A Coboundary Morphism For The Grothendieck Spectral Sequence

A COBOUND AR Y MOR PHISM F OR THE GR OTHENDIECK SPECTRAL SEQUENCE DA VI D BARAGLI A Abstract. Giv en an abeli an category A with enough injectiv es w e show t hat a short exac t sequence of c hain complexes of ob jects in A gi ves rise t o a short exact sequence of Cartan-Eilenberg resolutions. Using this we construct coboundary morphisms betw een Grothendiec k spectral s equences associated to ob jects in a short exact sequence. W e sho w that the coboundary preserves the filtrations associated with the sp ectral sequenc es and give an appli cation of these result to filtrations in sheaf cohomology . 1. Introduction Whenever sp ectral sequences occur in practice there is usually a corresp onding filtration on the ob jects o ne would like to compute. The a sso ciated graded ob j ects of the filtration coincide with the limit of the sp ectra l sequenc e . There ar e a lso situations wher e such filtrations ar e of int erest indep endent fro m the sp e c tral se - quence. F or example this is true of the filtration asso ciated to the Leray sp ectral sequence. In this pap er we es tablish a gene r al r esult allowing us to compar e the filtrations asso ciated to certain s pe c tral sequences. Sp ecifically we are concer ned with the behavior of the Grothendieck sp ectral sequence on short exa c t sequences and the implications this has o n the ass o ciated filtrations. Let us r ecall the conten t of the Grothendieck sp ectra l s equence. Let A , B , C b e ab elian catego ries, F : A → B , G : B → C left exact functor s. Suppo se A , B hav e eno ug h injectives and F sends injectiv e o b jects to G -acyclic ob jects. Giv en an ob ject A ∈ A there is a sp ectr al sequence { E p,q r ( A ) , d r } consis ting of ob jects in C a nd filtration F p R n ( G ◦ F )( A ) on R n ( G ◦ F ) such that the sp ectra l sequence conv er ges to the asso ciated graded ob jects and such that E p,q 2 ( A ) = R p G ( R q F ( A )). The details o f the s pe c tral sequence a nd filtration along with o ur notation ar e describ ed in Section 3. Now it is clear from the construction of the Grothendieck sp ectral sequence that given tw o ob jects A, B ∈ A and a morphism A → B the induced maps R n ( G ◦ F )( A ) → R n ( G ◦ F )( B ) resp ect the filtra tio ns a nd the maps induced on the asso ciated g raded o b jects come from a morphism E p,q r ( A ) → E p,q r ( B ) of sp ectral sequences. The maps E p,q 2 ( A ) → E p,q 2 ( B ) a re of course just the induced mo rphisms R p G ( R q F ( A )) → R p G ( R q F ( B )). W e thus ha ve a go o d under standing of how the sp ectral sequences of A a nd B are rela ted. Date : Septem ber 20, 2018. 2010 Mathematics Subje ct Classific ation. Pri m ary 18G40, 18G10; Secondary 55Txx. This w or k is supp orted by the Australian Researc h Council Di sco v ery Pro j ect DP110103745. 1 2 DA VID BARAG LIA Consider now a short exact s equence 0 → A → B → C → 0 in A . The ma in question we s eek to answer is ho w the sp ectral sequences and filtratio ns asso ciated to A and C are rela ted. Of course we g et corr esp onding morphisms E p,q r ( A ) → E p,q r ( B ) and E p,q r ( B ) → E p,q r ( C ) but the comp ositio n E p,q r ( A ) → E p,q r ( B ) → E p,q r ( C ) is trivial, so this do es not help to directly rela te E p,q r ( A ) and E p,q r ( C ). Our main result, Theorem 4.1 is that ther e is a kind o f cob ounda ry morphism of sp ectral sequences E p,q r ( C ) → E p,q +1 r ( A ) and c lo sely related to this is the fa c t tha t the cob oundary maps R n ( G ◦ F )( C ) → R n +1 ( G ◦ F )( A ) in the long exac t seque nce asso ciated to 0 → A → B → C → 0 and G ◦ F resp ects the filtrations. W e s ta te the full result: Theorem 1.1. L et 0 → A → B → C → 0 b e a short exact se quenc e in A . Ther e ar e morphisms δ r : E p,q r ( C ) → E p,q +1 r ( A ) for r ≥ 2 b etwe en the Gr othendie ck sp e ctr al se quenc es for C and A with the fol lowing pr op erties: • δ r c ommutes with the differ entials d r and the induc e d map at the ( r + 1) - stage is δ r +1 . • δ 2 : R p G ( R q F ( C )) → R p G ( R q +1 F ( A )) is t he map induc e d by the b ound- ary morphism R q F ( C ) → R q +1 F ( A ) in the long ex act se quenc e of derive d functors of F asso ciate d to 0 → A → B → C → 0 . • The b oun daries R n ( G ◦ F )( C ) → R n +1 ( G ◦ F )( A ) for the long exact se quenc e asso ciate d to G ◦ F send F p R n ( G ◦ F )( C ) to F p R n +1 ( G ◦ F )( A ) and t hus induc e maps E p,q ∞ ( C ) → E p,q +1 ∞ ( A ) . These maps c oincide with δ ∞ wher e δ ∞ denotes the limit of the δ r . In Section 5 we sp ecialize to the ca se o f the Ler ay sp ectra l s equence. Let X, Y be pa racompac t spac es and f : X → Y contin uous. Consider on X the ex po nen- tial sequence 0 → Z → C → C ∗ → 0, wher e A indicates the s heaf of contin uo us functions v alued in an ab elian gro up A . A well known fact is that the cob oundar y map δ : H n ( X, C ∗ ) → H n +1 ( X, Z ) is an isomor phism for n ≥ 1. Note how ever that the Ler ay sp ectra l sequences ass o ciated to the ma p f : X → Y determine filtrations on H n ( X, C ∗ ) and H n +1 ( X, Z ). It is natural then to ask how these t wo filtrations compare under the cob o unda ry δ . W e give the answer in Theo rem 5.2, as an application of Theor e m 4.1. Section 2 contains the main technical result needed for Theo rem 4.1. The result here is that given a sho r t exa ct sequence of chain complexe s 0 → A ∗ → B ∗ → C ∗ → 0 in an ab e lian c ategory with enoug h injectives one can c o nstruct Cartan-Eilenber g resolutions for A ∗ , B ∗ , C ∗ in such a w ay that they fit into a short exact sequence of double co mplexes. Section 3 re c a lls the neces sary details of the Grothendieck sp ectral sequence, Section 4 is the pro of of the ma in result and Section 5 the application to sheaf cohomo logy previously describ ed. 2. Car t an-Eilenberg resolutions Let A be an ab elian category . W e recall the no tion of a Cartan-E ilenberg reso- lution. Definition 2 .1. Let A ∗ be a co mplex in A . A Cartan-Eilenb er g r esolution [8 ],[5] of A ∗ is a sequence of complexes I 0 , ∗ , I 