A NOTE ON DETERMINANT FUNCTORS AND SPECTRAL SEQUENCES. 1. Introduction

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1 A NOTE ON DETERMINANT FUNCTORS AND SPECTRAL SEQUENCES OTMAR VENJAKOB Abstact. The aim of this note is to pove cetain compatibilities of deteminant functos with spectal sequences and (co)homology theeby extending esults of [3] and efining a desciption in [9]. It tuns out that the deteminant behaves as well as one would have expected in this egad, only that we wee not able to find efeences fo it in the liteatue. The esults ae cucial fo descent calculations in the context of Iwasawa theoy [12] o Equivaiant Tamagawa Numbe Conjectues [4, 5, 6]. 1. Intoduction As a genealisation o athe efinement of Eule-Poincae chaacteistics deteminant functos have been fist studied by Knudsen and Mumfod [10] in the context of pefect complexes of O X -modules on a scheme X. The theoy has been genealised by Deligne [7] and Knudsen [9] to deteminants on exact categoies with values in commutative Picad categoies. Apat fom applications in algebaic geomety deteminant functos ae a fundamental technical tool in Buns and Flach s fomulation of Equivaiant Tamagawa Numbe Conjectues [4, 5, 6], they have been used in Iwasawa theoy by Kato, Pein-Riou and Fontaine, Hube and Kings. In paticula, fo an associative ing R with unit, Fukaya and Kato pesent an adhoc constuction of deteminant functos on pefect complexes of R-modules in [8]. Witte used yet anothe appoach in his thesis [15, 14], see also the nice suvey aticle [11]. Deteminant functos on tiangulated categoies have been constucted by Beuning [2], while the pesent note is stongly based on [3], see also [1]. Ou motivation fo this note stems fom descent calculations in Iwasawa theoy involving deteminant functos. While fom the theoetical point of view it is quite elegant to wok with complexes, e.g. epesenting Galois cohomology like RΓ(G, T) whee G is the absolute Galois cohomology say of a global o local field and T denotes a big Galois epesentation, i.e., a module ove some Iwasawa algeba with an additional compatible action by G, it often tuns out in paxis that explicit descent calculations involve cohomology goups H i (G, T) instead. Thus it seems cucial to the autho to claify how to compae the descent, i.e., the application of some deived o hype To-functo to the complex and cohomology goups, espectively. Fom the technical point of view it concens the question how the coesponding To-spectal sequence (3.34) behaves with espect to deteminants and to the functo H which associates to a complex its (co)homology, see (3.35) subsection 3.1. In ode to obtain an adequate statement we Date: Septembe 17, The autho acknowledges suppot by the ERC.

2 2 OTMAR VENJAKOB fist ecall and develop a geneal desciption fo deteminants in spectal sequences in section 2. Acknowledgements: The autho is gateful to Malte Witte fo seveal helpful suggestions and ove all fo pointing out a blunde in an ealie vesion. 2. Deteminants and spectal sequences 2.1. Spectal sequences. Let C = (C n, n ) n Z be a filteed cochain complex with values in some abelian categoy E, i.e., an object in the categoy of cochain complexes togethe with a deceasing filtation F :... F i (C) F i+1 (C)... within this categoy. We shall assume hencefoth that the filtation is bounded, i.e., F t (C) = C and F s (C) = 0 fo some s, t Z, as well as that the complex C itself is bounded - then we shall say fo simplicity that C is a bounded filteed complex. We will now ecall some basic facts and notation concening the associated convegent (weakly convegent, biegula) cohomological spectal sequence E, = (E pq ) pq H p+q (C), 0 with a specific viewpoint fom the pespective of deteminants. In paticula it is often useful to conside a spectal sequence as a complex with espect to its total degee E = (E n, n ), E n := p E p,n p, n := p,n p. Recall that we have the canonical identification E pq 0 = F p C p+q /F p+1 C p+q, E n 0 = p F p C n /F p+1 C n and that the diffeentials pq ae induced fom n of C. In the sequel we shall need a couple of diffeent filtations. To this end we define A pq : = ke(f p C p+q C p+q+1 /F p+ C p+q+1 ) = F p C p+q 1 (F p+ C p+q+1 ), B pq : = A p +1,q+ 2 1 and identify (2.1) E pq = (A pq + F p+1 C p+q )/(B pq + F p+1 C p+q ) = A pq /(B pq + A p+1 1 ). The diffeentials of C induce diffeentials on A and E of degee such that the following diagam is commutative C p+q C p+q+1 A pq E pq A p+,q +1 E p+,q +1.

