SHEAVES. Contents 1. Presheaves 1 2. Sheaves 4 3. Direct and inverse images of sheaves 7 4. The Čech resolutions 10 References 15

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1 w SHEAVES LIVIU I NICOLAESCU Abstract We describe the Abelian categories of sheaves over topological spaces, natral fnctors between them, and several remarkable resoltions of sheaves Contents 1 Presheaves 1 2 Sheaves 4 3 Direct and inverse images of sheaves 7 4 The Čech resoltions 10 References 15 1 Presheaves A presheaf on is a correspondence S which associates to each nonempty open sbset U a nonempty set S(U) and to every inclsion U V a map r U V = r(s) U V : S(V ) S(U) sch that r U U = 1 S(U) and for any U V W we have a commtative diagram r V W S(W ) S(V ) [ [[ r U W [ [] S(U) r U V r U W = r U V r V W rv U is called the restriction map For simplicity we will often denote it by U The set S 0 (U) is referred to as the space of sections of S 0 over U and it is often denoted by Γ(U, S 0 ) A morphism of pre-sheaves S 0 and S 1 is a collection of maps φ = φ U : S 0 (U) S 1 (U); U open sbset of } Date: Janary, 2005 Notes for a Hodge Theory Seminar 1

2 w w 2 LIVIU I NICOLAESCU sch that for any inclsion U V we have a commtative diagram φ V S 0 (V ) S 1 (V ) r U V S 0 (U) φ U r U V S 1 (U) r U V φ V = φ U r U V (11) We obtain in this fashion a category psh(), of presheaves on If C is a sbcategory of Set then a pre-sheaf of C-objects is a presheaf sch that every set S(U) is an object in C and the restriction maps are morphisms in C A morphism of presheaves of C-objects is a morphisms of presheaves of sets φ sch that for any open set U the map φ U is a morphism in C We denote by psh C () the reslting sbcategory of presheaves of C-objects over If R is a commtative ring with 1, we denote by R Mod the category of R-modles and we set psh R () = psh R Mod We set Γ(, F) = 0, F psh R () Note that for any two presheaves of R-modles S 0, S 1 the space of morphisms Hom pshr (S 0, S 1 ) is natrally an R-modle We denote by R = R the constant presheaf on determined by the reqirement that Γ(U, R) consists of the continos maps U R, where R is eqipped with the discrete topology Eqivalently, the sections in Γ(U, R) are the locally constant maps U R For any morphism φ Hom pshr (S 0, S 1 ), any inclsion of open sets U V and any s S 0 (V ) we dedce from the commtative diagram (11) that φ V (s) = 0 = φ U (s U ) = 0 Ths the restriction maps of S 0 indce morphisms Hence the correspondence U : ker ( S 0 (V ) φ V S 1 (V ) ) ker ( S 0 (U) φ U S 1 (U) ) U (ker φ)(u) := ker ( S 0 (U) φ U S 1 (U) ) S 0 (U) defines a presheaf on called the kernel of φ and denoted by ker φ Arging in a similar fashion we dedce that the correspondence U (im φ)(u) := im ( S 0 (Y ) φ U S 1 (U) ) S 1 (U) is a presheaf called the image of φ and denoted by im φ Similarly we obtain presheaves coker φ = S 1 / im φ, Γ(U, coker φ) = S 1 (U)/φ U ( S 0 (U) ), coim φ = S 1 / ker φ, Γ(U, coim φ) = S 0 (U)/ ker φ U Note that we have a natral isomorphism of presheaves coim φ im φ This shows that psh R is natrally an Abelian category 1 1 Recall that a category is called additive if HomC (, Y ) has a natral strctre of Abelian grop, there exists a zero object (ie simltaneosly initial and terminal) and there exist finite direct sms A category is called Abelian if it is additive, every morphism φ has a kernel and cokernel, and the canonical map coim φ im φ is an isomorphism For more details we refer to [1, 4, 5]