1 , ∗ , I 2 , ∗ , . . . tog ether with chain ma ps I p, ∗ → I p +1 , ∗ and injective c hain map A ∗ → I 0 , ∗ such tha t • each o b ject I p,q is injectiv e, A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 3 • the sequence 0 → A ∗ → I 0 , ∗ → I 1 , ∗ → · · · is ex act, • let Z q ( A ) denote the degree q co cycles of A ∗ and Z p,q ( I ) the deg ree q co cycles of I p, ∗ . The induced sequence on co cycles Z q ( A ) → Z 0 ,q ( I ) → Z 1 ,q ( I ) → · · · is required to b e an injective resolution of Z q ( A ). In par tic- ular each Z p,q ( I ) is injective, • similarly for co b oundaries and cohomolo g y the induced seq ue nc e s are in- jective r esolutions. Our main res ult is that for a s hort exact se quence o f chain co mplex es in an ab elian categ ory with e nough injectives one c a n construct corresp onding sho rt exact sequence of Cartan-E ilenberg resolutions. Theorem 2.2. L et 0 → A ∗ → B ∗ → C ∗ → 0 b e a short exact se qu enc e of chain c omplexes in an ab elian c ate gory with enough inje ctives. Then t her e exists a short exact se quenc e 0 → I ∗ → J ∗ → K ∗ → 0 of chain c omplexes of inje ctives, to gether with inje ctions A q → I q , B q → J q , C q → K q for al l q forming a c ommut ative diagr am 0 / / A ∗ / /   B ∗ / /   C ∗ / /   0 0 / / I ∗ / / J ∗ / / K ∗ / / 0 L et Z ∗ ( A ) , B ∗ ( A ) , H ∗ ( A ) denote the c o cycles, c ob oundaries and c ohomolo gy for the c omplex A ∗ and use similar notation fo r the other c omplexes. W e may in ad dition make t he choi c es so t hat the Z ∗ ( I ) , B ∗ ( I ) , H ∗ ( I ) ar e al l inje ct ives, t he natura l maps Z ∗ ( A ) → Z ∗ ( I ) , B ∗ ( A ) → B ∗ ( I ) , H ∗ ( A ) → H ∗ ( I ) ar e inje ctive and such that the c orr esp onding st atements for J ∗ , K ∗ hold as wel l . Mor e over if A q = B q = C q = 0 whenever q < 0 then we may cho ose the I ∗ , J ∗ , K ∗ so that likewise I q = J q = K q = 0 fo r q < 0 . Before w e pr ov e Theorem 2.2 let us state the main result a nd show ho w it follows: Theorem 2.3. L et 0 → A ∗ → B ∗ → C ∗ → 0 b e a short exact se qu enc e of chain c omplexes in an ab elian c ate gory with enou gh inje ctives. Then ther e exists Cartan- Eilenb er g r esolutions 0 → A ∗ → I 0 , ∗ → I 1 , ∗ → · · · , 0 → B ∗ → J 0 , ∗ → J 1 , ∗ → · · · , 0 → C ∗ → K 0 , ∗ → K 1 , ∗ → · · · f or A ∗ , B ∗ , C ∗ and maps I q, ∗ → J q, ∗ → K q, ∗ forming a c ommu tative diagr am of chain c omplexes 0   0   0   0 / / A ∗ / /   B ∗ / /   C ∗ / /   0 0 / / I 0 , ∗ / /   J 0 , ∗ / /   K 0 , ∗ / /   0 0 / / I 1 , ∗ / /   J 1 , ∗ / /   K 1 , ∗ / /   0 . . . . . . . . . 4 DA VID BARAG LIA wher e in addition the r ows ar e exact. Mor e over if A q = B q = C q = 0 for q < 0 we may cho ose the chain c omplex so that in addi tion I p,q = J p,q = K p,q = 0 for q < 0 as wel l. Pr o of. W e use Theorem 2.2 to construct the fir st row 0 → I 0 , ∗ → J 0 , ∗ → K 0 , ∗ → 0 forming a commutativ e diagram with exact rows and co lumns 0   0   0   0 / / A ∗ / / α ∗   B ∗ / / β ∗   C ∗ / / γ ∗   0 0 / / I 0 , ∗ / / J 0 , ∗ / / K 0 , ∗ / / 0 Next take the cokernels of the vertical ma ps to obtain 0   0   0   0 / / A ∗ / / α ∗   B ∗ / / β ∗   C ∗ / / γ ∗   0 0 / / I 0 , ∗ / /   J 0 , ∗ / /   K 0 , ∗ / /   0 0 / / cok er ( α ∗ ) / /   cok er ( β ∗ ) / /   cok er ( γ ∗ ) / /   0 0 0 0 Note that the b ottom row is exa c t by the Nine lemma [7]. W e may now apply Theorem 2.2 to 0 → cok er ( α ∗ ) → cok er ( β ∗ ) → cok er ( γ ∗ ) → 0 to obtain the next row 0 → I 1 , ∗ → J 1 , ∗ → K 1 , ∗ → 0. Co nt inuin g in this fashion we co nstruct the desired double co mplex of chain complexes with exact columns and s ho rt ex act rows. It rema ins to s how that the res o lutions so o btained of A ∗ , B ∗ , C ∗ are indeed Cartan-E ilenberg r esolutions. W e will show that 0 → A ∗ → I 0 , ∗ → I 1 , ∗ → · · · is a Car tan-Eilenberg resolution. The pro ofs for B ∗ and C ∗ are ide ntical. By co n- struction we hav e that the Z p, ∗ ( I ) , B p, ∗ ( I ) , H p, ∗ ( I ) are all injective ob jects and that the maps Z ∗ ( A ) → Z 0 , ∗ ( I ), B ∗ ( A ) → B 0 , ∗ ( I ), H ∗ ( A ) → H 0 , ∗ ( I ) are all injectiv e. It r emains to that the seq ue nc e s · · · → Z p, ∗ ( I ) → Z p +1 , ∗ ( I ) → · · · , · · · → B p, ∗ ( I ) → B p +1 , ∗ ( I ) → · · · , · · · → H p, ∗ ( I ) → H p +1 , ∗ ( I ) → · · · are ex act. Consider the exact sequence · · · → I p − 1 , ∗ → I p, ∗ → I p +1 , ∗ → · · · of c hain complexes. Letting C p, ∗ denote the cokernel of I p − 1 , ∗ → I p, ∗ we g e t a co mm utative A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 5 diagram 0 # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ 0 # # ● ● ● ● ● ● ● ● ● C p, ∗ $ $ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ . . . / / I p − 1 , ∗ $ $ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ / / I p, ∗ < < ① ① ① ① ① ① ① ① / / I p +1 , ∗ / / # # ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ . . . C p − 1 , ∗ $ $ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ; ; ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ 0 9 9 t t t t t t t t t t t 0 where the diagonal sequences are exact. More over by the construction of the com- plexes I p, ∗ we kno w that the maps C p, ∗ → I p +1 , ∗ are injective on c o cycles, cob ound- aries and coho mo logy as in Theor e m 2.2. Let Z p, ∗ ( C ) , B p, ∗ ( C ) , H p, ∗ ( C ) denote the co cycles, c o b oundaries and cohomolo gy for the complex C p, ∗ . Let us chec k e x act- ness at I p, ∗ for cocycles . The kernel of Z p, ∗ ( I ) → Z p +1 , ∗ ( I ) is equal to the kernel of Z p, ∗ ( I ) → Z p, ∗ ( C ) which in turn is equal to Z p − 1 , ∗ ( C ). Now using the fac t that H p − 2 , ∗ ( C ) → H p − 1 , ∗ ( I ) is injectiv e w e immediately se e that Z p − 1 , ∗ ( I ) → Z p − 1 , ∗ ( C ) is surjective. It follows that the kernel of Z p, ∗ ( I ) → Z p +1 , ∗ ( I ) is exactly the imag e Z p − 1 , ∗ ( I ) → Z p, ∗ ( I ) as requir e d. A similar pro of sho ws exa c tness a t I p, ∗ for cohomolo gy and b oundar ies. The final statement about v anishing of negative degree terms follows from the similar result for Theor em 2.2.  