3 Conside also the cycle complex Z = Z(C) and bounday complex B = B(C) with thei induces filtation fom C as well as the cohomology complex H = H(C) (all with tivial diffeentials) with induced filtation via the fist of the two canonical complexes (2.2) 0 B Z H 0 and (2.3) 0 Z C B[1] 0. Hee, if C is a (cochain) complex, then C[1] denotes the shifted one, i.e., C[1] i = C i+1 with diffeential C[1] = C. In othe wods and F p H n (C) = F p Z + B n /B n = F p Z n /F p B n (2.4) g p H n = F p H n /F p+1 H n = F p Z n + B n /F p+1 Z n + B n = E pq, p + q = n. As suggested by Knudsen in [9, 3] the deived filtations F i (C) = DF i = DF i (C), 0, associated to F (C) ae pobably the most appopiate ones in the context of deteminants and spectal sequences. They ae given as follows F i (C n ) := A p,q whee the indices tansfom always as F i (C n+1 ) = A p+,q +1 p = i + n, q = n(1 ) i, n = p + q. Note that F 0 (C) = F (C). We wite g i (C) := g i F (C) fo the associated gaded complexes g i (C n ) := F i (C n )/F i+1 (C n ) g i (C n+1 ) = F i (C n+1 )/F i+1 (C n+1 ) and set g (C) := g(c). i i Then accoding to [9, pop. 3.5] it is easily checked that thee is a canonical quasiisomophism q : g (C) E of complexes which is induced by the natual pojections A p,n p /A p+1,n p 1 A p,n p /(B p,n p + A p+1,n p 1 1 ) of the summands of the individual objects. In paticula, thee is a canonical isomophism H(q ) : H(g (C)) = (E +1, 0) whee both complexes ae endowed with tivial diffeentials. Fo late puposes we calculate the kenel of q o athe its ith component ( ) K i := ke g(c) i E i+, (1 ) i such that ke(q ) = i Ki. Then K i is acyclic and looks like (2.5) (B p + A p +1 1 )/A p +1 (B p + A p+1 1 )/Ap+1 (B p+ + A p++1 1 )/A p++1 3

4 4 OTMAR VENJAKOB whee the middle tem is put in degee n and whee we omit now and sometimes late the 2nd supescipts (= n minus the fist one) fo bette eadability. We obtain canonical isomophisms (2.6) A p+1 1 /Ap+1 = K i,n /Z(K i,n ) and - on the level of complexes - = B(K i,n+1 ) = (B p+ + A p++1 )/A p++1, (2.7) K i /Z(K i ) = B(K i )[1] Deteminants. Recall that a (commutative) Picad categoy is a goupoid P with a poduct : P P P which satisfies compatible associativity, commutatitivity and unit constaints, and fo which evey object in P is invetible, see [9, Appendix A] fo moe details. In this note we will in geneal not explicitly mention the associativity and commutativity isomophisms; by the coheence theoem fo AC tenso categoies this will not cause any confusion. We fix a unit object 1 = 1 P which is unique up to unique isomophism. Now we choose a deteminant functo [9, 2] d : E iso P on E with values in some (commutative) Picad categoy and extend it [9, thm. 2.3] to the categoy of bounded cochain complexes C b (E) qis of objects in E whee we wite qis fo the quasi-isomophisms in C b (E). Roughly speaking we thus have by definition d(c) = i d(c i ) ( 1)i whence we obtain a canonical map e 0 : d(c) d(f ) = p d(f p C/F p+1 C) = d(e 0, 0 ) induced fom the filtation F on C. isomophism Following [3, pop. 3.1] thee is a canonical η C : d(c) d(h(c)) fo each C C b (E) which is induced by the sequences (2.2), (2.3) as well as by the canonical identification µ B : d(b)d(b[1]) = 1 as descibed in [3, lem. 2.3]. Thus we obtain canonical isomophisms fo all 0. η : d(e, ) η E d(h(e )) = d(e +1, 0) = d(e +1, ) On the othe hand, associated with the deived filtation F i C of C we have canonical maps (fo each 0) which identity the deteminant of C with the poduct ove the deteminants of all subquotients g i (C) of F C. They induce isomophisms e : d(c) = i d(gi (C)) = d(g (C)) d(q) d((e, ))