3 SHEAVES 3 Example 11 (a) Sppose is a smooth manifold For every open set U we denote by A p (U) (or Ap (U) if there is no confsion concerning ) the space of degree p, complex valed, smooth, differential forms on U In local coordinates (x 1,, x n ) on U sch a differential form has the description ω = α ω α dx α, (12) where the coefficients ω α are smooth, complex valed fnctions, the smmation is carried over all the ordered mlti-indices α = (α 1,, α p ) and dx α = dx α 1 dx α p The exterior differential indces a natral morphism of presheaves If ω is as in (12) then dω = d : A p Ap+1 n k=1 We obtain in this fashion a complex of presheaves (A, d) : A 0 ( dx k ω ) α x k dxα α d A p d A p+1 called the DeRham complex As the case = S 1 shows this complex may not be acyclic Note that we can identify the constant presheaf C with the kernel of the morphism d : A 0 A1 (b) Sppose is a complex manifold For every open set U we denote by A p,q (U) the vector space of complex valed, smooth forms of bi-degree (p, q) on U If we se holomorphic coordinates (z 1,, z n ) on U then ω A p,q (U) has the local description ω = α,β ω αβ dz α d z β, (13) where the smmation is carried over all ordered mlti-indices α = (α 1,, α p ), β = (β 1,, β q ) and We have natral morphisms of sheaves If ω is as in (13) then Note that dz α = dz α 1 dz α p, d z β = d z α 1 d z β q : A p,q ω = x ω = x A m = Ap+1,q, : A p,q Ap,q+1 ( dz k α,β ( d z k p+q=m α,β A p,q ω αβ z k dzα d z β ) ω αβ z k dzα d z β ) and d = +

4 w w w w w w 4 LIVIU I NICOLAESCU From the eqalities d 2 = 2 = 2 = 0 we dedce that we can form a doble complex A p,q+1 A p+1,q+1 (A,,, )) A p,q A p+1,q The rows and the colmns of this complex are simple complexes called the Dolbealt complexes The total complex of this doble complex is precisely the DeRham complex We denote by Ω p,q the kernel of : A p,q A p,q+1 Ω p,q is called the (pre)sheaf of holomorphic (p, q)-forms on Note that the sections of Ω 0,0 are precisely the (locally defined) holomorphic fnctions on This (pre)sheaf is sally denoted by O and it is called the strctral (pre)sheaf of the complex manifold Note that the seqence Ω 0,0 Ω 1,0 Ωp,0 Ω p+1,0 is a complex of (pre)sheaves called the holomorphic DeRham complex Note that on Ω m we have d = 2 Sheaves Sppose that S psh R () Given any family U = ( U i of open sbsets of we set )i I U := U i, U ij = U i U j, Γ(U, S) = (s i ) i I S(U i ); i, j : s i Uij = s j Uij } i I i I Note that we have a natral map U : Γ( U, S) Γ(U, S), s ( s Ui )i U i In general this map is neither injective, nor srjective A presheaf S is called a sheaf if for any family U of open sbsets of the map U is a bijection Example 21 Sppose is a set eqipped with the discrete topology (ie every sbset is open) For example cold be the set of integers eqipped with the topology indced from the Eclidean topology of the real axis We consider the presheaves F and ˆF of C-vector spaces on defined by Γ(S, F) = C, Γ(S, ˆF) = C s S s S The restriction maps are given by the natral projections ˆF is a sheaf, while F is not