Pr o of of The or em 2.2. Given a short exact s equence 0 → A ∗ → B ∗ → C ∗ → 0 of chain complexes in tro duce the following ob jects: co cycles Z ∗ ( A ) , Z ∗ ( B ) , Z ∗ ( C ), cob oundaries B ∗ ( A ) , B ∗ ( B ) , B ∗ ( C ), cohomolog ies H ∗ ( A ) , H ∗ ( B ) , H ∗ ( C ) and t wo more sets of ob jects W ∗ ( A ) , W ∗ ( B ) , W ∗ ( C ) , X ∗ ( A ) , X ∗ ( B ) , X ∗ ( C ). W e define W q ( A ) to be the k er nel of the induced map H q ( A ) → H q ( B ) and W q ( B ) , W q ( C ) are simila rly defined as kernels in the long exact sequence in cohomo logy . W e de- fine X q ( A ) to be the kernel of the comp osition Z q ( A ) → H q ( A ) → H q ( B ) and similarly define X q ( B ) , X q ( C ). The o b jects so defined fit in to a v ariety of short exact sequences as follows. First the long exact se quence in cohomolo gy we ha ve: 0 → W q ( A ) → H q ( A ) → W q ( B ) → 0 , (2.1) 0 → W q ( B ) → H q ( B ) → W q ( C ) → 0 , (2.2) 0 → W q ( C ) → H q ( C ) → W q +1 ( A ) → 0 . (2.3) F rom the definition of the o b jects X q ( A ) , X q ( B ) , X q ( C ) we ha ve: 0 → X q ( A ) → Z q ( A ) → W q ( B ) → 0 , (2.4) 0 → X q ( B ) → Z q ( B ) → W q ( C ) → 0 , (2.5) 0 → X q ( C ) → Z q ( C ) → W q +1 ( A ) → 0 . (2.6) 6 DA VID BARAG LIA F rom the definition of coho mology: 0 → B q ( A ) → Z q ( A ) → H q ( A ) → 0 , (2.7) 0 → B q ( B ) → Z q ( B ) → H q ( B ) → 0 , (2.8) 0 → B q ( C ) → Z q ( C ) → H q ( C ) → 0 . (2.9 ) Also from the definition of the X q ( A ) , X q ( B ) , X q ( C ): 0 → B q ( A ) → X q ( A ) → W q ( A ) → 0 , (2.10 ) 0 → B q ( B ) → X q ( B ) → W q ( B ) → 0 , (2.11) 0 → B q ( C ) → X q ( C ) → W q ( C ) → 0 . (2.1 2) F rom the definition of cycles a nd b oundaries: 0 → Z q ( A ) → A q → B q +1 ( A ) → 0 , (2.13) 0 → Z q ( B ) → B q → B q +1 ( B ) → 0 , (2.14) 0 → Z q ( C ) → C q → B q +1 ( C ) → 0 . (2.15) Finally a few more exa ct sequences that can b e ea sily shown: 0 → X q ( A ) → B q ( B ) → B q ( C ) → 0 , (2.16) 0 → Z q ( A ) → Z q ( B ) → X q ( C ) → 0 , (2.17) 0 → Z q ( A ) → X q ( B ) → B q ( C ) → 0 , (2.18) 0 → A q → B q → C q → 0 . (2.19) W e choose fiv e families of injective ob jects indexed b y the integer q whic h we suggestively denote as follows W q ( I ) , W q ( J ) , W q ( K ) , B q ( I ) , B q ( K ). By a s sump- tion A has enough injectives s o we can choose these ob jects together with injec- tions W q ( A ) → W q ( I ), W q ( B ) → W q ( J ), W q ( C ) → W q ( K ), B q ( A ) → B q ( I ), B q ( C ) → B q ( K ). W e aim ultimately to construct chain complexes I ∗ , J ∗ , K ∗ such that the ob jects W q ( I ) , W q ( J ) , W q ( K ) , B q ( I ) , B q ( K ) agr e e with the ob jects that their notation sugges ts . W e further define the following ob jects H q ( I ) = W q ( I ) ⊕ W q ( J ) , (2.20) H q ( J ) = W q ( J ) ⊕ W q ( K ) , (2.21) H q ( K ) = W q ( K ) ⊕ W q +1 ( I ) , (2.22) B q ( J ) = W q ( I ) ⊕ B q ( I ) ⊕ B q ( K ) , (2.23) X q ( I ) = W q ( I ) ⊕ B q ( I ) , (2.24) X q ( J ) = W q ( I ) ⊕ W q ( J ) ⊕ B q ( I ) ⊕ B q ( K ) , (2.25) X q ( K ) = W q ( K ) ⊕ B q ( K ) , (2.26) Z q ( I ) = W q ( I ) ⊕ W q ( J ) ⊕ B q ( I ) , (2.27) Z q ( J ) = W q ( I ) ⊕ W q ( J ) ⊕ W q ( K ) ⊕ B q ( I ) ⊕ B q ( K ) , (2.28) Z q ( K ) = W q ( K ) ⊕ W q +1 ( I ) ⊕ B q ( K ) , (2.29) I q = Z q ( I ) ⊕ B q +1 ( I ) , (2.30) J q = Z q ( J ) ⊕ B q +1 ( J ) , (2.31) K q = Z q ( K ) ⊕ B q +1 ( K ) . (2.32) The idea behind these definitions is that assuming the existence of the desired complexes I ∗ , J ∗ , K ∗ we hav e exa c t s equences like E quations (2.1)-(2.19). If in A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 7 addition all the cocy cles, cob oundaries a nd so forth ar e injective then a ll these sequences would b e split a nd the ab ove definitio ns would hold. In fact we will now give I ∗ , J ∗ , K ∗ as defined in (2.30)-(2.32) the structure of co chain complexes so that these assumptions a ctually hold true. First of all note tha t using inclusions and pro jections there are unique ways to define maps betw een the v arious ob jects so that the sho rt exact sequenc e s corr esp onding to (2.1 )-(2.19) hold (with I , J, K in pla ce of A, B , C ). F or example the exa ct sequence 0 → W q ( I ) → H q ( I ) → W q ( J ) → 0 is just the spit exact seq uence 0 → W q ( I ) → W q ( I ) ⊕ W q ( J ) → W q ( J ) → 0. Next we define a differential d : I q → I q +1 as the comp ositio n I q → B q +1 ( I ) → Z q +1 ( I ) → I q +1 . Note tha t the three maps in this comp osition corre s po nd to maps in the sequences (2.1 3),(2.7 ) a nd (2.13) again. Thus by (2.13) we see tha t d 2 = 0 so that this indeed defines a differential. W e can similarly define differe nt ials for J ∗ and K ∗ . W e claim that the ma ps I q → J q → K q as in (2.19) ar e in fact morphisms of complexes. T o see this one easily chec ks that the following dia gram commutes: I q / / ( 2 . 19 )   B q +1 ( I ) / / ( 2 . 10 )   Z q +1 ( I ) / / ( 2 . 17 )   I q +1 ( 2 . 19 )   X q +1 ( I ) ( 2 . 16 )   J q / / ( 2 . 19 )   B q +1 ( J ) / / ( 2 . 16 )   Z q +1 ( J ) / / ( 2 . 17 )   J q +1 ( 2 . 19 )   X q +1 ( K ) ( 2 . 6 )   K q / / B q +1 ( K ) / / Z q +1 ( K ) / / K q +1 The lab els o n the vertical arrows indicate that these ma ps are the same as the maps defined a s in the exact seq uence indicated b y the lab el. The horizontal rows are precisely the maps defining the differen tials for I ∗ , J ∗ , K ∗ so commutativit y o f this diagram implies the maps I ∗ → J ∗ → K ∗ are ch ain maps. Now that we have a shor t exact s equence 0 → I ∗ → J ∗ → K ∗ → 0 of chain complexes we get asso cia ted spaces of co cycles, cob ounda ries, cohomolog y a nd so on. One can check easily that the Z ∗ ( I ) , Z ∗ ( J ) , Z ∗ ( K ) a re indeed the co cycles, B ∗ ( I ) , B ∗ ( J ) , B ∗ ( K ) are indeed the cob oundar ies and so forth. Thus the ob jects W ∗ ( I ) , W ∗ ( J ) , W ∗ ( K ) , B ∗ ( I ) , B ∗ ( J ) and the v ar ious ob jects defined in E quations (2.20)-(2.29) coincide with their na mesake. The next part of the pro of is to construct c hain maps A ∗ → I ∗ , B ∗ → J ∗ and C ∗ → K ∗ . Such maps would induce ma ps betw ee n co cycles , cobounda ries and so on. Thus our strategy will be to construct these maps by w orking backwards, s