5 5 and a diagam (2.8) d(c) = d(g (C)) η g(c) d(h(g (C))) e d(q ) d(e ) η d(e +1 ) d(h(q )) which is commutative by [3, pop. 3.1]. Lemma 2.1. Fo all 0 we have upon identifying d((e, )) = d((e, 0)). e = η 1... η 0 e 0 We postpone the poof of the Lemma and conside fist the case = : We set e := e fo big enough such that = 0, whence E = E. Although the deived filtation will not stabilize fo big in the liteal sense, it stabilizes if we conside it up to eindexing and up to omitting o inseting tivial filtation steps. The esulting filtation class is denoted by F, fo which a epesenting filtation can be descibed as follows. Fist conside genealized good tuncations τ j n : C n 2 1 (F j (C n )) F j Z n 0 fo t j s and m 0 := min{n C n 0} n n 0 := max{n C n 0} with associated gaded complexes g (j,n) : 0 1 (F j (C n ))/ 1 (F j+1 (C n )) F j Z n /F j+1 Z n 0. Then it is easy to check that F can be epesented by F : C = τ t n 0 τ j n 0 τ s n 0 = τ t n 0 1 τ j n τ s m 0 = 0. Refining the latte by the filtation (2.9) 0 B Z C, o by the slightly fine one (2.10) 0 B... B + F p Z... Z C, we obtain the filtation with the popety that G :... τ j n Z(τ j n ) B(τ j n ) τ j+1 n... (2.11) Z(τ j j+1 n )/τ n = Z(g(j,n) ) and (2.12) B(τ j j+1 n )/τ n = B(g(j,n) ) as can be easily checked. Then the associated gaded complexes g G (C) ae given as follows (2.13) τ j n /Z(τ j n ) = 1 (F j (C n ))/ 1 (F j+1 (C n ))[ (n 1)],

6 6 OTMAR VENJAKOB (2.14) Z(τ j n )/B(τ j n ) = F j Z n /(F j B n +F j+1 Z n )[ n] = E i,n i [ n] = g j H n (C)[ n] and (2.15) B(τ j j+1 n )/τ n = (F j B n + F j+1 Z n )/F j+1 Z n [ n] = F j B n /F j+1 B n [ n], whee fo an object M E we wite M[ n] fo the obvious complex concentated in degee n (this convention is compatible with shifting in the following sense (M[0])[n] = M[n]). On the othe hand the efinement R of the filtation (2.10) by F = (τ j n ) looks like 0 B τm s B τn j... B... B + F p+1 Z B + F p+1 Z + (B + F p Z) τ p m 0 B + F p+1 Z + (B + F p Z) τ p m B + F p+1 Z + (B + F p Z) τ p n B + F p+1 Z + (B + F p Z) τ p n... B + F p Z... Z Z + τ s 1 m Z + τ j n... C with associated gaded complexes g? R (C) (2.16) F j B n /F j+1 B n [ n],..., g p H n (C)[ n],..., 1 (F j C n )/ 1 (F j+1 C n )[ (n 1)]. Now we ae eady to pove the following Poposition 2.2. Fo each bounded filteed complex C thee is a canonical commutative diagam (2.17) d(c) e d(e ) η C d(ψ) d(h(c)) d(f H) p d(gp H(C)) whee ψ : E = p gp H(C) is the canonical isomophism being pat of the convegent spectal sequence associated to C and induced by (2.4). Poof. Recall fist that η C is induced by the filtation (2.9) above using the exact sequences (2.3) and (2.2). Recall that d(g R (C)) = d( 1 (F j C n )/ 1 (F j+1 C n )[ n 1]) d(g p H n (C)[ n]) d(f j B n /F j+1 B n [ n]) p,n n,j n,j = j d( 1 (F j C)/ 1 (F j+1 C)) p d(g p H(C)) j d(f j B/F j+1 B) and set R := d(g /Z(g ))d(h(g ))d(b(g )) = d(b(g )[1])d(H(g ))d(b(g )).