5 SHEAVES 5 To any presheaf S psh R () we can associate a sheaf F + in the following canonical way First, for x define the stalk of S in x to be the R-modle S x = lim U x Γ(U, S), By definition, for any open neighborhood U of x we have a map [ ] x : Γ(U, S) S x, s [s] x [s] x is called the germ of s at x We now define Γ(U, S + ) to be the sbset of U S consisting of families of germs (s ) U sch that every U there exist an open neighborhood V in U and a section s Γ(V, S) sch that [ s] v = s v, v V It is not hard to check that F + is indeed a sheaf It is also called the sheafification 2 of F Note that by constrction Γ(U, F) Γ(U, F + ) In Example 21 ˆF is the sheafification of F We can organize the collection of sheaves as a category Sh R () The correspondence F F + is a fnctor psh R () Sh R () Note that Sh R () is by definition a fll sbcategory of psh R () and ths we have an inclsion fnctor i : Sh R () psh R () For any F psh R (), G Sh R () and φ Hom pshr (F, G) there exists an indced morphism φ + Hom ShR (F +, G) One can show that reslting map Hom pshr (F, G) Hom ShR (F +, G) is an isomorphism of R-modle We can write this as Hom ShR (F +, G) = n Hom pshr ( F, i(g) ), F pshr (), G Sh R (), where = n means the isomorphism in natral with respect to the variables F and G In modern terms it says that + is the left adjoint of i Given F, G Sh R and φ Hom ShR (F, G) we can define ker φ, coker φ, im φ, coim φ Sh R () by sheafifying the corresponding constrctions for presheaves One can show that Sh R () is an Abelian category Any φ Hom ShR (F, G) indces morphisms between stalks φ x Hom R (F x, G x ), x A short seqence F φ G ψ H of sheaves and morphism of sheaves is exact in Sh R () if and only if for every x the seqence F x φ x ψ x Gx Hx is exact in R Mod More explicitly this means that for every x, any neighborhood U of x and any section G Γ(U, G) sch that ψ U (g) = 0 there exists a neighborhood V U containing x and a section f Γ(V, F) sch that φ V (f) = g V If R is a sheaf of rings on then we can define in an obvios fashion the category Sh R () of sheaves of R-modles 2 We refer to [1, II412] or [7] for a more conceptal constrction of the sheafification

6 6 LIVIU I NICOLAESCU For any sheaf S Sh R (), any open set U and any s Γ(U, S) we define the spport of s to be spp s = U; [s] 0 0} Note that spp s is closed in U with respect to the sbspace topology on U Example 22 All the presheaves defined in Example 11 are sheaves of complex vector spaces If We denote by A = A 0 the sheaf smooth complex valed fnctions on then A is natrally a sheaf of C algebras and A m is a sheaf of A-modles Theorem 23 (a) (Smooth Poincaré Lemma) Sppose is a smooth manifold Then the smooth DeRham seqence A 0 A m d A m+1 is exact Sh C () (b) (Dolbealt Lemma) Sppose is a complex manifold Then the Dolbealt seqences and A,0 A 0, A,q Ap, A,q+1 A p+1, are exact (c) (Holomorphic Poincaré Lemma) Sppose is a complex manifold Then the holomorphic DeRham complex is exact Ω 0 Ω m Ω m+1 For a proof of this theorem we refer to [3] Let s rephrase the above facts in the langage of homological algebra Sppose A is an Abelian category and (A, d) := A n d A n+1, n Z is a co-chain complex We denote its homology objects by H (A, d A ) We denote by Com(A) the additive category of complexes of A objects Then a morphism of cochain complexes φ : (A, d A ) (B, d B ) is called a qasi-isomorphism, or a resoltion if it indces an isomorphism in homology Every object A A defines a complex [A] [A] n A if n = 0 = 0 if n 0 A complex (C, d) is called a resoltion of the object A if it is qasi-isomorphic to [A] Sppose is a smooth manifold The smooth Poincaré lemma is eqivalent to the statement that the natral inclsion of C in the DeRham complex (A, d) is a resoltion If is a smooth manifold then the Dolbealt lemma shows that the inclsion Ω p (Ap,, ) is a resoltion Similarly, the holomorphic Poincaré lemma shows that the inclsion is a resoltion C (Ω, )