tart- ing from the existing maps W q ( A ) → W q ( I ), W q ( B ) → W q ( J ) , . . . a nd working our wa y through the exa ct sequences (2 .1)-(2.19). In other words for each of the 8 DA VID BARAG LIA short exact sequences (2.1)-(2.19), we wan t to construct a corresp o nding co mmu- tative dia gram of short exact sequences. F or exa mple we start off with (2.1). The desired commutativ e diagr am is as follows: 0 / / W q ( I ) / / H q ( I ) / / W q ( J ) / / 0 0 / / W q ( A ) / / O O H q ( A ) / / O O W q ( B ) / / O O 0 where the maps W q ( A ) → W q ( I ), W q ( B ) → W q ( J ) ar e the pre v iously chosen injections. Since H q ( I ) = W q ( I ) ⊕ W q ( J ) we need to chose maps f : H q ( A ) → W q ( I ) and g : H q ( A ) → W q ( J ). T he diagram will commute if and only if the following diagr ams comm ute: W q ( I ) W q ( A ) : : t t t t t t t t t / / H q ( A ) f O O and W q ( J ) H q ( A ) g : : t t t t t t t t t / / W q ( B ) O O In the first case s uch a ma p f ex ists b ecause W q ( I ) is an injective o b ject a nd W q ( A ) → H q ( A ) is an injection. In the s e c ond c a se g exists just by defining it to be the comp osition. No te a ls o that the map H q ( A ) → H q ( I ) so defined is injective. Using the exact s ame rea soning we construct similar injections H q ( B ) → H q ( J ), H q ( C ) → H q ( K ) yielding co mmu tative diagrams corr esp onding to the se q uences (2.2) and (2.3). Similarly w e construct injections X k ( A ) → X k ( I ) and X k ( C ) → X k ( K ) yielding commut ative diagrams corresp onding to (2.10) and (2.1 2). F rom this we may co nstruct an injection B q ( B ) → B q ( J ) yielding c ommut ative diagra m corres p o nding to (2 .16). How ever when w e cons ider the construc tio n of a ma p Z q ( A ) → Z q ( I ) w e run into a co mplication, namely there are tw o exa ct sequences (2.4),(2.7) with Z q ( A ) as the middle term. W e would lik e to choose the map Z q ( A ) → Z q ( I ) to yield co mm utative dia grams cor resp onding to b oth o f thes e exact sequences. F or clarity we write out the tw o desir ed commutativ e diag r ams 0 / / X q ( I ) / / Z q ( I ) / / W q ( J ) / / 0 0 / / X q ( A ) / / O O Z q ( A ) / / O O W q ( B ) / / O O 0 and 0 / / B q ( I ) / / Z q ( I ) / / H q ( I ) / / 0 0 / / B q ( A ) / / O O Z q ( A ) / / O O H q ( A ) / / O O 0 A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 9 All the ma ps in these diagr ams are already determined except the desired map Z q ( A ) → Z q ( I ). By s equence (2.27) we hav e Z q ( I ) = W q ( I ) ⊕ W q ( J ) ⊕ B q ( I ), so we need to cho ose maps Z q ( A ) → W q ( I ), Z q ( A ) → W q ( J ), Z q ( A ) → B q ( I ). The comp onents Z q ( A ) → W q ( I ) a nd Z q ( A ) → W q ( J ) are determined by r equiring a commutativ e diagra m H q ( I ) Z q ( A ) : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ / / H q ( A ) O O in addition the commutativ e diagram H q ( I ) / / W q ( J ) H q ( A ) O O / / W q ( B ) O O which follows from the construction of the map H q ( A ) → H q ( I ) ensures that we also hav e a commutativ e diagram W q ( J ) Z q ( A ) : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ / / W q ( B ) O O Next we need a map Z q ( A ) → B q ( I ). L e t us use injectivity of B q ( I ) to choose suc h a map so that the following commutes B q ( I ) X q ( A ) : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ / / Z q ( A ) O O where X q ( A ) → B q ( I ) is the co mp o s ition o f the map X q ( A ) → X q ( I ) a nd the pro jection X q ( I ) → B q ( I ). Therefor e we hav e a co mm utative diagr am B q ( I ) B q ( A ) / / 5 5 ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ X q ( A ) : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ / / Z q ( A ) O O Finally we need to chec k that the comp osition X q ( A ) → X q ( I ) → W q ( I ) ag rees with X q ( A ) → Z q ( A ) → H q ( A ) → H q ( I ) → W q ( I ), where the last map H q ( I ) → W q ( I ) is the pro jection. But this is stra ightforw a rd since the comp osition X q ( A ) → Z q ( A ) → H q ( A ) equals the comp ositio n X q ( A ) → W q ( A ) → H q ( A ). F r om this is follows that the a bove t w o diag r ams co rresp onding to sequences (2.4) and (2.7) commute. By the exact same argument we c o nstruct an injection Z q ( C ) → Z q ( K ) yielding t wo co mm utative diag rams corr esp onding to (2.6) and (2.9). The nex t injection to constr uct is X q ( B ) → X q ( J ). A gain ther e is a complication since we w ant to 10 DA VID BARAG LIA choose the map to yield tw o co mmut ative diag rams cor resp onding to (2.11) and (2.18). F rom sequence (2.25) w e ha ve X q ( J ) = W q ( I ) ⊕ W q ( J ) ⊕ B q ( I ) ⊕ B q ( K ) so we need to define maps X q ( B ) → W q ( I ), X q ( B ) → W q ( J ), X q ( B ) → B q ( I ) and X q ( B ) → B q ( K ). Of these maps the ones into W q ( J ) a nd B q ( K ) ar e a lready determined b y commutativit y . Since W q ( I ) ⊕ B q ( I ) = X q ( I ) the r emaining tw o terms can be expressed as a map X q ( B ) → X q ( I ). F or co mmu tativity we need that this map fits into a commu tative diag ram as follows: X q ( I ) Z q ( A ) a : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ / / X q ( B ) O O B q ( B ) o o b d d ■ ■ ■ ■ ■ ■ ■ ■ ■ where the map a : Z q ( A ) → X q ( I ) is the comp os ition of the map Z q ( A ) → Z q ( I ) and the pro jection Z q ( I ) → X q ( I ) and the ma p b : B q ( B ) → X q ( I ) is defined in a similar manner. T o pro ceed le t i 1 : Z q ( A ) → X q ( B ) and i 2 : B q ( B ) → X q ( B ) b e the inclusions. W e observe that the kernel of ( i 1 , 0) + (0 , i 2 ) : Z q ( A ) ⊕ B q ( B ) → X q ( B ) is pr ecisely ( j 1 , − j 2 ) : X q ( A ) → Z q ( A ) ⊕ B q ( B ) where j 1 : X q ( A ) → Z q ( A ), j 2 : X q ( A ) → B q ( B ) ar e the inclusio ns. Next we observe that a ◦ j 1 = b ◦ j 2 since bo th maps are just the map X q ( A ) → X q ( I ) we hav e previously cons tructed. It follows that the map ( a, 0) + (0 , b ) : Z q ( A ) ⊕ B q ( B ) → X q ( I ) fac tors to a map Q → X q ( I ) wher e Q is the cokernel of ( j 1 , − j 2 ) : X q ( A ) → Z q ( A ) ⊕ B q ( B ). Obviously the map ( i 1 , 0) + (0 , i 2 ) : Z q ( A ) ⊕ B