7 By the efinement pinciple [9, pop. 1.9] applied to the filtations F and (2.10) we obtain the following diagam 7 d(c) d(f ) d(g (C)) η d(h(g (C))) d(e ) d(g G (C)) R µ B(g ) can d(g R (C)) d(c/z)d(z/b)d(b) d(c/z)d(g H(C))d(B) (?) d(ψ) d(b[1])d(h)d(b) d(b[1])d(g H(C))d(B) µ B d(h) µ B can d(g H) d(g H) in which all inteio ectangles clealy commute except possibly (?). Hee we wite g H fo the complex p gp H (with tivial diffeentials). Note that the left vetical map is just η C while the uppe hoizontal map is e by diagam (2.8), because η is the identity. Afte canceling g p H(C) and the pat coesponding to the middle pat of R between B and Z, espectively, and taking into account the identification 1 (F j (C n ))/ 1 (F j+1 (C n )) = F j B n /F j+1 B n the commutativity of (?) boils down to the commutativity of the following diagam (which is upside down in compaison with the pevious diagam!) d(b[1])d(b) µ B 1 µf i B n /F i+1 B n i,n d(f i B n /F i+1 B n [ n + 1]) i,n d(f i B n /F i+1 B n [ n]) 1 d(b(g )[1])d(B(g )) µ B(g ) 1 which follows immediately fom [3, lem. 2.3]. Hee the uppe squae uses (pat of) R, while the lowe one uses the identifications (2.12) and (2.15), i.e., B(g ) = j,n F i B n /F i+1 B n [ n]. Remak 2.3. By simila, but simple consideations egading the filtation (2.9) and the (usual) good tuncation filtation τ n (= τ t n ) one sees that η C can also be descibed

8 8 OTMAR VENJAKOB using the canonical filtation... τ n Z(τ n ) B(τ n ) τ n 1... with Z(τ n ) : C n 2 Z n 1 Z n 0 and B(τ n ) : C n 2 Z n 1 B n 0 togethe with the identifications C n 1 /Z n 1 = B n induced by. Now we come back to the Poof (of Lemma 2.1). Denote by R 0 the efinement of the filtation F (C) such that its gaded pieces coespond to and B(g i ) : g/z(g i ) i : A p /A p +1 (B p +1 + Ap +1 )/A p +1 H(g i ) : E p +1 A p /A p E p +1 (B p +1 + Ap+1 )/A p+1 0 E p+ +1, A p+ /A p+ +1, (B p Ap++1 )/A p++1. We wite R 1 fo the efinement of F +1 (C) by R 0. Then, by the efinement pinciple applied to F +1 (C) and R 0 the left uppe squae in the following diagam is commutative: d(c) d(g (C)) d(g R0 (C)) a d(h(g (C))) d(g +1 (C)) d(g R1 (C)) b d((e +1, 0)) d(g +1 (C)) d(q ) d((e +1, +1)) Note that the composite fom the left uppe cone to the ight lowe cone via the left lowe cone is just e +1. The uppe hoizontal line hee is defined such that it coincides with the uppe line of the diagam (2.8), in paticula, by the definition of η g the map a is induced by the identifications (2.18) A p /A p+1 +1 (B p +1 + Ap+1 )/A p+1, i.e., g/z(g i ) i B(g i )[1] plus µ B(g i ) as above. The map b makes by definition the lowe ectangle commutative, i.e., it is induced by the identifications of the canonical map d(ke(q )) d(h(ke(q ))) = 1 as the involved complex is acyclic. Thus b coesponds to the identifications given by (2.6) o (2.7), which ae the same as in (2.18) fo a whence the whole diagam is commutative and combined with diagam (2.8) the lemma is poved. Indeed, upon identifying d((e, )) = d((e, 0)) the composite fom the left uppe cone to the ight lowe cone via the ight uppe cone is nothing else than η e.