7 w SHEAVES 7 These resoltions of the constant sheaf se special featres of the topology of, sch as a smooth strctre or a complex strctre We describe below a constrction of a resoltion of an arbitrary sheaf on an arbitrary space Example 24 (The Godement resoltion) Let be a topological space For every sheaf F Sh R () we denote by D(F) the sheaf defined by Γ(U, D(F)) = U F In other words, the sections of D(f) are possibly discontinos sections of F Note that we have an inclsion F D(F) We can now describe indctively the Godement resoltion (F, d) We set F n = 0 for n < 0, F 0 = D(f) Sppose we have already constrcted d n 1 : F n 1 F n Then we set d n : F n coker d n 1 D(coker d n 1 ) =: F n+1 3 Direct and inverse images of sheaves Sppose f : Y is a continos map We want to constrct covariant fnctors f : Sh R () Sh R (Y ), f 1 : Sh R (Y ) Sh R (), called respectively the direct and inverse image fnctors Given F Sh R () we observe that the presheaf f F given by Y V Γ(V, f F) = Γ(f 1 V, F) is already a sheaf Next, for g Sh R (Y ) and any open set U we set Γ(U, f 1 G) = lim V f(u) Γ(V, G), where in the above limit V ranges over the open neighborhoods of f(u) in Y Then we define f 1 G to be the sheaf associated to the presheaf U Γ(U, f 1 G) Note that (f 1 G) x = Gx, Γ(V, f f 1 G) = Γ(f 1 V, f 1 G) From the definition Γ(f 1 V, f 1 G) = lim W Γ(W, G) f(f 1 V ) we dedce the existence of a natral morphism Γ(V, G) Γ(f 1 V, f 1 G) = Γ(V, f f 1 G) This defines a natral morphism of sheaves α : G f f 1 G called the adjnction morphism Given a sheaf F Sh R () and a morphism φ Hom ShR ()(f 1 G, f) we obtain a morphism φ v Hom ShR (Y )(G, f F) as the composition G α [ f f 1 G [[ φ v [] f F f (φ)

8 8 LIVIU I NICOLAESCU Proposition 31 The correspondence φ φ v defines a natral isomorphism Hom ShR ()(f 1 G, F) = n Hom ShR (Y )(G, f F), F, G (31) For a proof see [4, Thm 48] The above proposition states that f is the right adjoint of f 1 The next reslt follows easily from the definition Proposition 32 (a) f 1 is an exact fnctor while f is only left exact, ie for every exact seqence 0 S S S 0 in Sh R () the seqence 0 f S f S f S is exact in Sh R (Y ) (b) If f Y g Z are continos maps then we have eqivalences of fnctors (g f) = g f, (g f) 1 = f 1 g 1 Example 33 (a) Sppose c = c pt} is the collapse map Note that Sh R (pt) = R Mod so the direct image gives a fnctor c : Sh R () R Mod This is precisely the global sections fnctor Sh R () F Γ(, F) Note that c 1 R = R, c c 1 R = c (R) = Γ(, R) R Mod The adjnction morphism α : R Γ(, R) associates to each r R the constant section of R eqal to r (b) Consider the nit circle S 1 = z C; z = 1} and the doble cover p : S 1 S 1, z z 2 We wold like to nderstand p C Note that for any connected sbset U S 1 the preimage p 1 (U) consists of two connected components V ± and ths Γ(U, p C) = Γ(p 1 (U), C) = Γ(V, C) Γ(V +, C) = C 2 On the other hand p 1 (S 1 ) = S 1 and we dedce Γ(S 1, p C) = Γ(S 1, C) = C The sheaf p C is only locally constant Its stalks are all isomorphic to C 2 and for this reason we say it is locally constant sheaf of complex vector spaces of rank 2 For the classification of locally constant sheaves of vector spaces we refer to [4, IV9] (c) Sppose x and j x denotes the inclsion x} Then jx 1 F = F x, F Sh R () More generally, sppose S is a closed sbset and j : S denotes the natral embedding For any sheaf F on we set Note that for every open set V we have F S := j j 1 F Sh R () Γ(V, F S ) = Γ(V S, j 1 F) = lim W V S Γ(W, F) Hence we conclde that (F S ) x = Fx if x S 0 if x S