q ( B ) → X q ( B ) factors to a n injection Q → X q ( B ). No w s inc e X q ( I ) is injective there exists a map X q ( B ) → X q ( I ) yielding a commutativ e diag ram X q ( I ) Q / / < < ② ② ② ② ② ② ② ② ② X q ( B ) O O Moreov er it follows easily that we have a commutativ e dia gram as follows X q ( I ) Z q ( A ) / / a 6 6 ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ Q / / < < ② ② ② ② ② ② ② ② ② X q ( B ) O O and similarly with B q ( B ) in place of Z q ( A ). Thus the map X q ( B ) → X q ( I ) has the des ired prop erties. F rom this it follows ea sily tha t the co rresp onding map X q ( B ) → X q ( J ) w e hav e now construc ted yields co mm utative diagr ams co rre- sp onding to sequences (2.11) a nd (2.18). Along the same lines as has been descr ib ed so far one ca n find an injective map Z q ( B ) → Z q ( J ) yielding c o mmut ative diagra ms corre sp onding to (2.5),(2.8) and (2.17). Next one constructs injections A q → I q and B q → K q yielding comm utative diagrams co rresp onding to (2 .13) a nd (2.1 5). Finally o ne constructs an injection B q → J q . Once a ll of this is done we hav e commutativ e dia grams corr esp onding to Equa tio ns (2.1)-(2.19). F r o m the construction of the differentials on I ∗ , J ∗ and K ∗ we see tha t the maps A ∗ → I ∗ , B ∗ → J ∗ and C ∗ → K ∗ are chain ma ps. The A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 11 commutativ e diagra m co r resp onding to (2.19) is 0 / / I q / / J q / / K q / / 0 0 / / A q / / O O B q / / O O C q / / O O 0 Finally by co nstruction the induced ma ps Z q ( A ) → Z q ( I ), Z q ( B ) → Z q ( J ), Z q ( C ) → Z q ( K ) on co cycles are injective a nd similarly for cob oundaries a nd coho- mologies. F or the final statement ab o ut v anis hing in nega tive degrees supp ose A q = B q = C q = 0 for q < 0. It follows easily that W q ( A ) = W q ( B ) = W q ( C ) = B q ( A ) = B q ( C ) whenever q < 0. Therefor e in the ab ov e cons tructions we ma y choose W q ( I ) = W q ( J ) = W q ( K ) = B q ( I ) = B q ( K ) = 0 for q < 0 and it follows di- rectly that I q = J q = K q = 0 when q < 0.  3. The Grothendieck spectral sequence W e recall the constr uction of the Grothendieck sp ectra l sequence and estab- lish s o me of its ba sic prop erties. Let A , B , C b e ab elian categ o ries, F : A → B , G : B → C left exact functors . Supp ose A , B hav e enough injectives and F sends injectiv e o b jects to G -acyclic ob jects . Let A be an ob ject of A . Let M ∗ , A → M 0 be an injective reso lution o f A a nd set A ∗ = F ( M ∗ ). Let 0 → A ∗ → I 0 , ∗ → I 1 , ∗ → · · · be a Car tan-Eilenberg reso lution of A ∗ . Since B has eno ugh injectives it is well known that a Ca rtan-Eile nber g resolutio n exists, but w e can als o deduce this from Theorem 2 .3 using the complex 0 → A ∗ → A ∗ → 0 . Now set R p,q = G ( I p,q ) to obtain a double co mplex. Note that using Theo r em 2.3 we can als o assume tha t I p,q = 0 if p < 0 or q < 0 a nd so the same is true of R p,q . As usual for a double complex we hav e two natura l filtrations and thus tw o s p ectr al sequences ass o ciated to the double complex R p,q , b oth abutting to the coho mology of the asso cia ted single complex R ∗ . W e consider these t wo spe c tral sequences in turn. Consider fir st the spectral sequence cor resp onding to the filtration by q -degree, the terms of w hich we denote by ˜ E p,q r . By a s sumption each A p = F ( M p ) is G - acyclic since M p is injective. The ˜ E p,q 1 terms are obtained by taking cohomolog y of the double complex R p,q in the p direction so we find (3.1) ˜ E p,q 1 =  G ( A q ) p = 0 , 0 p > 0 . Note also that G ( A q ) = ( G ◦ F )( M q ) so that on passing to the next stage of the sp ectral sequence we ha ve ˜ E p,q 2 = ˜ E p,q ∞ =  R q ( G ◦ F )( A ) p = 0 , 0 p > 0 . W e deduce that the degree n cohomolo gy of the single complex a sso ciated to R p,q is given by R n ( G ◦ F )( A ). Consider now the second spectral sequence asso c ia ted to R p,q corres p o nding to the filtration F k R n by p - degree where F k R n = L p,q | p ≥ k R p,q . W e denote by E p,q r 12 DA VID BARAG LIA the asso ciated sp ectral sequence which as we now know abuts to R n ( G ◦ F )( A ). T o be more pre cise there is a filtratio n 0 = F n +1 R n ( G ◦ F )( A ) ⊆ F n R n ( G ◦ F )( A ) ⊆ · · · · · · ⊆ F 1 R n ( G ◦ F )( A ) ⊆ F 0 R n ( G ◦ F )( A ) = R n ( G ◦ F )( A ) of R n ( G ◦ F )( A ) such that E p,q ∞ ≃ F p R p + q ( G ◦ F )( A ) /F p +1 R p + q ( G ◦ F )( A ). In fa c t if w e let H n ( F p R ∗ ) denote the coho mology of F p R ∗ then F p R n ( G ◦ F )( A ) is the image o f H n ( F p R ∗ ) under the ma p H n ( F p R ∗ ) → H n ( R ∗ ) induced by the inclusion F p R ∗ → R ∗ . Getting back to the sp ectral sequence E p,q r , w e fir st co mpute E p,q 1 by taking the cohomolog y o f the double complex R p,q in the q direction. F or this we make use o f the fact that R p,q = G ( I p,q ) where I p,q is a Cartan-Eilenber g resolution of A q . If Z p,q , B p,q , H p,q denote the co cycles, cob oundaries and co homologies o f I p,q in the q direction then since I p,q is a Car ta n-Eilenberg resolution the Z p,q , B p,q , H p,q are all injectiv e and so the seq ue nc e s 0 → B p,q → Z p,q → H p,q → 0 and 0 → Z p,q → I p,q → B p,q +1 → 0 ar e s plit. Applying G it easily fo llows that the cohomolog y of R p,q in the q direction is G ( H p,q ) so that E p,q 1 = G ( H p,q ). Using the Ca rtan- Eilenberg prop er t y aga in we hav e that 0 → R q F ( A ) → H 0 ,q → H 1 ,q → · · · is an injective reso lutio n of R q F ( A ) so that the E 2 stage o f this spectra l sequence is given by E p,q 2 = R p G ( R q F ( A )) . This is the Gr othendie ck sp e ctr al se quenc e [5],[8]. One can show that fr o m the E 2 stage o nw ards the sp ectral sequence do es not dep end on the c hoice of Cartan- Eilenberg resolution [11]. 