9 Lemma 2.4. Let : 0 A B C 0 be a stictly exact sequence of bounded filteed complexes,i.e., 0 F p A F p B F p C 0 is exact fo any p. Then the sequence of complexes E 0 ( ) : 0 E 0 (A) E 0 (C) E 0 (C) 0 is also exact and gives ise to the bottom line in the following commutative diagams 9 d(b) d( ) d(a)d(c) e i (B) fo i = 0, 1 with d(e 1 ( )) = d(h(e 0 ( ))). e i (A)e i (C) d(e i (B)) d(e i( )) d(e i (A))d(E i (C)) Poof. The exactness fo the E 0 -tems follows immediately fom the stict exactness while the commutative diagam fo i = 0 is anothe easy consequence of the efinement pinciple [9, pop. 1.9] applied to the filtations and 0 A B F B, the details of which we leave to the eade. Fo i = 1 simply apply the functoiality [3, thm. 3.3] of η with espect to shot exact sequences (see also diagam (3.24)). Ou teatment of deteminants of spectal sequences should be compaed to that in [9, 3] whee apat fom the deived also the spectal filtation of C is used. As in (loc. cit.) ou desciption also genealizes immediately to exact categoies if the admissibility (loc. cit., def. 1.6) and compatibility (loc. cit., def. 1.8) of all involved filtations is ganted. Also note that analogous esults hold fo homological spectal sequences, of couse. 3. (Co)homology and spectal sequences Let F : E D be a ight exact, additive functo between two abelian categoies with finite homological dimension assuming that E has sufficiently many pojectives. In the following we will not distinguish between chain and cochain complexes as they can be identified using the usual enumeation. We wite D b (E) fo the tiangulated deived categoy of bounded complexes in E. Assume that d E : D b (E) qis P E and d D : D b (D) qis P D ae F -compatible deteminant functos (fo deteminant functos of tiangulated categoies we efe the eade to [2]), i.e., such that thee exists a commutative diagam of functos (modulo a natual isomophism q) D b (E) qis d E P E LF D b (D) qis d D P D ψ E D

10 10 OTMAR VENJAKOB in which ψd E is a AC-tenso functo in the language of [9, Appendix A] and such that we have a mophism of deteminants (ψ E D d E ) (d D LF ) in the sense of (loc. cit., def. 1.11) o [2, def. 3.2]. Hee the deived functo LF has to be calculated by a finite esolution of F -acyclic objects, if necessay. Fo simplicity we assume hencefoth that E has finite pojective dimension. Now let (3.19) : 0 A B C 0 be an exact sequence in E (o moe geneally in C b (E) using hype-deived functos LF below). Applying LF gives a distinguished tiangle (3.20) LF ( ) : LF (A) LF (B) LF (C) LF (A)[1] which in tun induces the long exact LF -sequence (3.21) L i F (A) L i F (B) L i F (C) δ F (A) F (B) F (C) 0, in which L i F (A) = L i F (B) = L i F (C) = 0 fo lage i by assumption. additivity elation (3.22) d E ( ) : d E (B) = d E (A)d E (C) induces via ψd E the additivity elation (3.23) d D (LF ( )) : d D (LF (B)) = d D (LF (A))d D (LF (C)) coesponding to (3.20) by the F -compatibility. Lemma 3.1. Thee is a commutative diagam Then the d D (LF (B)) d D (LF ( )) d D (LF (A))d D (LF (C)) η LF (B) i d D(L i F (B)) ( 1)i d D (H(LF ( ))) η LF (A) η LF (C) i d D(L i F (A)) ( 1)i i d D(L i F (C)) ( 1)i whee the bottom line is induced by the long exact sequence (3.21). Poof. [3, thm. 3.3]. On the othe hand the same efeence applied to (3.19) (fo complexes) leads to a commutative diagam (3.24) d E (B) d E ( ) d E (A)d E (C) η B d E (HB) d E (H( )) η A η C d E (HA)d E (HC)

11 whee again the bottom line is induced by the long exact homology sequence attached to (3.19). Now we may apply LF to each homology complex like HA which boils down to fom L i F (H j A) fo all i, j. Recall that fo a bounded complex A thee is a convegent (homological) spectal sequence (3.25) E 2 pq = L p F (H q (A)) LF p+q (A) Moe pecisely, conside a Catan-Eilenbeg esolution of A, i.e., double complexes D A, D Z, D B and D H consisting of pojective objects togethe with shot exact sequences of double complexes 11 (3.26) (3.27) 0 D B D Z D H 0 0 D Z D A D B [1] h 0 We set whee [1] h means shift in the hoizontal diection, i.e., in the diection paallel to B. augmentation maps D A A, D Z Z etcetea such that tot(d A ) A etcetea ae quasi-isomophisms. Moeove, in the vetical diection one has { H v A, i = 0; i (D A ) = 0, othewise. and similaly fo B, Z and H = H(A). (3.28) (3.29) A = totf (D A ) = LF (A) Z = totf (D Z ) B = totf (D B ) H = totf (D H ) = LF (H) = j LF (H j [ j]) and ecall that H i (A) = L i F (A) H i (H) = L i F (H) = j L i j F (H i )) as the diffeentials of H and hence the hoizontal diffeentials h of D H ae all tivial. Note also that (3.25) is just the spectal sequences associated to the double complex F (D A ) using the filtation by ows [13, def ]. With (3.26) also 0 totd B totd Z totd H 0 is exact and consists of pojectives, whence (3.30) 0 B Z H 0 and similaly (3.31) 0 Z A B[1] 0 is also exact. The latte two sequences give ise to an isomophism (3.32) ϑ : d D (A) = d D (H).