9 SHEAVES 9 The adjnction morphism α : F j j 1 F is called in this case the attachment map We will denoted it by a S The indced morphism a S x : F x (F S ) x satisfies a S 1Fx if x S x = 0 if x S (32) We set U = \ S, F U := ker α Sh R () Note that (F U ) x = Fx if x U 0 if x U We have a fndamental exact seqence of sheaves 0 F \S F F S 0 (33) Note that if Z is either closed, or open, and F is a sheaf on we have defined a sheaf F Z whose stalks are trivial otside Z, and coincide with the stalks of F on Z The correspondence F F Z is an exact covariant fnctor Sh R () Sh R () In general if is a topological space and Z a sbset of then the following are eqivalent (i) The sbset Z is locally closed, ie it is the intersection of a closed set in with an open sbset of (ii) For every sheaf F on there exists a sheaf F Z on sch that 0 if x \ Z (F Z ) x = F x if x Z For a proof of this fact we refer to [2, II29] For example, the set z C; Im z > 0} 0} is not locally closed (d) Sppose U is an open set and i : U denotes the natral inclsion For any sheaf F Sh R () the inverse image i 1 F is the restriction of F to U and it is often denoted by F U We set Γ U F := i i 1 F Sh R () For any open set V we have This implies Γ ( V, Γ U F ) = Γ(V U, F) (Γ U F) x = Fx if x U 0 if x \ Ū If x Ū \ U then it is possible that (Γ UF) x 0 Take for example = R 2, U = (x, y); xy 0}, x = (0, 0), F = A R 2 (34) The fnction 1 xy is a section of Γ UA on R 2 bt it is not eqal to a section of A in any neighborhood of zero If U is an open sbset of U then we have a natral morphism Γ U F Γ U F defined by Γ(V, Γ U F) = Γ(U V, F) f f U V Γ(U V, F) = Γ(V, Γ UF) We will denote this morphism as a restriction map, U U or U We set S = \U and we denote by Γ S F Sh S () the kernel of the adjnction morphism α : F i i 1 F The example (34) shows that α need not be onto Since the fnctor i is left exact and i 1 is exact we dedce that for any open set U the fnctor Γ U : Sh R () Sh R () is left exact

10 10 LIVIU I NICOLAESCU The sheaf Γ S F can be alternatively characterized by Γ(U, Γ S F) = f Γ(U, F); spp f S } We leave it as an exercise to prove that the fnctor Γ S : Sh R () Sh R () is left exact The fnctors Γ S, Γ U : Sh R () Sh R () defined above play an important role in local cohomology 4 The Čech resoltions Given a family (F j ) j J of sheaves of R-modles we denote by j J F j the sheaf defined by Γ(U, F j ) := Γ(U, F J ) j J j J ( We want to point ot that the natral morphism j J j) F j J (F j) x need not be an x isomorphism Sppose U = (U i ) i I is an open cover of the topological space We fix a total order 3 on I For every sbset α I we define U α = i α U i The nerve of the cover is the collection of all finite sbset α I sch that U α For α = α 0,, α p } N(U) we set dim α := p We will refer to sch a set as a p-simplex The 0-simplices are called vertices We will write α = (α 0,, α p ) when we want to indicate that the vertices of α are arranged in increasing order We set N p (U) = α N(U); dim α = p } The nerve is a simplicial complex in the sense that if β N(U) and α β then α N(U) For every α = (α 0,, α p ) N p (U) and 0 i p we define its i-th face, or the face opposite to the vertex α i to be the (p 1)-simplex Now define 4 C p (U, F) = and where F i α = α \ α i } N p 1 (U) α N p (U) Γ Uα F Sh R () δ : C p (U, F) C p+1 (U, F), δ = p+1 δ β = ( 1) i δ F iβ i=0 β, β N p+1 (U) δ β 3 We can even choose a well ordering on a set, that is a total order sch that every nonempty sbset has minimal element The existence of sch orderings on an arbitrary sbset is eqivalent to the axiom of choice 4 Or definition differs from that in [8, vol I, 413] That definition is incorrect since it ses direct sms as opposed to direct prodcts

11 w and δ F iβ β is the composition α N p (U) SHEAVES 11 Γ Uα F π Fi β ( (( () δ F i β β Γ UFi β F Γ Uβ F U β U Fi β If V is an open sbset of then Γ ( V, C p (U, F) ) = α N p(u) Γ(V U α, F) We can represent s Γ ( V, C p (U, F) ) as a collection } s(α); α N p (U), s(α) Γ(V U α, F) Then we can represent δs as a collection δs (β); β N p+1 (U) } where p+1 δs (β) = ( 1) i s (F i β) Uβ = ( 1) i s (β 0,, ˆβ i,, β p+1 ) U Uβ Γ(V U β, F) i=0 p+1 i=0 Using the total ordering < on I we define for any vertex v β the integers ν(v, β) = # j β; j < v }, ɛ(v, β) = ( 1) ν(v,β) The integer ν(v, β) describes the position of the vertex v in the set β arranged in increasing order The definition of δ can be rewritten δs (β) = v β ɛ(v, β) s (F v β) Uβ Note that we have a natral morphism of complexes r : [F] C (U, F), Γ(V, F) ( f V Ui ) i I C 0 (U, F) Example 41 Sppose U = U 0, U 1 } Then C 0 (U, F) = Γ U0 F Γ U1 F, C 1 (U, F) = Γ U01 F For any open set V we have Γ ( V, C 0 (U, F) ) = Γ(V U 0, F) Γ(V U 1, F), Γ ( V, C 1 (U, F) ) = Γ(V U 01, F) and δ : Γ ( V, C 0 (U, F) ) Γ ( V, C 1 (U, F) ) is given by δ(s 0, s 1 ) = s 0 V U01 s 1 V U01 Theorem 42 Sppose U is an open cover of the topological space F Sh R () then the natral morphism of complexes [F] C (U, F) is a resoltion called the Čech resoltion determined by the open cover U In other words the long seqence is exact in Sh R () 0 F r C 0 (U, F) C p δ (U, F) C p+1 (U, F)

12 12 LIVIU I NICOLAESCU Proof The exactness at F and C 0 follows from the properties of a sheaf, namely that for any open set the map Γ(V, F) Γ(V U, F) is bijective Let s prove the exactness at C p, p 1 It sffices to show that for every x the seqence C p 1 δ (U, F) p 1 x C p δ (U, F) p x C p+1 (U, F) x is exact Let x and s x C p (U, F) x sch that δ p s x = 0 We can represent the germ s x by a section s Γ(V, C p (U, F), ), where V is an open neighborhood of x By shrinking V if needed we can assme that V is contained in some open set U l U Set St(x, U) := α N(U); V U α } St(x, U) is a simplicial sbcomplex of N(U) Note also that α St(x, U) = U l U α = l} α N(U) This sbcomplex is the star of the vertex l in N(U) and in particlar it is contractible This is essentially the reason for the acyclicity of the Čech complex We denote by St m (x, U) the set of m-simplices in St(x, U) and by St m (x, U) the set of m-simplices which do not contain l as a vertex Note that we have a natral map St p 1 (x, U) α l α := l} α St p (x, U) Geometrically, l α is the cone on α with vertex l The germ s x is represented by a collection of sections s (α) Γ(V U α, F), α St p (, U) The condition δ p s x = 0 is eqivalent to Now define σ We want to prove that p+1 ( 1) i s (F i β) V Uβ = 0, β St(x, U) p+1 i=1 β St p 1 (x,u) σ (β) = Γ(V, Γ Uβ F) = β St p 1 (x,u) 0 if l β ɛ(l, l β) s (l β) if l β δσ (α) = s (α), α St(x, U) We distingish two cases l α so that α = l, β 1,, β p } = l β Then l α = (α 0,, α p ) Then p δσ (α) = ɛ(j, α)σ (F j α) V Uα = j α δσ (α) = ɛ(l, α)σ (β) = s (l β) = s (α) Γ(V U β, F) p ɛ(j, α) ɛ(l, l F i α) s (l F i α) V Uα i=0 = s (α) δs (l α) = s (α) Consider for example the sitation depicted in Figre 1 where p = 2, the simplex α is

13 SHEAVES Figre 1 A three-dimensional simplex (α 0, α 1, α 2 ) = (0, 2, 3) (marked in green), and the vertex l is l = 1 (marked in red) The condition that s is a cocycle translates into Note that for j > 1 while for j > i > 1 we have Then 0 = δs (0, 1, 2, 3) = s, (1, 2, 3) s (0, 2, 3) + s (0, 1, 3) s (0, 1, 2) σ, (0, j) = s (0, 1, j), σ (i, j) = s (1, i, j) δσ (0, 2, 3) = σ (2, 3) σ (0, 3) + σ (0, 2) = s (1, 2, 3) + s (0, 1, 3) s (0, 1, 2) In view of the cocycle condition the last expression is eqal to s (1, 2, 3) Remark 43 As explained in [5, 28], the Čech complex of sheaves associated to an open cover and a sheaf is locally the cone complex of a morphisms between complexes This explains its acyclicity Sppose now S = (S i ) i I is a locally finite closed cover of This means that the sets S i are closed, their nion is and any point x has a neighborhood which intersects only finitely many of the sets S i Fix a total order on I The nerve N(S) of this cover is defined as before For every simplex α N(S) we set as before S α = v α S v Note that if α β N(S) then S α S β and we have a natral isomorphism F Sβ = (FSα ) Sβ In particlar we have a natral attachment map a β α : F Sα F Sβ Now define C p (S, F) := F Sα and δ p : C p (S, F) C p+1 (S, F), δ = α N p (S) where δ β = ɛ(v, β)a β F v β v β Note that we have a natral attachment map a S : F C 0 (S, F), a S = S S a S β N p+1 (S) δ β,

14 14 LIVIU I NICOLAESCU Theorem 44 Sppose is a T 3 -space 5 If S = (S i ) i I is a locally finite closed cover of then the long exact seqence of sheaves 0 F as C 0 (S, F) δ0 δp 1 C p (S, F) δp is exact In other words, the morphism of complexes a S : [F] C (S, F) is a resoltion of F called the Čech resoltion determined by the closed cover S Proof The injectivity of a S follows from the injectivity of the explicit description (32) of the attachment maps The fact that im a S = ker δ 0 follows from the srjectivity of the attaching maps a S We need to check the exactness of C p 1 (S, F) x δ C p δ (S, F) x C p+1 (S, F) x, for any x Let s x C p (S, F) x sch that δs x = 0 We can represent s x as the germ of a section s Γ ( U, C p (S, F) ), where U is a very small neighborhood of x which intersects finitely many S i s Since is a T 3 we can choose U small enogh so that the set S i intersects U if and only if it contains the point x We let I x := i I; x S i }, S x = (S i ) i Ix, l = min I x Note that the simplicial complex N(S x ) is an elementary simplex In particlar it is contractible The section s is described by a family Now define by setting σ s(α) Γ(U, F Sα ), α N p (S x ) β N p 1 (S x) σ(β) = Γ(U, F Sβ ) Γ(U, C p 1 (S, F)) 0 if l β s(l β) if l β A comptation identical to the one in the proof of Theorem 42 shows that δσ = s Sppose U is an open cover of the topological space The Čech resoltion constrction associates to each sheaf F Sh R () a complex C (U, F) Com(Sh R () ) This constrction is fnctorial in the sense that any morphism of sheaves Φ : F 0 F 1 indces a morphism of complexes of sheaves Φ U : C (U, F 0 ) C (U, F 1 ) satisfying all the fnctorial reqirements In particlar, to every complex (F, d) we can associate a doble complex ( ) C (U, F ), d I, d II, d p I = δp : C p (U, F ) C p+1 (U, F ) d q II = ( 1)p d : C p (U, F q ) C p (U, F q+1 ) 5 We se the conventions in [6, Chap 4] so is T3 if it is Hasdorff and for every point x and any closed set S not containing x there exist disjoint neighborhoods of x and C in For example, all paracompact spaces ar reglar

15 SHEAVES 15 References [1] SI Gelfand, YI Manin: Methods of Homologcal Algebra, Springer Monographs in Mathematics, 2003 [2] R Godement: Topologie Algébriqe et Théorie des faisceax, Hermann 1958 [3] PA Griffiths, J Harris: Principles of Algebraic Geometry, John Wiley& Sons, 1978 [4] B Iversen: Cohomology of Sheaves, Universitext, Springer Verlag, 1986 [5] M Kashiwara, P Schapira: Sheaves on Manifolds, Gründlehren der mathematischen Wissenschaften,vol 292, Springer Verlag, 1990 [6] JL Kelley: General Topology, Gradate Texts in Mathematics, vol 27, Springer Verlag, 1975 [7] P Schapira: Sheaves, [8] CVoisin: Hodge Theory and Complex Algebraic Geometry, Cambridge Stdies in Adv Math vol 76-77, Cambridge University Press, 2002 Department of Mathematics, University of Notre Dame, Notre Dame, IN address: nicolaesc1@nded