4. Beha vior on shor t exa ct sequences F or any ob ject A in A let F p R n ( G ◦ F )( A ) denote the filtration on R n ( G ◦ F )( A ) corres p o nding to the Grothendieck sp ectral sequence ( E p,q r ( A ) , d r ). In particu- lar E p,q ∞ ( A ) ≃ F p R n ( G ◦ F )( A ) /F p +1 R n ( G ◦ F )( A ). Recall also that E p,q 2 ( A ) = R p G ( R q F ( A )). Theorem 4.1. L et 0 → A → B → C → 0 b e a short exact se quenc e in A . Ther e ar e morphisms δ r : E p,q r ( C ) → E p,q +1 r ( A ) for r ≥ 2 b etwe en the Gr othendie ck sp e ctr al se quenc es for C and A with the fol lowing pr op erties: • δ r c ommutes with the differ entials d r and the induc e d map at the ( r + 1) - stage is δ r +1 . • δ 2 : R p G ( R q F ( C )) → R p G ( R q +1 F ( A )) is t he map induc e d by the b ound- ary morphism R q F ( C ) → R q +1 F ( A ) in the long ex act se quenc e of derive d functors of F asso ciate d to 0 → A → B → C → 0 . • The b oun daries R n ( G ◦ F )( C ) → R n +1 ( G ◦ F )( A ) for the long exact se quenc e asso ciate d to G ◦ F send F p R n ( G ◦ F )( C ) to F p R n +1 ( G ◦ F )( A ) and t hus induc e maps E p,q ∞ ( C ) → E p,q +1 ∞ ( A ) . These maps c oincide with δ ∞ wher e δ ∞ denotes the limit of the δ r . A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 13 Pr o of. L e t M ∗ , N ∗ , P ∗ be injective resolutions of A, B , C resp ectively . By the Horsesho e lemma [1 2] we can choose M ∗ , N ∗ , P ∗ so that there exists a commu- tative diagr am of o b jects in A wher e the columns are short exact sequences 0 / / A   / / M 0 / /   M 1   / / · · · 0 / / B   / / N 0 / /   N 1   / / · · · 0 / / C / / P 0 / / P 1 / / · · · W e th us hav e a sho rt exact sequence 0 → M ∗ → N ∗ → P ∗ → 0 of chain com- plexes. The s equence 0 → F ( M ∗ ) → F ( N ∗ ) → F ( P ∗ ) → 0 is als o e xact since the M q are injective. Thus we ma y apply Theor em 2.3 to find Cartan-Eile n b erg resolutions 0 → F ( M ∗ ) → I 0 , ∗ → I 1 , ∗ → · · · , 0 → F ( N ∗ ) → J 0 , ∗ → J 1 , ∗ → · · · , 0 → F ( M ∗ ) → K 0 , ∗ → K 1 , ∗ → · · · with the prop erties de s crib ed in Theor e m 2.3, in particular we ca n assume I p,q = J p,q = K p,q = 0 if p < 0 o r q < 0. Next w e g et double c omplexes R p,q = G ( I p,q ), S p,q = G ( J p,q ), T p,q = G ( K p,q ) with terms in C by applying the functor G . If w e let R ∗ , S ∗ , T ∗ denote the asso ci- ated s ingle complexes then we know that the degr ee n cohomolo g y H n ( R ∗ ) of R ∗ coincides with R n ( G ◦ F )( A ). Similar ly fo r S ∗ and T ∗ . By assumption F tak es injective ob jects of A to G -acy clic ob jects in B . It follows that the natural s equences 0 → R p,q → S p,q → T p,q → 0 and 0 → R n → S n → T n → 0 are exact. W e cla im that the long exact sequence 0 → H 0 ( R ∗ ) → H 0 ( S ∗ ) → H 0 ( T ∗ ) → H 1 ( R ∗ ) → H 1 ( S ∗ ) → . . . coincides with the long exact sequence 0 → ( G ◦ F )( A ) → ( G ◦ F )( B ) → ( G ◦ F )( C ) → R 1 ( G ◦ F )( A ) → R 1 ( G ◦ F )( B ) → . . . . Indeed we will show tha t the map ( G ◦ F )( M ∗ ) → R ∗ obtained from the co mp o s ition ( G ◦ F )( M ∗ ) → G ( I 0 , ∗ ) → R ∗ is a quasi-iso mo rphism, similar ly for B , C . W e th us hav e a commutativ e diagr am of short exact sequences of co mplex es in C 0 / / ( G ◦ F )( M ∗ ) / /   ( G ◦ F )( N ∗ ) / /   ( G ◦ F )( P ∗ ) / /   0 0 / / R ∗ / / S ∗ / / T ∗ / / 0 such tha t the vertical arrows are quasi-isomorphisms. The top row is exact s inc e F takes injectives to G -acy c lic o b jects. W e th us g et a chain isomorphism betw een the cor resp onding long exact s e q uences. Note that the degree n cohomo logy of ( G ◦ F )( M ∗ ) is precisely R n ( G ◦ F )( A ) and similar ly for B , C . Now to finish the claim w e must show that ( G ◦ F )( M ∗ ) → R ∗ is a quas i-isomorphis m. F or this we int ro duce filtr a tions F ′ k ( G ◦ F )( M ∗ ) o n ( G ◦ F )( M ∗ ) a nd F ′ k R ∗ on R ∗ . W e set F ′ 0 ( G ◦ F )( M ∗ ) = ( G ◦ F )( M ∗ ) and F ′ k ( G ◦ F )( M ∗ ) = 0 if k > 0. F or R ∗ we take F ′ k R ∗ = L p,q | q ≥ k G ( I p,q ). The map ( G ◦ F )( M ∗ ) → R ∗ is easily seen to pr eserve the filtra tions so induces a map b etw een the sp ectral sequences for these filtra tions. One finds easily tha t the map at the E 1 -stage is an isomorphism. Indeed E p,q 1 for the filtra tion on R ∗ is given by Equation (3.1), wher e A q denotes F ( M q ). F ro m 14 DA VID BARAG LIA this we indeed find that the map is an isomor phism at the E 1 -stage. This is enough to show that ( G ◦ F )( M ∗ ) → R ∗ is a quasi- isomorphism. Int ro duce a filtra tion F k R ∗ on R ∗ by setting F k S ∗ = L p,q | p ≥ k R p,q . This filtra- tion y ie lds a corres p o nding filtration F k H n ( R ∗ ) on the cohomolog y H n ( R ∗ ) b y let- ting F p H n ( R ∗ ) b e the image in H n ( R ∗ ) of the degree n cohomolo gy of F p R ∗ under the natur a l inclusion F p R ∗ → R ∗ . W e hav e a sp e c tral seq ue nce ( E p,q r ( R ∗ ) , d r ) cor - resp onding to the filtration s o E p,q ∞ ( R ∗ ) ≃ F p S p + q /F p +1 S p + q . Similarly fo r S ∗ , T ∗ we have filtratio ns and s p ectr al s equences defined in the same manner. In fact since I p,q , J p,q , K p,q are Carta n- Eilenberg resolutions we hav e as shown in Section 3 that the res ulting sp ectral sequences ( E p,q r ( R ∗ ) , d r ), ( E p,q r ( S ∗ ) , d r ), ( E p,q r ( T ∗ ) , d r ) are the Gr othendieck spectral sequences co rresp onding to A, B , C . W e thus hav e E p,q 2 ( R ∗ ) = R p G ( R q F ( A )), E p,q 2 ( S ∗ ) = R p G ( R q F ( B )) a nd E p,q 2 ( T ∗ ) = R p G ( R q F ( C )). The sp ectr al sequences fo r R ∗ and T ∗ are determined by cor resp onding exact couples ( A 1 ( R ∗ ) , E 1 ( R ∗ )), ( A 1 ( T ∗ ) , E 1 ( T ∗ )) where A p,q 1 ( R ∗ ) = M p,q H p + q ( F p R ∗ ) E p,q 1 ( R ∗ ) = M p,q H p + q ( F p R ∗ /F p +1 R ∗ ) and similar ly for T ∗ . T o define the exact co uple ( A 1 ( R ∗ ) , E 1 ( R ∗ )) we m ust a lso give maps i : A 1 ( R ∗ ) → A 1 ( R ∗ ), j : A 1 ( R ∗ ) → E 1 ( R ∗ ) and k : E 1 ( R ∗ ) → A 1 ( R ∗ ). W e take i : A p,q 1 ( R ∗ ) → A p − 1 ,q +1 1 ( R ∗ ) to be the map in cohomo logy induced by the inclusions F p R ∗ → F p − 1 R ∗ , j : A p,q 1 ( R ∗ ) → E p,q 1 ( R ∗ ) induced by the pro jectio n F p R ∗ → F p R ∗ /F p +1 R ∗ and k : E p,q 1 ( R ∗ ) → A p +1 ,q 1 ( R ∗ ) to be the co b o undary in the long exact sequence for 0 → F p +1 R ∗ → F p R ∗ → F p R ∗ /F p +1 R ∗ → 0. Define similar maps i, j, k in the case of T ∗ . W e define a map δ : ( A 1 ( T ∗ ) , E 1 ( T ∗ )) → ( A 1 ( R ∗ ) , E 1 ( R ∗ )) betw een exact cou- ples. By this we mean a pair of mor phisms δ : A 1 ( T ∗ ) → A 1 ( R ∗ ), δ : E 1 ( T ∗ ) → E 1 ( R ∗ ) intert wining the maps i, j, k of the exact couples. More precisely we will define maps δ : A p,q 1 ( T ∗ ) → A p,q +1 1 ( R ∗ ), δ : E p,q 1 ( T ∗ ) → E p,q +1 1 ( R ∗ ) as fol- lows. The short exa ct sequence 0 → R ∗ → S ∗ → T ∗ → 0 yields corresp ond- ing sho rt ex a ct seq uences 0 → F p R ∗ → F p S ∗ → F p T ∗ → 0 o n the filtrations and th us we obtain bounda r y maps H n ( F p T ∗ ) → H n +1 ( F p R ∗ ). This defines the map δ : A p,q 1 ( T ∗ ) → A p,q +1 1 ( R ∗ ). By the Nine lemma we g e t ex act seq uences 0 → F p R ∗ /F p +1 R ∗ → F p S ∗ /F p +1 S ∗ → F p T ∗ /F p +1 T ∗ → 0 and th us b o undary maps H n ( F p T ∗ /F p +1 T ∗ ) → H n +1 ( F p R ∗ /F p +1 R ∗ ) which w e take as the definition of δ : E p,q 1 ( T ∗ ) → E p,q +1 1 ( T ∗ ). One needs to show that the maps δ so defined int ertwine 1 the maps i, j, k and then we hav e a morphism b etw een exact couples. When one passes to the derived exa ct co uples the δ ma ps induce corr esp onding maps on the der ived exact couples. Thus we get maps δ r : E p,q r ( T ∗ ) → E p,q +1 r ( R ∗ ) which are the ma ps in the statement of the theorem. By definition of the filtrations F p H n ( R ∗ ), F p H n ( T ∗ ) we see that the b oundar y map H n ( T ∗ ) → H n +1 ( R ∗ ) sends F p H n ( T ∗ ) to F p H n +1 ( R ∗ ). Upo n identification of H n ( R ∗ ) with R n ( G ◦ F )( A ) 1 strictly speaking the maps δ only intert wine i, j, k up to certain irrelev ant sign factors A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 15 and H n ( T ∗ ) with R n ( G ◦ F )( C ) w e se e a s c la imed that the natural bounda ry map R n ( G ◦ F )( C ) → R n +1 ( G ◦ F )( A ) sends F p R n ( G ◦ F )( C ) to F p R n +1 ( G ◦ F )( A ) and that the induced maps E p,q ∞ ( T ∗ ) → E p,q +1 ∞ ( R ∗ ) coincide with δ ∞ . T o finish the pr o of we must show tha t δ 2 has the exp ected for m. Fir st note that E p,q 1 ( R ∗ ) can b e identified with G ( H q ( I p, ∗ )) and similar ly for T ∗ , while the bo undary ma ps E p,q 1 ( T ∗ ) → E p,q +1 1 ( R ∗ ) are eas ily seen to aris e from the b oundar y maps H q ( K p, ∗ ) → H q +1 ( I p, ∗ ). T o pa s s from E 1 to E 2 one then takes cohomolog y with res pe ct to the differential induced by the maps I ∗ ,p → I ∗ ,p +1 . Since I ∗ , ∗ is a Cartan-E ilenberg resolutio n we ha ve that { H q ( I p, ∗ ) } p ≥ 0 is an injective resolutio n of H q ( F ( M ∗ )) = R q F ( A ). On apply ing G a nd taking cohomolo g y in the p directio n we get E p,q 2 ( R ∗ ) = R p G ( R q F ( A )). As usua l a similar statement holds for C . Now o bserve tha t H q ( K p, ∗ ) → H q +1 ( I p, ∗ ) is a map be tw een injective resolutions of H q ( F ( P ∗ )) = R q F ( C ) and H q +1 ( F ( M ∗ )) = R q +1 F ( A ) co mm uting with the bo undary map R q F ( C ) → R q +1 F ( A ) so o n applying G and taking c o homology in the p direction we see that the induced map E p,q 2 ( C ) → E p,q +1 2 ( A ) is indeed the map R p G ( R q F ( C )) → R p G ( R q +1 F ( A )) induced b y the boundary R q F ( C ) → R q +1 F ( A ).  5. Applica tion The main application o f Theorem 4.1 we ha ve in mind is to the Leray s pe c tr al sequence for sheaf cohomolog y . Let X , Y b e top olo gical spaces and f : X → Y a contin uous map. W e tak e A , B to be the categor ies of s heav es o f ab elia n groups on X and Y r esp ectively and C the categ o ry of abelian g roups. W e take the functor F : A → B to b e the push-forward F = f ∗ under f a nd G : B → C to be the global sections functor G = Γ. Note that G ◦ F is the globa l sections functor for shea ves on X . T o derive the Lera y sp ectral sequence from the Grothendieck spectra l sequence one only needs to note that the category of sheav es of ab elia n groups on a top ologica l space has enoug h injectiv es [3, Lemma 1 .1.13] and that the push-for ward functor F = f ∗ actually sends injectives to injectives [3, Lemma 1 .6.3]. F or any sheaf A on X we thus obtain the Leray sp ectral sequence, a sp ectr al sequence { E p,q r , d r } whic h abuts to the sheaf cohomo logy H ∗ ( X, A ) and s uch that E p,q 2 ( A ) = H p ( Y , R q f ∗ A ). As with the Grothendieck spectr a l sequence there is a natur al filtration 0 = F n +1 ,n ( A ) ⊆ F n,n ( A ) ⊆ · · · ⊆ F 1 ,n ( A ) ⊆ F 0 ,n ( A ) = H n ( X, A ) related to the sp ectra l seq uence b y E p,q ∞ ( A ) ≃ F p,p + q ( A ) /F p +1 ,p + q ( A ) . Theorem 4.1 translated to this s itua tion b ecomes Theorem 5.1 . L et X , Y b e top olo gic al sp ac es and f : X → Y a c ontinuous m ap. L et 0 → A → B → C → 0 b e a short exact se quenc e of she aves. Ther e ar e morphisms δ r : E p,q r ( C ) → E p,q +1 r ( A ) for r ≥ 2 b etwe en the L er ay sp e ctr al se quenc es for C and A with t he fo l lowing pr op ert ies: • δ r c ommutes with the differ entials d r and the induc e d map at the ( r + 1) - stage is δ r +1 . 16 DA VID BARAG LIA • δ 2 : H p ( Y , R q f ∗ C ) → H p ( Y , R q +1 f ∗ A ) is the map induc e d by the b oundary morphism R q f ∗ C → R q +1 f ∗ A in the long ex act se qu enc e of higher dir e ct image functors of f asso ciate d to 0 → A → B → C → 0 . • The b oundaries H n ( X, C ) → H n +1 ( X, A ) for the long exact se quenc e of she af c ohomolo gy sen d F p,n ( C ) to F p,n +1 ( A ) and thus induc e maps E p,q ∞ ( C ) → E p,q +1 ∞ ( A ) . These maps c oincide with δ ∞ wher e δ ∞ denotes the limit of the δ r . Let us now consider a sp ecific application of this result we ha ve in mind. Hence- forth we assume that the top olo gical spaces X , Y are paracompa ct. F urthermor e assume that every subspace of X is parac o mpact. This is the case for instance if X is a metric space [6, Theorem 5.1 3]. W e use the notation C , C ∗ to denote the sheav es o f contin uo us functions with v alues in C , C ∗ , where C ∗ is the non-z ero complex num b ers. Theorem 5.2. L et f : X → Y b e a map b etwe en sp ac es X , Y , let E p,q r ( Z ) , E p,q r ( C ∗ ) b e the L er ay sp e ct ra l se quenc es asso ciate d to the she aves Z , C ∗ and let F p,n ( Z ) , F p,n ( C ∗ ) b e t he asso ciate d filtr ations on H n ( X, Z ) , H n ( X, C ∗ ) . Then • The c ob oundary δ : H n ( X, C ∗ ) → H n +1 ( X, Z ) r estricts to morphisms δ : F p,p + q ( C ∗ ) → F p,p + q +1 ( Z ) which ar e isomorphisms whenever p + q ≥ 1 and surje ctive fo r p = q = 0 . • The induc e d quotient maps E p,q ∞ ( C ∗ ) → E p,q +1 ∞ ( Z ) ar e isomorphisms for q ≥ 1 and surje ctions for q = 0 . Pr o of. W e apply Theorem 5.1 to the expo nent ial sequence 0 → Z → C → C ∗ → 0. Thu s the cob oundar y δ : H n ( X, C ∗ ) → H n +1 ( X, Z ) restricts to morphisms δ : F p,p + q ( C ∗ ) → F p,p + q +1 ( Z ) and induces quotient maps δ ∞ : E p,q ∞ ( C ∗ ) → E p,q +1 ∞ ( Z ). Next we obser ve that E p,q 2 ( Z ) = H p ( Y , R q f ∗ Z ) and E p,q 2 ( C ∗ ) = H p ( Y , R q f ∗ C ∗ ). The na tur al maps R q f ∗ C ∗ → R q +1 f ∗ Z in the long exact sequence of higher direct image sheav es are isomorphisms fo r q ≥ 1 , since R q f ∗ C = 0 for q ≥ 1. W e therefor e hav e that δ 2 : E p,q 2 ( C ∗ ) → E p,q +1 2 ( Z ) is a n isomorphism for q ≥ 1. W e will now show that for q ≥ 1 the ma ps δ ∞ : E p,q ∞ ( C ∗ ) → E p,q +1 ∞ ( Z ) ar e injectiv e. In fact we star t by s howing that the ma ps δ 3 : E p,q 3 ( C ∗ ) → E p,q +1 3 ( Z ) are injective for q ≥ 1 . Let x ∈ E p,q 3 ( C ∗ ) b e such that δ 3 ( x ) = 0. Cho ose a representative ˜ x ∈ E p,q 2 ( C ∗ ) for x . Then δ 3 ( x ) = 0 means that δ 2 ( ˜ x ) = d 2 ˜ y for some ˜ y ∈ E p − 2 ,q +2 2 ( Z ). W e can then find ˜ z ∈ E p − 2 ,q +1 2 ( C ∗ ) s o that ˜ y = δ 2 ( ˜ z ) and thus δ 2 ( ˜ x ) = d 2 ˜ y = d 2 δ 2 ˜ z = δ 2 d 2 ˜ z . By injectivity of δ 2 we have ˜ x = d 2 ˜ z and thus x = 0 proving injectivity o f δ 3 . Pro ceeding by induction we find that δ r : E p,q r ( C ∗ ) → E p,q +1 r ( Z ) is injective for all r ≥ 2 and q ≥ 1. Next we show that the maps δ ∞ : E p,q ∞ ( C ∗ ) → E p,q +1 ∞ ( Z ) are surjective for all p, q . T o b egin let x ∈ E 0 ,q +1 ∞ ( Z ) and lift x to a class ˜ x ∈ F 0 ,q +1 ( Z ) = H q +1 ( X, Z ). Since δ : H q ( X, C ∗ ) → H q +1 ( X, Z ) is surjective w e can find ˜ y ∈ H q ( X, C ∗ ) so that δ ( ˜ y ) = ˜ x . P ro jecting ˜ y to a class y ∈ E 0 ,q ∞ ( C ∗ ) we find that δ ∞ ( y ) = x proving surjectivity in the p = 0 case. Now we pro ceed b y induction on p , so as s ume that δ ∞ : E k,q ∞ ( C ∗ ) → E k,q +1 ∞ ( Z ) is surjective for all q and all k ≤ p − 1 , where now p > 0. Given x ∈ E p,q +1 ∞ ( Z ) lift x to a clas s ˜ x ∈ F p,p + q +1 ( Z ) ⊆ H p + q +1 ( X, Z ). Then s ince δ : H p + q ( X, C ∗ ) → H p + q +1 ( X, Z ) is surjective we can find ˜ y ∈ H p + q ( X, C ∗ ) such that δ ˜ y = ˜ x . T o pro ceed we need to a rgue that ˜ y ∈ F p,p + q ( C ∗ ). Notice that ˜ y ∈ A COBOUND AR Y MORPHISM F OR THE GROTHENDIECK SPECTRAL SEQUENCE 17 F 0 ,p + q ( C ∗ ) = H p + q ( X, C ∗ ). Using the fact that δ ∞ : E 0 ,p + q ∞ ( C ∗ ) → E 0 ,p + q +1 ∞ ( Z ) is injectiv e (since p + q > 0) and that the image of ˜ x in E 0 ,p + q +1 ∞ ( Z ) is zero we see that corresp onding ly the imag e o f ˜ y in E 0 ,p + q ∞ ( C ∗ ) is zero and thu s ˜ y ∈ F 1 ,p + q ( C ∗ ). Contin uing in this fashion using the injectivity of the δ ∞ : E a,b ∞ ( C ∗ ) → E a,b +1 ∞ ( Z ) for b ≥ 1 we get that ˜ y ∈ F 2 ,p + q ( C ∗ ), ˜ y ∈ F 3 ,p + q ( C ∗ ) , . . . and even tually g et that ˜ y ∈ F p,p + q ( C ∗ ). Pro jecting ˜ y to y ∈ E p,q ∞ ( C ∗ ) we immediately see that δ ∞ ( y ) = x proving surjectivity of δ ∞ . Using the fac t that the δ ∞ : E p,q ∞ ( C ∗ ) → E p,q +1 ∞ ( Z ) are surjective for all p, q we easily see that the maps δ : F p,p + q ( C ∗ ) → F p,p + q +1 ( Z ) surject onto the quo- tien ts F p,p + q +1 ( Z ) /F p + q +1 ,p + q +1 ( Z ) for all p, q . Given this we can show the maps δ ∞ are s urjective b y showing that every class x ∈ F p + q +1 ,p + q +1 ( Z ) has the form x = δ ( y ) for some y ∈ F p + q,p + q ( C ∗ ). Since δ : H p + q ( X, C ∗ ) → H p + q +1 ( X, Z ) is surjective we can find y ∈ H p + q ( X, C ∗ ) so that x = δ ( y ). If w e pro ject y to a class in E 0 ,p + q ∞ ( C ∗ ) and use injectivit y of δ ∞ : E 0 ,p + q ∞ ( C ∗ ) → E 0 ,p + q +1 ∞ ( Z ) (if p + q ≥ 1) we find that the pro jection of y to E 0 ,p + q ∞ ( C ∗ ) is zer o so that y ∈ F p + q − 1 ,p + q ( C ∗ ). Contin uing o n in this fashion we w ork our wa y do wn the filtration and ultimately find that y ∈ F p + q,p + q ( C ∗ ) as claimed. Finally to see that the maps δ : F p,p + q ( C ∗ ) → F p,p + q +1 ( Z ) a re injectiv e when- ever p + q ≥ 1 we just need to note that δ : H p + q ( X, C ∗ ) → H p + q +1 ( X, Z ) is injectiv e, indeed an iso morphism whenever p + q ≥ 1 .  R emark 5.3 . Observe that in the pro of of Theorem 5.2 the only pr op erty of the exp onential sequence 0 → Z → C → C ∗ → 0 that is us e d is tha t the sheaf C on X has the pro pe r ty that its restriction to any op e n subs et of X is acyclic. Thu s the result ca rries o ver to many other short e x act sequences o f sheav es. F or example given ǫ ∈ H 1 ( X, Z 2 ), tensoring by the co rresp onding lo cal system Z ǫ yields an exact sequence 0 → Z ǫ → C ǫ → C ∗ ǫ → 0 that also sa tisfies this condition. Theorem 5 .2 has applica tions to topolo gical T-duality . Supp os e that f : X → Y is a pr incipa l T n -bundle ov e r Y where T n = R n / Z n is the n -torus and supp ose G is a bundle gerb e on X [9 ],[10] which up to stable is omorphism is classified b y its Dixmier -Douady class h = [ G ] ∈ H 2 ( X, C ∗ ) = H 3 ( X, Z ). Bunke, Rumpf and Schic k [4 ] g ive a definition of topo logical T -duality for the data ( f : X → Y , G ) building up on the definition of to po logical T -duality first intro duced in [2]. In [4] it is established that ( X, G ) admits a T-dual (in the sense of [4]) if and only if h = [ G ] ∈ F 2 , 3 ( Z ). F rom Theor em 5.2 we see that this is equiv alent to h ∈ F 2 , 2 ( C ∗ ) if we regard h a s an ele men t of H 2 ( X, C ∗ ). Next we observe that there is a natural surjection E 2 , 0 2 ( C ∗ ) → F 2 , 2 ( C ∗ ). Th us the T-dualizable class e s o n X ar e precisely the imag e of E 2 , 0 2 ( C ∗ ) = H 2 ( Y , f ∗ ( C ∗ )) under the natur al map H 2 ( Y , f ∗ ( C ∗ )) → H 2 ( X, C ∗ ). 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