12 12 OTMAR VENJAKOB By the functoiality 3.1 of η we obtain the canonical commutative diagam (3.33) d D (A) ϑ d D (H) η A η H d D (HA) H(ϑ) d D (HH). Now we will use the homological vesions of the esults of section 2.1. Theoem 3.2. The following diagam commutes d D (LF (A)) e η ϑ d D (H) = i d D(LF (H i (A))) ( 1)i e 2 (H)=e (H) η H η d D (E (A)) j d D(L j F (A)) ( 1)i e 2 d D (E 2 (A)) i,j d D(L j F (H i (A))) ( 1)i whee the top line is induced by (3.32) and (3.29), while the lowe line e 2 := η 1... η 2 is induced fom the spectal sequence (3.25) using the notation fom section 2.1 and assuming E = E. Moeove the diagam is compatible with (3.33) and the diagams aising fom Poposition 2.2. Poof. Fist note that (E 1 (H), ) = (E 1 (A), ) as complexes (with E 1 pq = D H qp ) and E 2 pq(h) = E 2 pq(a) = LF p (H q ) on objects (but in geneal not as complexes!). Applying Lemma 2.4 to the exact sequences (3.30),(3.31) (taking into account that F applied to the sequences (3.26) again gives exact sequences which gant the hypothesis of the lemma) leads to the commutative diagam d(a) ϑ d(h) e 1 d(e 1 (A)) η d(e 2 (A)) E 1 (ϑ) H(E 1 (ϑ)) e 1 d(e 1 (H)) η d(e 2 (H)) e 2 d(f H) invese d(ψ) d(e (A)) d D (HA) H(ϑ) d(e (H)) d D (HH) d(f H) invese d(ψ) whee accoding to the above calculation the isomophisms E 1 (ϑ) and hence H(E 1 (ϑ)) may be consideed as a natual identification ( )(only this identification defines the latte isomophism). Note that the oute diagam is just diagam (3.33). Taking all these identifications into account the esult follows.

13 3.1. Application to the To-Functo. Let R, R be two egula ings and Y a (R, R )-bimodule, which is finitely geneated and pojective as R -module. Taking fo E and D the categoy of finitely geneated R- and R -modules, espectively, fo d R : D b (E) qis P R and d R : D b (D) qis P R the deteminant functo of Fukaya and Kato [8, 1], setting L( ) = Y R and letting ψr R := ψ E D : P R P R be the canonical functo induced by L we ae in the situation of the pevious subsection. In paticula thee is a convegent (homological) spectal sequence (3.34) E 2 pq = To R p (Y, H q (A)) To R p+q(y, A) and the canonical identification d R (A) = d R (HA) = i d R (H i (A)) ( 1)i induces by (3.22), (3.23) and Theoem 3.2 the fist line of the following commutative diagam 13 (3.35) d R (Y L A) i d R (Y L H i (A)) ( 1)i j d R (ToR j (Y, A))( 1)j i,j d R (ToR j (Y, H i (A)) ( 1)i+j ) whee the bottom ow is induced by the spectal sequence (3.34). Remak 3.3. Stictly speaking Fukaya and Kato s deteminant functo goes fom C b (Pmod(R)) qis to P R whee Pmod(R) denotes the exact categoy of finitely geneated pojective (say left) R-modules. But due to ou egulaity assumption on R we may choose finite pojective esolutions fo each finitely geneated R-module and conside the functo E C b (Pmod(R)) qis P R which then extends also to give a deteminant on C b (E) qis. Note also that by [10, co. 2 of thm. 2] in this situation we have canonical isomophism fo all distinguished tiangles (instead of shot exact sequences of complexes) whence we obtain the desied deteminant functo fom D b (E) qis to P R. This should also be compaed with [2, co. 5.3] applied to the tiangulated deived categoy D b (E) of bounded complexes in E. In this way moe natual examples aise whee the esults of this section apply Compatibility of the Snake lemma and Spectal sequences. Finally we just state a esult concening the compaison of spectal sequences vesus long exact sequences: Conside a shot exact sequence (3.36) 0 A f B g C 0 of bounded cochain complexes concentated in degee geate o equal to 0. Setting D 0 := A, d 0 v : = d A, D 1 := B, d 1 v : = d B, D 2 := C, d 2 v : = d C, D i := 0 othewise,

