CPu=l/Iu, Proof: We put X:=l/I-cp. The sheaf X(!/') is of finite type at x (Exam• ple 2), by assumption X(!/')x=Ox' i.e. x¢SuPPX(!/'). Hence there is a neigh• borhood U of x such that X(!/')u=Ou, i.e. CPu=l/Iu' D Especially we just proved that the kernel of an d -morphism cP: !/' -+!/, where !/' is of finite type, vanishes in a neighborhood of x if the kernel of CPx is zero. The verification of the following statements is left as an exercise to the reader. 1) Let !/',!/" be subsheaves of an d -sheaf !/'. Assume that !/' is of finite type at x and that!/; c !/;'. Then Y:; c Y:;' for an open neighborhood U of x. 2) Let l/I: !/ -+!/" be an d -homomorphism. Assume that !/" is of finite type at x and that l/I x: ~ -+!/;' is surjective. Then there exists an open neighborhood U of x such that l/Iu: YU-+Y:;' is surjective. 3) Let cP: !/' -+!/ be an d -epimorphism. Assume that $:e-i cP is of finite type at x. Then an d-submodule ff of !/ is of finite type at x if and only if the d-submodule cp-1(ff) of !/' is of finite type at x. 2. Sheaves of Relation Finite Type. If a: d~-+YU is an du-homo• morphism determined by sections S1' ... ,spe!/(U) the sheaf of relations of S1> ••• , sp is defined by ~d(S1' ... ,sp):=$:e-ia= U {(a 1X , ... ,apx)ed:: fa;xs;x=O}; xeU 1 §3. Coherent Sheaves 235 obviously this is an du-submodule of d{}. An d-sheaf g is called of relation finite type at xeX, if for every rmite system S1' ... , sp of sections over an open neighborhood U of x the sheaf of relations P&t(S1' ... , sp) is of finite type at x. This is the case if and only if all kernels ~-t a of all homomor• phisms a: d{}--+Yu are of finite type at x. An d-sheaf g is called of relation finite type on X, if g is of relation finite type at all points of X. Examples: 1) All sheaves (9D of germs of holomorphic functions are of relation rmite type. This is aKA'S Coherence Theorem which is proved in 2.5.2. 2) If g' --+ g is a monomorphism and if g is of relation finite type at xeX, then g' is of relation finite type at x too. Thus subsheaves of sheaves of relation finite type are again of relation finite type. 3) Quotient sheaves of sheaves of relation finite type are not necessarily of relation finite type. Consider the quotient sheaf g:=(9D/f; where f denotes the ideal of (9D introduced in Example 4) of the preceding para• graph. We have g;=0. for zeU and g;=(9. for zeD ...... U. Though (9D is of relation finite type the sheaf g is not: we have rIld(s)=f for the section seg(D) given by s.:=O for zeU and s.:=1 for zeD ...... U, and the sheaf f is not of finite type. 3. Coherent Sheaves. A sheaf of d-modules g on X is called coherent if g is of finite and of relation finite type on X. The notion of coherence is local: g is coherent if it is coherent at every point xeX, i.e. if every xeX has an open neighborhood U such that Yu is coherent. A subsheaf g of a coherent sheaf is coherent if and only if g is of finite type. A quotient sheaf g of a coherent sheaf is coherent if and only if g is of relation finite type. One easily proves the following Remark: If g is a coherent d-module then, for every xeX, there exists an open neighborhood U of x and an exact du-sequence The sheaf of rings d is called coherent if d is a coherent d-module, this is precisely the case if d is of relation finite type. A sheaf of ideals f in d is said to be coherent if f is a coherent d-submodule of d. If d is coherent then the product f· f' of coherent ideal sheaves is likewise a coherent sheaf of ideals (since f· f' is of finite type). The notion of a coherent sheaf depends heavily on the underlying sheaf d of rings, therefore it would be more precise to talk about "d-coherent" sheaves. We only use this notion when we have to consider different sheaves of rings simultaneously (e.g. in 4.3). 236 A. Theory of Sheaves. Notion of Coherence § 4. Yoga of Coherent Sheaves It is not clear at all from the definition of coherence that coherent sheaves really exist (except trivial ones like zero sheaves). Nevertheless it seems advisable to develop the calculus for such sheaves independently of the existence problem, the lack of immediate convincing applications should not discourage the reader. By X we always denote a ringed space with structure sheaf d. The most useful device in all coherence theory is a simple statement from [FAC], p. 208, which we call 1. Three Lemma: Let O--+g" ~g' ~g''' --+0 be an exact sequence of s#• sheaves. Then all sheaves g", Y, g'ft are coherent if two of them are coherent. Proof: 1) Let //, g''' be coherent. Then locally we have an epimorphism x: d q--+!/'. Since g''' is coherent, the kernel" of ljJoX: dq--+g''' is of finite type. Therefore X(") c Y' is of finite type (Example 1.2)) and hence coherent since g' is coherent. Now L'l IjJ = §m cp, therefore cp induces an isomor• phism g" ~ X("), thus the sheaf g" is coherent. 2) Let Y}f, g' be coherent. Then g'ft is of finite type (Example 1.2)), therefore we only have to show that the kernel of each homomorphism p: d{j --+ g'J' is of finite type at each point XE U. There exists a neighborhood V c V of x and homomorphisms IT: dt: --+ Y'v, r: dJ --+ Y'v such that ljJyo IT = py and r(dJ) = cp(g")y. Then offe,z py = IT- 1 (offel ljJy)= IT- 1 (cpyW")) = IT- 1 (r(dJ)). Now offe,z(lT(f;r)cdr q is of finite type at x since g' is coherent. Therefore the image of L,z(lT(f;r) under the natural projection n: dr q --+sl{: is of finite type at x. Evidently n(offe,z(lTE8r)) = L,z py. 3) Let g", g''' be coherent. To each point XEX we choose a neigh• borhood Wand sections s~, ... ,S~Eg"(W), s~, ... ,S~Eg'''(W) generating g" resp. g''' over W. If we choose W small enough we have sections Sl, ... ,SqEg'(W) such that ljJ(s;)=s;'. Then clearly cp(s~), ... ,CP(S~),Sl, ... ,Sq generate g'w' It remains to show that each relation sheaf ~et(tl"'" tp)' t 1 , .•• ,tpEg'(U), is of finite type at each point XEU. Since g''' is coherent there exists a neighborhood VC V of x and sections (Jj\ ... ,fjP)EdP(V), 1-::;;,j-::;;'q, which generate ~et(ljJ(tl)' ... , ljJ(tp))y. The sections Uj :=Jjltl+JjZtZ+ ... +fjPtpEg'(V), 1-::;;,j-::;;,q, belong to cp(y}f)(V) since ljJ(u)=O and L,zIjJ=cp(g"). Now g" is coherent and cp(g")c:o:g". Therefore, after having chosen V small enough, we have §4. Yoga of Coherent Sheaves 237 sections (g~, ... , gz)Ed'q(V), 1 ~ k~r, generating P&I'(u l , ... , uq). Then Sk:= ctJ/ gl, ... ,J//gl)EP&I'(tl , ... , tp) (V), 1 ~k~r; we claim that Sl, ... ,Sk generate P&I'(tl, ... ,tp)y. Take a germ (hl, ... ,hp) in p P&I'(t l, ... , tp)y, yE V. Since '[, hi t/!(ti)y=O, we can write i=l q (hl,···,hp)= '[,cj(f], ... ,f/)y, CjEd'y. j=l Now we conclude Therefore (c l' ... , cq) is a linear combination of the germs (g!, ... , gZ}y and consequently (hI' ... ,hp) is such a combination of the germs Sly' ... , Sry with coefficients in d'y. This proves that [I' is a sheaf of relation finite type. 2. Consequences of the Three Lemma. We collect several consequences of the Three Lemma which are applied again and again in function theory. Consequence 1: The WHITNEY sum of finitely many coherent sheaves is a coherent sheaf. Proof: Assume [1'=[1"$[1''' with coherent sheaves [1",[1'''. Consider the canonically induced exact sequence 0-+[1"-+[1'-+[1'''-+0 and apply the Three Lemma. The generalization to any finite number of coherent sheaves follows by induction. D Consequence 2: Let qJ: [I' -+ /Y be an d'-homomorphism between coherent sheaves. Then the sheaves .Fm qJ, ~-i qJ and C(/oie-i qJ are coherent. Proof: Since [I' is coherent .Fm qJ is of finite type and hence coherent (cf. 3.3). The coherence of ~-i qJ resp. C(/oie-i qJ now follows by applying the Three Lemma to the exact sequence O-+~-iqJ-+[I'-+.FmqJ-+O resp. O-+.FmqJ-+/Y -+C(/oie-iqJ-+O. D The coherence of C(/oie-i qJ may be rephrased as follows: If [1" ~[I' -+[1''' -+0 is exact and if [1" and [I' are coherent then the sheaf [1''' is coherent. Proof: Clear since [1''' ~ [I' /.Fm qJ = C(/oie-i qJ. D Consequence 3: If a sequence [I"...!4 [I' -!4 fI'" of coherent sheaves is exact at a point XE X (i.e. if the sequence Y'; ~.~ ~~" is exact), then there 238 A. Theory of Sheaves. Notion of Coherence exists an open neighborhood U of x such that the sequence :J7J.!'!.!4 Yu~ :J7J' is exact. Proof: The sheaf :J7'/L,z(l/Jocp) is coherent and zero at x. Hence this sheaf is zero in an open neighborhood U of x, i.e. ~mcpucL,zl/Ju. Now L,zl/Ju/~mcpu is coherent on U and zero at x. Hence L,zl/Ju=~·mcpu if U is chosen small enough. D Consequence 4: Let :J7' and :J7" be coherent d-subsheaves of a coherent sheaf fI'. Then the sheaves :J7' +:J7" and :J7' n:J7" are coherent. Proof: Since :J7' +:J7" is a subsheaf of /I' of finite type the coherence of :J7' +:J7" is clear. The sheaf :J7' n:J7" is the kernel of the canonical homo• morphism :J7" ----7:J7/:J7' and hence coherent by Consequence 2. D We easily get a converse of Remark 3.3. Consequence 5: Let the structure sheaf d be coherent. Then an d-sheaf :J7 is coherent if for every point XEX there exists a neighborhood U of x and positive integers p, q and an exact sequence ~«IS~.YiJ-+Yu-+O. Proof: The sheaves ,YiP, d q are coherent and Yu ~ rcQi.e,z cpo D Finally we derive a result about the behaviour of coherent sheaves when "changing rings". Consequence 6: Let d be coherent and let ~ be a coherent "«I -ideal. Then an d/~-module:J7 on X is c«l/~-coherent if and only if :J7 is d-coherent. In particular, d /~ is a coherent sheaf of rings on X. Proof: Since the map d----7d/~ is surjective the d/~-sheaf :J7 clearly is of finite type if and only if the d-sheaf :J7 is of finite type. Now let (J: (d/~)~ ----7 Yu be an ("«I/~)u-homomorphism over an open set U in X. Denoting by n the projection s#P----7(d/~)P we have Lz (J= nu(Lz((Jonu))' Now, if :J7 is "«I-coherent, the d-sheaf x.ez((Jonu) is of finite type at all points of U, hence the d/~-sheaf x.ez(J is of finite type everywhere in U. This proves that d-coherence implies c«l/~-coherence, especially we see that d /~ is a coherent sheaf of rings. Finally suppose that :J7 is d /~ coherent. Then locally :J7 is the cokernel of an d/~-homomorphism cp: (d/~)P----7(d/~)q (Remark 3.3). Since d/~ is .Yi-coherent the sheaf rcQiM cp is d-coherent by Consequence 2. This proves the d-coherence of .'7'. § 4. Yoga of Coherent Sheaves 239 3. Coherence of Trivial Extensions. In this paragraph Y denotes a closed subspace of X and I: Y"""""* X the injection. For every sheaf 5 of abelian groups on Y the image sheaf 1*5 is a sheaf of abelian groups on X. We call 1* 5 the trivial extension of 5 to X, this sheaf is characterised by the fact that 1*51 Y=5 and 1*5IX" Y=O. If PJ is a sheaf of rings on Y and if 5 is a PJ-sheaf, then 1* f!J is a sheaf of rings on X and 1* 5 is a 1* PJ-module. Remark: A PJ-sh('u! jJ on Y is of f!J1inite type resp. PJ-coherent if and only if 1* 5 is of 1* .Yf-/lllite type resp. 1* f!J-coherent. Proof: Clearly finiteness resp. coherence of 1* 5 implies finiteness resp. coherence of !Y. In order to prove the converse let V be open in X and put V:=UnY. Any 1*f!J-homomorphism cp: (z*f!J)f,"""""*(1*5)u can be uniquely written as cp = 1* (t/I), where t/I: PJt """""* g;;. is a f!J-homomorphism. Clearly cp is surjective if and only if t/I is surjective. From this the assertion follows easily. 0 Now we are able to describe a procedure which is of utmost importance in the theory of coherent analytic sheaves. Let (X, d) be a Proposition (Extension Principle for Coherent Sheaves): Let d be co• herent and let § be an d-ideal of finite type. Then a f!J-module 5 on Y is f!J-coherent if and only if the trivial extension 1*5 of f'/- to X is d• coherent. Especially f!J itself is a coherent sheaf of rings. 4. Coherence of the Functors ;Yf~A and @. We consider the dx-homo• morphism Lemma: If !J" is coherent then the map p is bijective. Proof: 1) Fix XEX. Let cp: 9"~"""""*Yu be an du-homomorphism in a neighborhood V of x such that the germ CPx: 9";"""""*9". is zero. Since 9'" is of finite type at x it follows from 3.1 that cp is zero in a neighborhood of x, which proves the injectivity of p. 2) The proof of the surjectivity of p needs a little more work. Take an ow'x-homomorphism 0": 9";"""""* 9".. There exists a neighborhood V of x and 240 A. Theory of Sheaves. Notion of Coherence sections s~, ... ,S~E9"'(U) generating 9"~ as well as sections t 1 , ... ,tp E9"(U) such that u(S;,)=tix' 1 ~i~p. Since 9l:=9let(s~, ... ,s~) is of finite type at x we can find relations (fl, ... ,fj)EdP(U), 1 ~j~q, which generate 9l over U (eventually we have to pass to a smaller U). Then (fl, ... ,fj)E9let(t1 , ••• , t p ) for all j and small enough U since (~f/ ti)x = ~!;~U(S;x)=u (~!;~S;x) =0. Therefore there exists a well-defined du-homomorphism cp: 9"~--+Yu map• ping P P by:= L aiyS;yEYy' onto cp(by) = LaiytiyEYy, YEU. 1 1 Since CPx=U by construction, we conclude that p is surjective. o As an important consequence of the Lemma we derive now Proposition: If 9'" and 9" are coherent then Jt'omd(9"',9") is coherent. Proof: The problem being local we may assume that we have an exact sequence d Q--+dP--+9"'--+O (cf. Remark 3.3). Now the induced sequence 0--+ Jt'omd(9"', 9")--+ Jt'omd(dP, 9")--+ Jt'omd(dQ, 9") is exact, since Jt'omd(9"', 9")x~ HomdX<~"~) by our Lemma (cf. end of 2.4). Since all sheaves Jt'omd(dn, 9")~9"n are coherent (Paragraph 2, Con• sequence 1), we conclude that Jt'omd(9"',9") is coherent as kernel of a morphism 9"P--+9"Q between coherent sheaves (Consequence 2 of Para• graph 2). 0 For each d-module 9" we may consider the d-sheaves 9"*: = Jt'omd(Y, d), 9"**: = (9"*)*. We call 9"* the dual and 9"** the bidual sheaf of fI'. We have canonical stalk homomorphisms ux: ~--+~**. If d and 9" are coherent these maps give rise, due to the Lemma above, to a canonical d -homomorphism u: 9" --+ 9"**; the kernel of u is very important for the theory of torsion sheaves. As a matter of fact in 3.3.3 we make essential use of the Corollary: If d and 9" are coherent, the dual 9"* and the bidual 9"** and the kernel of the canonical homomorphism u: 9" --+9"** are coherent. We now turn to the tensor functor <8>. Here we immediately get: Proposition: If 9'" and 9" are coherent, then 9'" <8>d 9" is coherent. §4. Yoga of Coherent Sheaves 241 Proof: Locally we have exact sequences dP-+dL-+!/'-+O. Since dn(f!)Ji/!/=!/n we conclude (right exactness of (f!): !/p-+!/q-+!/'(f!)!/-+O. Hence !/'(f!)!/ is coherent (as a cokernel). 5. Annihilator Sheaves. For any d-sheaf!/ we define An~:= UxEdx: fx~=O} and dn!/= UAn~. x We call dn!/ the annihilator of !/ and claim If !/ is of finite type the annihilator of !/ is an ideal sheaf. Proof: "We only have to show that the set dn!/ed is open in d. Take a section fEd(U) in an open set U eX and assume fpEAn g;, for a certain point pEU. Then the kernel of the homothety morphism Yu-+Yu, sxHfxsx, induced by f is g;, at p. H~nce this kernel coincides with -% in an open neighborhood Ve U of p (cf. 3.1). This means fv-%=O, i.e. fxEAn~ for all XE V. Thus dn!/ is open in A. D Proposition: For each coherent d -module !/ there exists the canonical homomorphism (*) rx: d-+JromJi/(9;!/), fxH (sxHfx sx)' with .Yt.e'trx=dnf/. Thus if d and !/ are coherent the annihilator of !/ is a coherent ideal. Proof: We define rxx: dx-+Hom.,,)~,~) by fxH(SxHfxsx) for all points XEX. Then clearly rxx is an dx-homomorphism with .Yt.e'trxx=dn~. Now due to Lemma 4 we may identify HomJi/,,(~'~) with JromJi/(9; !/)x. Then it is easily seen that the maps rxx from d x into JromJi/(9; !/)x, XEX, determine an d-homomorphism rx: d -+JromJi/(9;!/) with .Yt.e'trx=dnf/. The last as• sertion now follows from Proposition 4 and Consequence 2 in Paragraph 2. D If (X, d) is a Monograpbs [CAR] Cartan, H.: Collected Works, vol. II, ed. R. Remmert and J-P. Serre, Springer-Verlag 1979 [ENS] Cartan, H.: Seminaire Ecole Normale Superieure 1960/61 [CAG] Fischer, G.: Complex Analytic Geometry, LNM 538, Springer Verlag 1976 [TF] Godement, R.: Theorie des Faisceaux, Act. Sci. Ind. 1252, Hermann Paris 1958 [AS] Grauert, H. and R. Remmert: Analytische Stellenalgebren, Grund!. 176, Springer• Verlag 1971 [TSS] Grauert, H. and R. Remmert: Theorie der Steinschen Riiume, Grund!. 227, Springer• Verlag 1977; Theory of Stein Spaces, transl. by A. Huckleberry, Grund!. 236, Springer-Verlag 1979 [TAG] Hirzebruch, F.: Neue Topologische Methoden in der Algebraischen Geometrie, Erg. Math. Springer-Verlag 1956; translated and extended 1966 by R.L.E. Schwarzenber• ger, Topological Methods in Algebraic Geometry, last edition 1978 [TAS] Narasimhan, R.: Introduction to the Theory of Analytic Spaces, LNM 25, Springer• Verlag 1966 [OKA] Oka, K.: Mathematical Papers, transl. by R. Narasimhan; with comments by H. Cartan; ed. R. Remmert, Springer-Verlag 1984 [OSG] Osgood, w.P.: Lehrbuch der Funktionentheorie II. 1,2. Aufl. Teubner Verlag 1929 [FAC] Serre, J-P.: Faisceaux algebriques coherents, Ann. Math. 61, 197-278 (1955) [GAGA] Serre, J-P.: Geometrie a1gebrique et geometrie analytique, Ann. Inst. Fourier 6, 1-42 (1955-56) Articles [FoKn] Forster, O. and K. Knorr: Ein Beweis des Grauertschen Bildgarbensatzes nach Ideen von B. Malgrange, Manuscripta Math. 5, 19-44 (1971) [GrI] Grauert, H.: Charakterisierung der holomorph vollstiindigen komplexen Riiume, Math. Ann. 129, 233-259 (1955) [Grz] Grauert, H.: Ein Theorem der analytischen Garbentheorie und die Modulriiume komplexer Strukturen, Publ. Inst. Hautes Etudes Sci. N° 5,233-292 (1960) [GrRetJ Grauert, H. and R. Remmert: Komplexe Riiume, Math. Ann. 136,245-318 (1958) [GrRez] Grauert, H. and R. Remmert: Bilder und Urbilder analytischer Garben, Ann. Math. 68, 393-443 (1958) [KoSp] Kodaira, K. and D.C. Spencer: On Deformations of complex analytic Structures, 1- III. Ann. Math. 67, 328-466 (1958) and 71, 43-76 (1960) [ReI] Remmert, R.: Meromorphe Funktionen in kompakten komplexen Riiumen, Math. Ann. 132,277-288 (1956) [Rez] Remmert, R.: Holomorphe and meromorphe Abbildungen komplexer Riiume, Math. Ann. 133,328-370 (1957) Bibliography 243 [ReSt] Remmert, R. and K. Stein: Ober die wesentlichen Singuiaritiiten analytischer Men• gen, Math. Ann. 126, 263-306 (1953) [Rii] Riickert, W.: Zum Eliminationsproblem der Potenzreihenideale, Math. Ann. 107, 259-281 (1933) [Se] Serre, J-P.: Prolongement DeFaiseaux Analytiques Coherents, Ann. lnst. Fourier 16, 363-374 (1966) The vast collection of articles concerned with coherent analytic sheaves is overwhelming. Any attempt to provide a reasonably complete list of such papers is a "Love's Labour's Lost". Index of Names Behnke, H. 12, 215 Hopf, H. 12 Remmert, R. 65, 128, 215 Hurwitz, A. 187 Riemann, B. 10, 12, 131, Caratheodory, C. 12 133, 139 Cartan, H. 2, 8, 12, 13, Kiehl, R. 188 Rothstein, W. 183 45, 58, 75, 81, 84, 128, Kneser, H. 187 Ruckert, W. 43, 44, 159, 162, 167, 174, 183, Knorr, K. 188 75,82 187,214,218 Kodaira, K. 207 Chow, w.L. 187 Krull, W. 43, 105 Serre, J-P. 13,111,218 Clement, G.R. 165 Siegel, C.L. 217 Lagrange, J.L. 142 Spencer, D.C. 207 Dedekind, R. 123, 142 Lasker, E. 43 Stein, K. 12, 183, 214, 215 Leray, J. 228 Stickel berger, L. 38 Forster, O. 188 Levi, E.E. 187 TeichmulJer, O. 12 Grauert, H. 65, 128, Malgrange, B. 188 Thimm, W. 217 214,215 ThuIlen, P. 12, 183 Grothendieck, A. 8, Narasimhan, R. 184 13, 188 Noether,E. 110 van der Waerden, B.L. 2 Verdier, J.L. 188 Hartogs, F. 2 Oka, K. 2, 8, 12, 58, Hermans, A. 89 65, 72, 75, 84, 128, Weierstra13, K. 38,72, Hermes, H. 174 159, 162 75, 145, 162, 174, 187, 217 Hirzebruch, F. 223 Osgood, w.F. 75, 165 Weyl, H. 142 Index Absolute Maximum Principle 110 CHEVALLEY dimension 95 active closed complex subspace 15 - function 97 - holomorphic embedding 20 - germ 97 - map 47 sheaf of - germs 97 coboundary 34 Active Lemma 100 - map 34 admissible sheaf 156 cochain 34 affine algebraic cone 162 cocycle 34 affine rational curve 152 coherence of cohomology sheaves 202 algebraic dimension 95 - of direct image sheaves 207 algebraically dependent 216 - of ideal sheaves 84 analytic - of image sheaves for fmite holomorphic - closure 220 maps 64 - covering 133 - of (Pc- 59 - dimension 93 - of radical sheaves 86 - image sheaf 18 - of the normalization sheaf 158 - inverse image sheaf 18 - of the structure sheaf 60 - restriction 20 - of torsion sheaves 69 - set 76 coherent sheaf 235 - sheaf 8 cohomology of a complex 192 - ZARISKI topology 211 CECH - 34 analytically dependent 216 cokernel 230 analytically normal vector bundle 148 common denominator 119 analyticity of the singular locus 117 complex atlas 10 annihilator sheaf 241 - chart 10 arcwise connected 178 - direct product 24 - image space 60 - inverse image space 19 base space 217 - manifold 8 bidual sheaf 240 rigid -- 212 biholomorphic map 7 - space 7 branch locus 134 holomorphically convex -- 33, 221 bundle holomorphically separable -- 142,176 - map 31 holomorphically spreadable -- 175 holomorphic vector - 31 irreducible -- 167, 168 HOPF - 33, 147, 150 normal -- 8, 13, 125 hyperplane - 33, 219 pure dimensional -- 106 tangent - 208,212 reduced - - 8 - value 8 canonical presheaf 225, 226 complex of polydisc modules 191 - product 25 cohomology of a - 192 CEcH cohomology group 34 cone - 191 characteristic polynomial 138 morphism of -es 191 246 Index component divisor 30 connected - 171 double point 4,8,9, 17, 107 prime - 80 dual sheaf 240 pure dimensional - 173 cone 4, 8, 146, 147, 148, 150, 151 - complex 191 embedding dimension 113 affine algebraic - 162 closed holomorphic - 20 SEGRE - 102, 151 exactness of the functor f* for finite connected 132 maps 50 - component 171 Existence Theorem for Finite Open arcwise - 178 Maps 107 constant sheaf 223 -- for Open Analytic Coverings 136 continuity of roots 52 extension principle 17, 239 corank 90 holomorphic - 130 Covering Lemma 136 meromorphic - 130 analytic - 133 trivial - 239 b-sheeted - 135 Extension Theorem for Analytic STEIN - 35 Sets 147, 181 WEIERSTRASS - 135, 153 First RIEMANN -- 131 Criterion of Activity 98 LEVI -- 185 - of Connectedness 133, 145 RIEMANN -- on Complex Manifolds 132 - of Normality 127, 144 RIEMANN -- on Locally Pure Dimensional - of Openness 69 Complex Spaces 143 - of Purity 106 RIEMANN -- on Normal Complex - of Reducedness 89 Spaces 144 - of Reducibility 81 Second RIEMANN -- 132 - of Smoothness 116 critical locus 134 cross section 31 factorial ring 44, 119, 124 factorization of a finite hoi om orphic decent vector bundle 147 map 65 decomposition of a complex space 172 - of a holomorphic map 16, 60, 90 Global Decomposition Theorem 172 STEIN - 213 LASKER-NoETHER - 78 STEIN Factorization Theorem 213 local - 79 universal property of the STEIN - 214 Local Decomposition Lemma 79 filtered family 111 WEIERSTRASS - 41 finite map 47 decreasingly filtered family 111 local description of - -s 48 deformation 207 Finite Mapping Theorem 64 denominator 120 Finiteness Criterion 176 common - 119 - Lemma F(q) 193 sheaf of -s 120 - Theorem 146 universal - 141 dimension 93 algebraic - 95 germ 224 analytic - 93 active - 97 CHEVALLEY - 95 prime - 163 embedding - 113 set - 80 direct image sheaf 36 Global Decomposition Theorem 172 Direct Image Theorem 207 Global Existence Theorem 161 direct product 24 Global Maximum Principle 174 canonical -- 25 Gluing Lemma 10 discriminant 140 graph 30 Division Theorem 40 Graph Lemma 29 Generalized WEIERSTRASS -- 53 - map 28,30 Index 247 - of a system of meromorphic local existence of universal functions 216 denominators 159 - space 30 Local Existence Theorem 156 normal- 217 locally free sheaf 31, 90 locally irreducible 8 HAUSDORFF sheaf 224 Local Maximum Principle 109 henselian algebra 45 HENSEL'S Lemma 44 holomorphic cross section 31 map - function 9 biholomorphic - 7 - line bundle 32 closed - 47 - map 2,7 finite - 47 - vector bundle 31 graph - 28, 29 weakly - function 144 holomorphic - 2, 7 holomorphically convex 33, 221 open - 52 - separable 142, 176 OsGOOD - 68 - spreadable 175 presheaf - 225 Jf__ -functor 231 proper - 175 HOPF bundle 33, 147, 150 reduction - 88 HOPF u-process 147 stalk - 223 hyperplane bundle 33,219 WEIERSTRASS - 52, 135 Maximum Principle Absolute -- 110 ideal (sheaf) 230 Global- - 174 Identitiitssatz 3 Local -- 109 - for meromorphic functions 170 meromorphic function 119 Identity Lemma 167 sheaf of germs of - -s 119 image functor 227 model space 4 - sheaf 227 modification 214 analytic - sheaf 18 module complex - space 60 d- - 229 direct - sheaf 36 polydisc - 189 Inclusion Lemma 170 torsion - 68 increasingly filtered family 111 morphism of complexes 191 integral closure 123 - of CC-ringed spaces 6 integrally closed ring 124 - of polydisc modules 191 Intersection Inequality 102 - of sheaves 223 inverse image - system 191 analytic -- sheaf 18 d- - 229 complex -- space 19 irreducible 8, 78 - complex space 167, 168 locally - 8 natural functor 201 NEIL'S parabola 4, 8, 15, 107, 121, 143, 152 n-fold point 4 JACOBI Criterion 114 nilradical 86 NOETHER Lemma 111 LASKER-NOETHER decomposition 78 normal complex space 8, 13, 125 LEVI Extension Theorem 185 - graph 217 Lifting Lemma 154 - point 8, 124 line bundle 32 - ring 124 linking isomorphism 160 normalization 161 local decomposition 79 - sheaf 123 Local Decomposition Lemma 79 existence of a - 161 Local Description Lemma 72 uniqueness of a - 164 248 Index NuIlstellensatz reducible 8, 79 RUCKERT'S - 67 reduction 88 RUCKERT'S - for ideal sheaves 82 - map 88 numerator 120 Reduction Theorem for Holomorphically sheaf of -s 120 Convex Complex Spaces 221 regular point 8 OKA-CARTAN Theorem 84 - family 207 OKA'S Coherence Theorem 59 RIEMANN Extension Theorem 130 open map 52 -- on Complex Manifolds 132 Open Mapping Theorem 107,109 - on Locally Pure Dimensional Criterion of Openness 69 Complex Spaces 143 OSGOOD map 68 - on Normal Complex Spaces 144 - space 185, 215-220 First --- 131 Second --- 132 point of indeterminacy 121, 216, 217 RIEMANN surface 12, 129, 141, 149 polar set 120 rigid 212 polydisc module 189 ringed space 1 - sheaf 200 annihilator - 241 tangent bundle 208, 212 .91- - 229 tensor product 232 bidual- 240 Theoreme B 35 coherent - 235 Theorem of CHow 184 cohomology - 202 -- HURWITZ-WEIERSTRASS 186 constant - 223 -- Integral Dependence 138 direct image - 36 -- OKA 128 dual- 240 -- Primitive Element 140 finitely generated - 233 thin 132 HAUSDORFF - 224 Three Lemma 236 image - 227 torsion free sheaf 69 locally free - 31, 90 - module 68 morphism of sheaves 223 - sheaf 69 normalization - 123 trivial extension 239 polydisc - 200 trivialization 31 quotient - 230 structure - 1 torsion free - 69 unbranched 134 torsion - 69 Union Formula 101 WmTNEY sum of - 224 u-process 147 simple point 8 singular point 8, 116 vector bundle 31 - locus 116 analytically normal -- 148 analyticity of the - locus 117 decent -- 147 smooth point 8, 116 Criterion of Smoothness 116 smoothing 197 weakly holomorphic function 144 stalk 223 WEIERSTRASS Convergence Theorem 145 - map 223 - covering 135, 153 stationary family 111 - decomposition 41 STEIN covering 35, 206, 208 - Division Theorem 40 STEIN factorization 213 - isomorphism 55 STEIN Factorization Theorem 213 - map 52,135 universal property of the -- 214 - polynomial 41 STEIN space 33-36,205,208,212, 22lf. - Preparation Theorem 42 structure sheaf 1 Generalized - Division Theorem 53 submodule 229 WmTNEY sum 224, 229 subspace closed complex - 15 open CC-ringed - 7 zero section 146-149,228 subsheaf 224 - set 120 support 228 - set of an ideal sheaf 13