14 14 OTMAR VENJAKOB we obtain a (cochain) double complex D with hoizontal diffeentials d 0 h = f, d 1 h = g and 0 othewise. Lemma 3.4. In the above situation the total complex tot(d) is acyclic and the isomophism e 1 := η 1... η 1 (3.37) d(h i (A)) ( 1)i d(h i (B)) ( 1)i+1 d(h i (C)) ( 1)i = d(tot(d)) = 1 i i i aising fom the spectal sequence E pq 1 = Hq v(d p ) H p+q (tot(d)) = 0 coincides with the isomophism aising fom the long exact cohomology sequence associated with (3.36). The easy poof is left to the eade. In paticula, identifications via the Snake Lemma ae compatible with spectal sequences. Remak 3.5. If one shifts the double complex to diffeent degees, the diffeentials may change thei signs, but these changes ae compatible with usual sign conventions fo shifting and fo the elation between double complexes and cochain complexes ove the categoy of cochain complexes [13, Sign tick 1.2.5]. Refeences 1. M. Beuning, Deteminants of pefect complexes and Eule chaacteistics in elative K 0-goups, AXiv e-pints (2008). 1 2., Deteminant functos on tiangulated categoies, J. K-Theoy 8 (2011), no. 2, , 2.2, 3, M. Beuning and D. Buns, Additivity of Eule chaacteistics in elative algebaic K-goups, Homology, Homotopy Appl. 7 (2005), no. 3, (document), 1, 2.2, 2.2, 2.2, 2.2, 3 4. D. Buns and M. Flach, Tamagawa numbes fo motives with (non-commutative) coefficients, Doc. Math. 6 (2001), (document), 1 5., Tamagawa numbes fo motives with (noncommutative) coefficients. II, Ame. J. Math. 125 (2003), no. 3, (document), 1 6., On the equivaiant Tamagawa numbe conjectue fo Tate motives, pat II, Doc. Math. Exta Vol. (2006), , John H. Coates Sixtieth Bithday. (document), 1 7. P. Deligne, Le déteminant de la cohomologie, Cuent tends in aithmetical algebaic geomety (Acata, Calif., 1985), Contemp. Math., vol. 67, Ame. Math. Soc., Povidence, RI, 1987, pp T. Fukaya and K. Kato, A fomulation of conjectues on p-adic zeta functions in non-commutative Iwasawa theoy, Poceedings of the St. Petesbug Mathematical Society, Vol. XII (Povidence, RI), Ame. Math. Soc. Tansl. Se. 2, vol. 219, Ame. Math. Soc., 2006, pp , F. F. Knudsen, Deteminant functos on exact categoies and thei extensions to categoies of bounded complexes, Michigan Math. J. 50 (2002), no. 2, (document), 1, 2.1, 2.2, 2.2, 2.2, F. F. Knudsen and D. Mumfod, The pojectivity of the moduli space of stable cuves. I. Peliminaies on det and Div, Math. Scand. 39 (1976), no. 1, , F. Muo, A. Tonks, and M. Witte, On deteminant functos and K-theoy, AXiv e-pints (2010) O. Venjakob, On the non-commutative Local Main Conjectue fo elliptic cuves with complex multiplication, pepint (2011). (document) 13. C. A. Weibel, An intoduction to homological algeba, Cambidge Studies in Advanced Mathematics, vol. 38, Cambidge Univesity Pess, , 3.5

15 M. Witte, On the equivaiant main conjectue of Iwasawa theoy, Acta Aith. 122 (2006), no. 3, , Noncommutative Iwasawa main conjectues fo vaieties ove finite fields, Ph.D. thesis, Univesität Leipzig, Univesität Heidelbeg, Mathematisches Institut, Im Neuenheime Feld 288, Heidelbeg, Gemany. addess: venjakob@mathi.uni-heidelbeg.de URL: