On diagram-chasing in double complexes

ON DIA GRAM-CHASING IN DOUBLE COMPLEXES GEOR GE M. BER GMAN Abstract. W e construct, for any double complex in an ab elian category , certain “short-dista nce” maps, and an exa ct sequence inv olving these, instance s of which can be pieced together to g ive the “long-dis tance” maps and exact sequences of results suc h as the Snake Lemma. F urther applications are giv en. W e also note what the building blo cks of an analogo us study of triple complexes w ould b e. In tro duction Diagram-chasing a rgumen ts frequen tly lead to “mag ical” relations b et w een distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually comp o sites o f short “unmagical” connections, but the latter, and the ob jects they join, are not visible in the pro ofs. This note is aimed at remedying that situation. Giv en a double complex in an ab elian category , w e will consider, for eac h ob ject A o f the complex, the familiar horizon tal and v ertical homology ob jects at A (whic h w e will denote A · and A · ) , and t w o other ob jects, A and A (whic h we name the “donor” and the “receptor” at A ) . F or eac h arro w of the double complex, w e pro v e in § 1 the exactness of a certain 6 -term sequenc e of maps b et w een t hese ob jects (the “Salamander Lemma”). Standard results suc h as the 3 × 3 -Lemma, the Snak e Lemma, and the long exact sequence of homology associated with a short exact sequenc e of complexes are reco v ered in §§ 2 -4 as easy applications of this result. In § 5 w e generalize the last of these examples, g etting v a rious exact diagrams fro m double complexes with all but a few row s and columns exact. The total homolog y of a double complex is examined in § 6. In § 7.2 w e ta k e a brief lo ok a t the w orld of triple complexes , and in § 7.3 a t the relation b et w een the metho ds of this note and J. Lam b ek’s homological formulation of Goursat’s Lemma [8]. W e end with a couple of exercises . arXiv: 1108.0958. After publication of this no te, upda tes, errata , rela ted refer ences etc., if found, will be r ecorded at http://ma th.be rkeley.edu/ ~ gbergm an/pa pers/ . 2000 Mathematics Sub ject Classiﬁcatio n: Prima ry: 1 8G35. Secondary: 18E1 0.. Key words and phrases: double complex, e x act seq uence, diagr am-chasing, Salamander L e mma, total homology , tr iple complex. c  George M. Bergman, 2012. Permissio n to copy for priv ate use grant ed. 1 2 1. Deﬁnitio n s, and the Sal amander Lemm a W e shall w ork in an ab elian catego ry A . In the diagrams w e dra w, capital letters and b oldface dots · will represen t arbit r a ry o b jects of A . (Th us , suc h a dot do es not imply a zero ob ject, but simply an ob ject w e do not name.) Lo w er-case letters and arr o ws will denote morphisms. When we giv e examples in categories of mo dules, “mo dule” can mean left or r ig h t mo dule, as the reader prefers. A double c omplex is an array of ob jects and maps in A , of t he form · ✲ ❄ · ✲ ❄ · ✲ ❄ · ✲ ❄ ✲ · ✲ ❄ · ✲ ❄ · ✲ ❄ · ✲ ❄ ✲ · ✲ ❄ · ✲ ❄ · ✲ ❄ · ✲ ❄ ✲ · ✲ ❄ · ✲ ❄ · ✲ ❄ · ✲ ❄ ✲ ❄ ❄ ❄ ❄ (1) extending inﬁnitely on all sides, in whic h eve ry ro w and ev ery column is a complex (i.e., success iv e arrow s comp ose to zero), and all squares commute . Note that a “ partial” double complex suc h as · ✲ ❄ · ✲ · ✲ ❄ · ✲ · ✲ ❄ · ✲ · ❄ · (2) can b e made a double complex b y completing it with zero es on all sides; o r by writing in some k ernels and cok ernels, a nd then zero es b ey ond these. Th us, r esults on double complexes will b e applicable to suc h ﬁnite dia g rams. T opo logists often prefer double complexes with anti comm uting squares; but either sort of double complex can b e turned in to the other b y rev ersing the signs of the arrows in ev ery o t her ro w. In the theory of sp ectral sequence s, vertical arrow s generally go up w ard, while in results like the F our Lemma they are generally dra wn down w ard; I shall follow the latter con v en tion. 1.1. Definition . L et A b e an obje ct o f a double c omplex, with ne arby map s lab ele d as shown b elow: · ✲ · ✲ h · ❄ ❄ ❄ f g · ✲ d A ✲ e · ❄ ❄ ❄ b c · ✲ a · ✲ · , and let p = ca = db, q = g e = hf . (3) Then we deﬁne 3 A · = K er e / Im d, the horizontal homolo gy obje ct at A, A · = K er f / Im c, the vertic al hom o l o gy obje ct at A, A = ( K er e ∩ Ker f ) / Im p, which we shal l c al l the receptor at A, A = K er q / (Im c + Im d ) , which we shal l c al l the donor at A. F rom the inclusion relations among the kernels and images in Deﬁnition 1.1, w e get 1.2. Lemma . F or ev ery obje ct A of a double c omplex, the identity map of A induc es a c ommuting diagr am of m aps: A ❘ ✠ A · A · ✠ ❘ A (4) 1.3. Definition . We sha l l c al l the maps sh o wn in (4) the in tramural maps ass o ciate d with the obje ct A. When w e draw the diag ram o f a double complex, the donor and receptor at an ob ject will generally b e indicated by small squares to t he lo w er right and upp er left of the dot or letter represen ting that ob ject, as in (5) b elo w. Th us, the direction in whic h the square is displaced from the letter is t ow ard the most distan t p oin t of the diagra m inv olv ed in the deﬁnition o f the ob ject; in (3), the domain or co domain of the comp osite a r r o w p o r q . (Of course, if one prefers to dra w double complexes with arrows going upwar d and to the right, one should write A a nd A for the receptor and dono r at A. ) I will o ccasionally indicate horizontal or v ertical homology ob j ects in a diagram b y marks · and · placed at the lo cation of the ob ject; but this requires suppres sing the sym b o l for the o b jec t itself. The next result, whose pro of is a g ain straigh tforward, motiv a t es the names “dono r” and “receptor”. 1.4. Lemma . Each arr ow f : A → B in a double c omplex induc es an arr ow A → B : ✲ ✲ ✲ A B ❃ ❄ ❄ ❄ ❄ ❄ ❄ ❄ A B ✴ ✲ ✲ ✲ ✲ (5) 1.5. Definition . We shal l c al l the morphism of L emma 1.4 the extramural m ap ass o - ciate d with f . 4 The globa l picture of the extram ural maps in a double complex is ✲ ✒ ✲ ✒ ✲ ✒ ✲ ✒ · · · ✲ ✒ ✲ ✒ ✲ ✒ ✲ ✒ · · · ✲ ✒ ✲ ✒ ✲ ✒ ✲ ✒ · · · ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ ❄ ✠ (6) In most of this note, I shall not use an y notation for these in tram ural and extram ural maps. Betw een any t w o o f the ob jects w e ha v e constructed, w e will not deﬁne more than one map, so w e shall b e a ble to get by with an unlab eled arrow, represen ting the unique map constructed b et w een the ob jec ts named. (In § 6, where w e will hav e more tha n one map b etw ee n the same pair o f ob jects, I will intro duce sym b ols for some of t hese.) T o sho w a c omp osite of the maps w e ha v e deﬁned, w e may use a long arrow mark ed with dots indicating the in termediate ob jects in v olv ed, as in the stateme n t of the follo wing easily ve riﬁed lemma. 1.6. Lemma . I f f : A → B is a horizon tal arr ow of a d ouble c omplex ( as in the ﬁrst diagr am of (5)) , then the natur al induc e d map b etwe en v ertical homolo gy o bje cts, A · → B · , is the c omp osite of one extr amur al, and two intr amur a l maps: A · ✲ A · B · B · . (7) Similarly, for a ve rtical map f : A → B ( as in the se c ond diagr am of (5)) , the natur al induc e d map o f horizontal homolo gy obje cts is giv e n by A · ✲ A · B · B · . (8) W e no w come to our mo dest main result. W e will again state b oth the horizontal and v ertical cases, since w e will ha v e n umerous o ccasions to use eac h. T he v eriﬁcations are trivial if one is allow ed to “c hase elemen ts”. T o get the result in a general ab elian category , one can use o ne of the concretization theorems referred t o at [13, Notes to Chapter VII I], or the metho d of generalized “mem bers” dev elop ed in [1 3, § VII I.3], or the related metho d discuss ed at [2]. (Regarding p oint (vi) of [1 3, § VI I I.4, Theorem 3 ], w e note that this might b e replaced by the f ollo wing more con v enien t statement, clear from the pro of: Given g : B → C , and x, y ∈ m B with g x ≡ g y , ther e exis t x ′ ≡ x, y ′ ≡ y with a c ommon domain , such that g x ′ = g y ′ . ) W e giv e our result a name a nalogous to that of the Snak e Lemma. 5 ✲ C ❄ A ✲ B ❄ D ❄ C ✲ A ❄ B ✲ D · ❫ · s ✯ ❫ · ✙ · · s · ❫ ☛ ❘ · ✕ · · ❄ · ✲ · ❄ · (9) 1.7. Lemma . [Salamander Lemma] S upp ose A → B is a ho riz ontal arr ow in a double c omplex, and C , D a r e the obje cts ab ove A and b elo w B r esp e ctively, as in the upp e r left diagr am of (9) . Then the fol low ing se quenc e ((9) , upp er right ) , forme d f r om intr a mur al and extr amur al maps, is ex act: C ✲ A · A · ✲ A ✲ B ✲ B · ✲ B · D . (10) Likewise, if A → B is a vertic al arr ow (( 9 ) , lower left ) , we have an exact se quenc e ((9) , lower right ) : C ✲ A · A · ✲ A ✲ B ✲ B · ✲ B · D . (11) In eac h case, w e shall call the sequence displa y ed “ the 6-term exact sequence associated with the map A → B of the g iv en double complex”. Remark: F or A → B a horizon tal arr o w in a double complex, and C , D as in the upp er left diagram of (9 ), not only (10) but also (11 ) mak es sens e, but o nly the former is in general exact. Indeed, b y Lemma 1.6, the middle three maps of (11) comp ose in that case to the natural map from A · to B · , rather than to zero as they would if it w ere an exact sequence. Lik ewise , if A → B is a ve rtical map, then (10) a nd (11) b oth mak e sense, but o nly the latter is in general exact. 2. Sp ecial cases, a nd easy a pplicat ions Note that the extram ural arrows in (6) stand head-to-head and tail- to-tail, and so cannot b e comp osed. This diﬃcult y is remov ed under a ppropriate conditions by the following 6 corollary to Lemma 1.7, which one gets on assuming the t w o homolo gy o b jects in (10) or (11) to b e 0 : 2.1. Cor ollar y . L et A → B b e a horizontal ( r esp e ctively, vertic al ) arr ow in a double c omplex, and supp ose the r ow ( r esp., c olumn ) c ontaining this map is exact at b oth A and B . The n the induc e d extr amur al map A → B is an isomorphism. Using ano t her degenerate case of Lemma 1.7, we g et conditions for intr amur al maps to b e isomorphisms, allowing us to identify donor and receptor ob jects with classical homology ob j ects. 2.2. Cor ollar y . In e ach of the s ituations sh o wn b elow, i f the diagr am is a double c omplex, an d the d a rkene d r ow or c olumn ( the r ow or c olumn thr ough B p erp e n dicular to the arr ow c on n e cting it with A ) is exac t at B , then the two intr amur al maps at A indic ate d ab ove the dia g r am ar e isom orphisms. A ✲ ∼ = A · A · ✲ ∼ = A 0 ✲ B ✲ · ❄ ❄ 0 ✲ A ✲ · ❄ ❄ A ✲ ∼ = A · A · ✲ ∼ = A ✲ · ✲ · ❄ ❄ ✲ A ✲ B ❄ ❄ 0 0 A · ✲ ∼ = A A ✲ ∼ = A · ✲ · ✲ A ✲ 0 ❄ ❄ ✲ · ✲ B ✲ 0 ❄ ❄ A · ✲ ∼ = A A ✲ ∼ = A · 0 0 ❄ ❄ ✲ B ✲ A ❄ ❄ ✲ · ✲ · ❄ ❄ (12) Proo f . W e will pro v e the isomorphism statemen ts for the ﬁrst diagra m; the pro of s fo r the remaining diagrams are obta ined by rev ersing the roles of r ows and columns, and/or rev ersing the directions of all ar ro ws. In the ﬁrst diagram, Coro llary 2.1, applied to the arrow 0 → B , giv es B ∼ = 0 = 0 . The exact seque nce (10) a sso ciated with the map 0 → A therefore ends 0 → A → A · → 0 , while the exact seque nce (11) asso ciated with the ma p A → B b egins 0 → A · → A → 0 . This gives the tw o desired isomorphisms. R emarks. Corollaries 2.1 and 2.2 are easy to pro v e directly , so in these tw o pro ofs, the Salamander Lemma w as no t a necessary to ol, but a guiding principle. One should not b e misled b y apparen t further dualizations. F or instance, one migh t think t hat if in the leftmost diagram of (12), one assumed r ow-exactnes s at A rather than at B , o ne w ould get isomorphism s B ∼ = B · and B · ∼ = B b y rev ersing the directions of v ertical arrows , a nd applying the result pro v ed. How ev er, there is no principle that allo ws us to rev erse some arro ws (the v ertical ones) without rev ersing others (t he horizontal arro ws); and in fact, t he asserted isomorphisms fail. E.g., if the v ertical arro w down from 7 B is the iden tit y map of a nonzero ob ject, and all o ther ob jects of the double complex are zero, then ro w-exactnes s do es hold at A (since the ro w con taining A consists of ze ro- ob jects), but the maps B → B · and B · → B b oth ha v e the form 0 → B . (And if one conjectures instead that o ne or b oth o f the other tw o in tramu ral maps at B should b e isomorphisms, the diagram with the horizon tal ma p out of B an identit y map and all other ob jects 0 b elies these statemen ts.) The common feature of the four situations of ( 1 2) t ha t is not shared b y the results of reve rsing only v ertical or only horizon tal arro ws is that the arro w conne cting A with B , a nd the arro w connecting a zero ob ject with B , ha v e the same orien tation relativ e to B ; i.e., either b oth go in to it, or b oth come out of it. Most o f the “small” diagram-chas ing lemmas of homological algebra can b e obta ined from the ab o v e t w o coro llaries. F or example: 2.3. Lemma . [The Sharp 3 × 3 Lemma] In the diagr am b el o w (excluding , r esp e ctively including the p ar enthesize d arr ows; and ign oring for now the b oxes and dotte d lines, which b elong to the pr o of ) , if al l c olumns, and a l l r ow s but the ﬁ rst, ar e exact, then the ﬁrs t r ow ( again excluding, r esp e ctively including the p ar enthesize d arr ow ) is also e xact. 0 ✲ ❄ ✲ ❄ ✲ ❄ 0 ✲ ❄ ✲ ❄ ✲ ❄ 0 ✲ ❄ ✲ ❄ ✲ ❄ 0 0 0 ( ✲ 0) ( ✲ 0) ⌣ 0 ❄ ⌢ A ′′ A A ′ B ′′ B B ′ C ′′ C C ′ (13) Proo f . W e ﬁrst note that in view of the exactness of the columns, the ﬁrst row of (13) consists of subob jects of the ob j ects of the second row , and restrictions of the maps among these, hence it is, at least, a complex, so the whole diagram is a double complex. Corollaries 2.1 and 2.2, combined with the exactness h yp otheses not in v olving the paren thesized a rro ws, now giv e us A ′ · ∼ = A ′ ∼ = A ′ · = 0 , B ′ · ∼ = B ′ ∼ = B ∼ = A ∼ = A · = 0 (short dotted path in (13)), (14) and, assuming also the exactness conditions in v olving the parenthes ized arrows, C ′ · ∼ = C ′ ∼ = C ∼ = B ∼ = B ′′ ∼ = A ′′ ∼ = A ′′ · = 0 (long dotted path in (13 )). (15) (As a n example of ho w to determine whic h statemen t of whic h of those corollaries to use in eac h case, consider the ﬁrst isomorphism of the ﬁrst line of (1 4). Since it concerns an intram ural isomorphism, it m ust b e an a pplication of Corollar y 2.2. Since the one of 8 A ′ , A ′ that it in v olv es faces a w a y from the zero es of the diagram, it must come f r o m the second row of isomorphisms in tha t corollar y; and giv en that fact, since it inv olv es a horizon tal homology ob ject, it m ust call on exactness at an o b j ect horizon tally displaced from A ′ , hence m ust come from the second or f o urth diagram of (1 2). Lo oking at the placemen t of t he zero es, w e se e that it must come from the second diagram, a nd that the needed h yp othesis, v ertical exactness at the ob ject to the right of A ′ , is indeed pr esen t. The isomorphisms other than the ﬁrst and the last in each line of (14) and in (15 ), corresp onding to extram ural maps, f ollo w from Corollary 2.1.) The consequen t triviality of the tw o (r esp ectiv ely three) horizontal homolog y ob jects with whic h (14) (and (15)) b egin giv es the desired exactness of the top row of (13). The diagonal c hains of donor s and receptors whic h we follo w ed in the ab ov e pro of fulﬁll the promise that “long” connections w ould b e reduced to comp osites o f “short” ones. The pro of of the next lemma con tinu es this theme. 2.4. Lemma . [Snak e Lemma, [1, p. 23], [4], [10, p. 158], [12, p. 50]] I f, in the c ommuting diagr am at left b elow, b oth r ows ar e exact, and we app end a r ow of kernels and a r ow of c okernels to the vertic al ma ps, as in the diagr am at right, ✲ ✲ ✲ ✲ ❄ ❄ ❄ 0 ✲ ✲ 0 Y 1 Y 2 Y 3 X 1 X 2 X 3 ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ 0 ✲ ✲ 0 C 1 C 2 C 3 Y 1 Y 2 Y 3 X 1 X 2 X 3 K 1 K 2 K 3 (16) then those two r o w s ﬁt to gether into an exact se quenc e ✲ ✲ ✲ ✲ ✲ K 1 K 2 K 3 C 1 C 2 C 3 . (17) Proo f . W e extend the right diagram of (16) to a double complex b y at t a c hing a ke rnel X 0 to the second ro w and a cok ernel Y 4 to the third, and ﬁlling in zero es ev erywhere else. In this complex, the three columns sho wn in (16) are exact, and we ha v e horizontal exactness at X 1 , X 2 , X 3 , Y 1 , Y 2 and Y 3 . The exactnes s of (17) at K 2 , i.e., the trivialit y of K 2 · , no w follows from t he follow ing isomorphisms (the ﬁrst a case of Corolla r y 2.2 , the next four of Corolla ry 2 .1; cf. second equation of (1 4)): K 2 · ∼ = K 2 ∼ = X 2 ∼ = X 1 ∼ = Y 1 ∼ = 0 = 0 . (18) Exactness at C 2 is sho wn similarly . W e now w ant to ﬁnd a connecting map K 3 → C 1 making (17) exact at these tw o ob jects. This is equiv alen t to an isomorphism b et w een Cok( K 2 → K 3 ) = K 3 · and 9 Ker( C 1 → C 2 ) = C 1 · . And indeed, such an isomorphism is giv en b y the comp osite K 3 · ∼ = K 3 ∼ = X 3 ∼ = X 2 ∼ = Y 2 ∼ = Y 1 ∼ = C 1 ∼ = C 1 · (19) of t w o intram ural and ﬁve extram ural maps sho wn as the dotted path in (16), whic h are again isomorphisms b y Coro lla ries 2.2 and 2.1. The next result, whose pro of b y the same metho d we leav e to the reader, establishes isomorphisms b etw een inﬁnitely man y pairs of homology ob jects in a double complex b ordered b y zero es, either in t w o parallel, or t w o p erp endicular directions. Before stat ing it, we need to mak e some c hoices ab o ut indexing. 2.5. Convention . When the o bje cts of a double c omplex ar e indexe d by numeric al subscripts, the ﬁrst subsc ri p t wil l sp e ci f y the r ow a n d the se c ond the c olumn, and these wil l incr e a se downwar ds, r esp e ctively, to the right ( as in the numb ering of the entries of a matrix; but not as in the standar d c o or dinatization of the ( x, y ) -plane ) . Since our arrows also p oint do wn w ard and to the righ t, our complexes will b e double c o chain complexes; i.e., the b oundary morphisms will go from low er- to higher-indexed ob jects. How ev er, we will con tin ue to call the constructed ob jects “homology ob jects”, rather than “ cohomology ob j ects”. Here is the promised result, whic h the reader can easily prov e by the metho d used fo r Lemmas 2.3 and 2.4. 2.6. Lemma . If in the left-han d c o m plex b elow, al l r ows but the ﬁrst r ow shown ( the r ow of A 0 ,r ’s ) and al l c olumns but the ﬁrst c olumn shown ( the c ol umn o f A r, 0 ’s ) , ar e exact, then the hom olo gies of the ﬁ rs t r ow a nd the ﬁrst c olumn ar e isomorphic: A 0 ,r · ∼ = A r, 0 · . ( A nd anal o gously for a c omplex b or de r e d by z e r o es on the b ottom and the righ t . ) If in the right-hand c ommuting diagr am b elow, al l c olumns ar e exact, and al l r ows but the ﬁrst and last ar e e xact, then the ho m olo gies of those two r ows agr e e with a shift of n − m − 1 : A m,r · ∼ = A n,r − n + m +1 · . ( A nd analo gously for a c omplex b or de r e d on the left and right by zer o es. ) 0 ✲ ✲ ✲ ✲ ✲ 0 ✲ ✲ ✲ ✲ ✲ 0 ✲ ✲ ✲ ✲ ✲ 0 ✲ ✲ ✲ ✲ ✲ 0 ❄ ❄ ❄ ❄ ❄ 0 ❄ ❄ ❄ ❄ ❄ 0 ❄ ❄ ❄ ❄ ❄ 0 ❄ ❄ ❄ ❄ ❄ A 30 A 20 A 10 A 00 A 31 A 21 A 11 A 01 A 32 A 22 A 12 A 02 A 33 A 23 A 13 A 03 ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ 0 ❄ ❄ . . . ❄ ❄ 0 0 ❄ ❄ . . . ❄ ❄ 0 0 ❄ ❄ . . . ❄ ❄ 0 0 ❄ ❄ . . . ❄ ❄ 0 A m, − 1 A m, 0 A m, 1 A m, 2 · · · · · · · · A n, − 1 A n, 0 A n, 1 A n, 2 (20) 10 In con trast to t he ﬁrst of these results, if w e form a double complex b ordered o n the top a nd the right (or on the b ottom and the left ) b y zero es, and we again assume all rows and columns exact except those adjacen t to the indicated row and column of zeroes, there will in general b e no relatio n b et w een t heir ho mo lo gies. F or a coun terexample, one can tak e a double complex in whic h all ob j ects a r e zero except f or a “staircase” of isomorphic ob jects running up w ard to the righ t till it hits one of the “b orders”. The place where it hits that b o rder will b e the only place where a nonzero homology ob ject o ccurs. (If o ne tries to construct a similar counterex ample t o the ﬁrst assertion o f the ab ov e lemma b y running a staircase o f isomorphisms up w ard to the left , one ﬁnds that the resulting arra y of ob jects and morphisms is not a commu tative diagram.) 3. W eakly b ounded do uble complexes Before exploring uses of the full statemen t of the Salama nder Lemma, it will b e instructiv e to consider a mild generalization of our last result. Supp ose that as in the right-hand diagram o f (20) we ha v e a double complex with exact columns, b ounded ab o v e the m -th ro w and b elo w the n -th ro w b y zeros. But r a ther than assuming exactness in a ll but the m -th and n -th rows, let us assume it in a ll ro ws but the i -th and j -th, for some i and j with m ≤ i < j ≤ n. I claim it will still b e true that the ho molo gies o f these row s agr ee up to a shift: A i,r · ∼ = A j,r − j + i +1 · . (21) Indeed, ﬁrst no t e that by comp osing extram ural isomorphisms as in the preceding section, we get A i,r ∼ = A j,r − j + i +1 . (22) So the pro blem is to strengthen Corollary 2.2 to sh ow that the o b jects of (22 ) are isomor- phic (b y the intram ural maps) to their coun terparts in (21). The required generalization of Corollary 2.2 is quite simple. 3.1. Cor ollar y . Supp ose A is an obje ct of a double c omplex, and the ne arby donor and r e c eptor obje cts m arke d “ ” in one of the diag r ams b elow ar e zer o. · ✲ ✲ · ❄ ❄ A · · · ✲ ✲ · ❄ ❄ A · · A ∼ = A · A · ∼ = A A ∼ = A · A · ∼ = A (23) Then the two intr amur al is o morphisms indic ate d b elow that dia gr am hold. 11 Proo f . T o get the ﬁrst isomorphism of the ﬁrst diagram, a pply the Salamander Lemma to the arr ow coming v ertically into A ; to get the second, apply it to the arrow coming horizon tally out of A. In the second diagram, similarly apply it t o the tw o arro ws bearing the “ ”s. No w in the situation of the ﬁrst paragraph of this section, our double complex is exact horizon tally ab o v e the i -th r o w, and v ertically ev erywhere, so w e can use Corolla ry 2.1 to connect any donor ab ov e the i -th row, or a n y receptor at or ab o v e t ha t ro w, to a donor or receptor abov e the m -th ro w, pro ving it zero. Corollary 3.1 then show s the left-hand side of (21) to b e isomorphic to the left-hand side of (22). Similarly , using exactness b elo w the j -th ro w, a nd v anishing b elow t he n -th, w e ﬁnd that the righ t-hand sides of those displa ys are isomorphic. Th us, (22) yields (21), as desired. What if w e hav e a double complex in whic h all columns, and all but the i -th and j -th ro ws, ar e exact, but we do no t a ssume that all but ﬁnitely man y ro ws are zero? Starting from the receptor at an y ob ject of the i -th row, w e can still g et an inﬁnite c hain of isomorphisms going upw ard and to the right: A i,s ∼ = A i − 1 ,s ∼ = A i − 1 ,s +1 ∼ = A i − 2 ,s +1 ∼ = . . . , (24) but w e can no longer assert that the common v alue is zero; and similarly b elo w the j -th ro w. Ho w ev er, there are certainly w eak er hypotheses tha n the one we w ere using ab ov e that wil l allow us to sa y this common v alue is 0; e.g., the existen ce of zero quadra nts (rather than half-planes) on the upp er right and lo w er left. Let us make , still more generally , 3.2. Definition . A double c omple x ( A r,s ) wil l b e c al le d weak ly b ounded if for every r and s, ther e exi s ts a p ositiv e inte ger n such that A r − n,s + n or A r − n − 1 ,s + n is zer o, and also a negativ e inte ger n with the sam e pr op ert y. The ab ov e discussion no w yields the ﬁrst statemen t of the next corollary; t he ﬁnal statemen t is seen to hold by a similar arg umen t. 3.3. Cor ollar y . [to pro of of Lemma 2.6] L et ( A r,s ) b e a w eakly bounded double c om- plex. If al l c olumns ar e exact, and al l r ows b ut the i -th and j -th ar e exact, wher e i < j, then the homolo gies of these r ows ar e isomorphi c with a shift: A i,r · ∼ = A j,r − j + i +1 · . The analo gous statement holds if al l r ow s and al l but two c o l umn s ar e e x a ct. If al l r ows but the i -th, and al l c olumns but the j -th ar e exact ( i and j arbitr ary ) , then the i -th r ow and j -th c olumn have isomorphic homolo gies: A i,r · ∼ = A r − j + i,j · . T o see that the ab ov e corolla r y fails without the h yp othesis of w eak b oundedness, consider ag a in a double complex that is zero except for a “ staircase” o f copies of a nonzero 12 ob ject and iden tit y maps b et w een them: ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ✲ 0 ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ❄ ❄ ❄ ✲ ❄ 0 ✲ ❄ 0 A A A A A 0 0 ❄ ❄ ✲ ✲ (25) All ro ws and all columns a re then exact except the r ow containing the low est “ A ”. Considering that ro w and any other row , w e get a con tradiction to the ﬁrst conclusion of Corolla ry 3.3. Considering that ro w and any column gives a contradiction to the ﬁnal conclusion. W e remark that if, in Corolla ry 3.3, we ma ke the substitution s = r + i in the subscripts, then our isomorphisms take t he forms A i,s − i · ∼ = A j,s − j +1 · and A i,s − i · ∼ = A s − j,j · . These form ulas are more symmetric t ha n those using r , but I ﬁnd them a little less easy to t hink ab out, b ecause the v ariable index s nev er app ears alone. But in later results, Lemmas 4.4 and 5.1, where the analog of t he r - indexing w ould b e mess ier than it is here, w e shall use the a nalog of this s -indexing. 4. Long exact sequences A t this p oin t it w ould b e easy t o apply Lemma 1.7 and Corollaries 2.1 and 2.2 to give a quic k construction of the long exact sequ ence of homolo gies asso ciated with a short exact sequence s of complexes; the reader ma y wish to do so for him or her self. But w e shall ﬁnd it more instructiv e to examine ho w the six-term exact “ salamander” sequenc es w e ha v e a sso ciated with the arrows of a do uble complex link to gether under v arious w eak er h yp otheses, and see that the ab o v e long exact sequence is the simplest in teresting case of some more general phenomena. Let B b e an y o b jec t of a double complex, with some neigh b oring ob jects lab eled a s 13 follo ws. D E A B C F G , ❄ ❄ ✲ ✲ ❄ ❄ ✲ ✲ (26) and let us consider the six-term exact sequ ences asso ciated with the four arrow s in to and out of B . These piece together a s in the follo wing diagram, whe re the cen tral sq uare and eac h of the four triangular we dges comm ute: D E · E B B · B · B C C · G ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ D A · A F F · G ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ✲ ✲ (27) W e now note what happ ens if (26 ) is v ertically or horizon tally exact at B . 4.1. Lemma . Supp ose in (26) that B · = 0 , o r that B · = 0 , or m or e gener al ly, that the intr amur al map B → B is zer o. Then the fol lowing two 9 -term se quenc e s ( the ﬁrst 14 obtaine d fr om the “left-hand ” and “b ottom” br anches of ( 2 7) , the se c ond fr o m the “top” and “right-hand” b r anches ) ar e exa ct: ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ D A · A B B · B C C · G, (28) ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ D E · E B B · B F F · G. (29) Proo f . The exactness o f the 6-term sequences of whic h (27) is comp osed giv es the ex- actness of (28) and (29) ev erywhere but at the middle terms, B · and B · . That the comp osite map through that middle term equals zero is, in each case, our hypothesis on the in tram ural map B → B . That, con v ersely , the ke rnel of the map out of that middle ob ject is con tained in the imag e of the map going in to it can b e seen from (27); e.g., in the case of (28), we see from (2 7) that the k ernel o f the map B · → B is the image of the map E → B · , and that map f actors t hro ugh the ma p B → B · . (This is t he comm utativit y of the topmost triangula r w edge in ( 27). A dual pro of can b e gotten using the right-hand w edge.) In noting applications of the ab ov e result, we shall, for brevit y , restrict o ur selv es to (28); the corresp onding consequences of (29) follo w by symmetry . Of the alternative h yp otheses of Lemma 4.1, the condition B · = 0 mak es (28) degenerate, while the more general statemen t t hat B → B is zero do es not corresp ond to an y condition in the standard language of homolog ical algebra; so in the following corollary , we f o cus mainly on the condition B · = 0 . 4.2. Cor ollar y . If i n a double c omplex, a pie c e of which is l a b ele d as in (26) , the v ertical homolo gies ar e zer o for al l ob j e cts in the row · · · → A → B → C → . . . ( or mor e gener al ly, if the intr amur al map fr om r e c eptor to donor is zer o fo r e ach obje ct of that r ow ) , then the fo l lowing se quenc e of horizontal homolo gy obje cts, donors and r e c eptors, and intr amur al and extr amur al m aps, is e x act: ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ · · · A A · A B B · B C C · C · · · . (30) Proo f . At each ob ject o f the indicated ro w of (26), write down t he exact sequence corresp onding to (28), lea ving oﬀ the ﬁrst and last terms. The remaining parts of these sequence s o v erlap, giving (30). When the v ertical homology in our do uble complex is everywher e zero, the exact sequence s (3 0) arising fr o m successiv e ro ws ar e linke d, at ev ery third po sition, by isomor- phisms giv en b y Corolla ry 2.1, as describ ed in 15 4.3. Cor ollar y . If in a double c omplex W X Y Z P Q R S K L M N A B C D ✲ ❄ ✲ ❄ ✲ ❄ ✲ ✲ ❄ ✲ ❄ ✲ ❄ ✲ ✲ ❄ ✲ ❄ ✲ ❄ ✲ ✲ ❄ ✲ ❄ ✲ ❄ ✲ ✲ ❄ ❄ ✲ ❄ ❄ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ✲ (31) al l columns ar e exact, then the rows ind uc e long exact se quenc es, which ar e link e d by isomorphisms: W W · W X X · X Y Y · Y Z P Q Q · Q R R · R S S · S K · K L L · L M M · M N N · A A · A B B · B C C · C D ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ (32) Note that in these long exact sequenc es, the classical homology ob jects fo rm every thir d term – the terms o f (32) that are not connected either ab ov e or b elo w b y isomorphisms. Supp ose now tha t in (31 ), in addition to all columns b eing exact, some row is exact. This means that in the system of long exact sequenc es (32), t he corresp onding ro w will ha v e eve ry third term zero; and so the maps connecting the remaining terms will b e isomorphisms: ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · ✲ · · · · · · (33) W e see that these, together with the v ertical isomorphisms, tie together the pr e c e din g and fol low i n g exact seq uence to giv e a system essen tially lik e (32), except f or a horizontal shift by one step. If n successiv e rows of the double complex are exact, w e get a similar diagram with a shift b y n steps. 16 If all ro ws are exact ab ov e a certain p oin t, then w e get inﬁnite c hains of isomorphisms going up w ard and to the righ t. If the complex is also w eakly bounded (Deﬁnition 3 .2), the common v alue along those c hains will b e zero; hence ev ery third term of the long exact sequence correspo nding t o the top nonexact ro w of our double complex will b e zero, so w e again hav e isomorphisms b et w een pairs of remaining terms; though not the same pairs as b efore: in eac h isomorphic pair, one of the mem b ers is no w a horizon tal homology o b ject. If w e regard classical homology ob jects as more in teresting than donors and receptors, w e ma y use these isomorphisms and the isomorphisms joining this ro w to the next to insert these homology ob jects in that ro w, in place of a ll the receptors. F or instance, if all row s of (31) ab ov e the top one sho wn are exact, and the complex is w eakly b ounded ab ov e, then in the top tw o row s of (32) w e get ✲ ✲ ✲ ✲ ✲ K · K L L · A · A , whic h we can rewrite ✲ ✲ ✲ ✲ ✲ K · K A · L · . Of course, if, say , the second row o f (31) (unlik e the ﬁrst) happ ens to b e exact, then the ob jects K · , L · etc. in the ab ov e exact sequence are zero, allo wing us t o pull the homology ob jects from the ﬁrst row do wn y et another step, and insert them in to the long exact seque nce a rising from the third ro w. Con tin uing in this w a y as long a s we ﬁnd exact ro ws in (3 1), w e get a link ed system of long exact sequences, o f whic h the top sequence has, as two out of ev ery three terms, horizontal ho mo lo gy ob jects, and a r ises from the t w o highest non- exact row s of (31) (assuming there are at least tw o ). The ob vious analogous situation holds if, instead, all rows b el o w some p oin t are exact. If our original double complex has o nly three nonexact rows, then w e can see that, w orking in this w a y fro m b oth ends, w e get a single long exact sequence with horizontal homology ob j ects for all its terms: 4.4. Lemma . Supp ose we ar e given a we akly b ounde d double c o mplex, with obje cts A h,r , al l c olumns exac t, and al l r ows exact exc ept the i -th, j -th and k -th, w h er e i < j < k . Then we get a long exact se quenc e · · · → A i, s − i − 1 · → A j, s − j · → A k , s − k +1 · → A i, s − i · → A j, s − j +1 · → A k , s − k +2 · → · · · . (34) ( R e gar ding the indexin g, cf. last p ar agr aph of § 3. ) What if w e hav e four rather than three non-exact row s (again in a we akly b o unded double complex with exact columns)? Assuming for concreteness that our double complex is (3 1), and that all rows but the four sho wn there are exact, w e ﬁnd that (32) collapses to W · Q Q · X · R R · Y · S S · Z · K · K A · L · L B · M · M C · N · ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ · · · · · · · · · · · · . (35) 17 5. Some rows, a nd some columns W e hav e just seen what happ ens when all columns, and all but a ﬁnite n um b er of ro ws of a w eakly b ounded double comple x are exact; the corresp onding results hold, of course, when all rows and all but a ﬁnite n um b er of columns are exact. One can lo ok, mor e generally , at the situation where All but m rows , and all but n columns are exact. (36) In this section w e shall examine the sort of b eha vior that this leads to. F or the ﬁrst result, I g iv e a formal statemen t, Lemma 5.1, and a sk etc h of t he pro of, of whic h the reader can chec k the details, followin g the tec hnique of the preceding section. The pro ofs of the remaining results discussed use t he same ideas. The case m + n ≤ 2 of (36 ) is cov ered b y Corolla ry 3.3 ab o v e. The case m + n = 3 is co v ered (up to row -column rev ersal) by Lemma 4.4 , together with 5.1. Lemma . Supp ose we have a we akly b ounde d double c omplex with obje cts A h,r , a l l r ows exact but the i -th and j - th, wh e r e i < j, and al l c olumns ex a ct but the k - th. Then we have a lon g exact se quenc e · · · → A i,s − i − 1 · → A j,s − j · → A s − k ,k · → A i,s − i · → A j,s − j +1 · → A s − k +1 ,k · → · · · . (37) Sketch of Proof . W rite out the salamander exact sequences corresp onding to the horizontal arrow out of A i,r for eac h r < k , to the vertic al arrow out of A h,k for h = i, . . . , j − 1 , and to the horizon tal ar r o w out o f A j,r for eac h r ≥ k . Except where w e come to a corner, these exact sequences piece together (due to ex- actness of all other rows a nd columns) as in (30). When we do turn a corner, we g et a diﬀeren t sort of piecing together; e.g., if w e tak e the maps A i,k − 1 → A i,k → A i +1 ,k for the arrows A → B → F of (26), then the E and C of (26) are b oth zero, due to w eak b oundedness, so that in ( 2 7), the tw o horizon tal exact sequence s collapse into one 8 -term exact sequence. So the path of arro ws in our double complex describ ed in t he preceding parag raph leads to a single long exact sequence of homology ob jects, donors, and receptors. F or eac h donor or receptor in this sequence, we now use a string of extram ural isomor- phisms ( consequences o f Corollar y 2.1 and the exactness of all but our three exceptional ro ws and columns) to connect it with a receptor or donor at an ob ject of o ne of the o ther t w o non-exact row s or columns. (In eac h case, there is only one direction w e can go b y extram ural isomorphisms from our dono r o r receptor ob ject, without crossing the non- exact row or column we are on, and this indeed leads to an ob j ect of a nother non-exact ro w or column.) W e know from weak b oundedness that the donor and receptor ob jects on the other side of the ro w or column w e hav e arrived at are zero, and so conclude b y Corollary 3.1 tha t the receptor or donor we ha v e reac hed is isomorphic to a v ertical or horizon tal homology ob j ect in that ro w or column. Th us, w e get an exact sequence in whic h all ob jects are v ertical or horizon tal homolog y ob jects. 18 So far, the general case of (36) has giv en results as nice as when m or n is 0 . But no w consider m + n = 4 . W e sa w in (35) what happ ens when n = 0 ; let us compare this with the case of a double complex with three not necessarily exact ro ws and one not necessarily exact column, suc h as the follo wing (where w e ha v e darke ned the arrows in the not necessarily exact ro ws and columns). ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ ✲ · · · · ✲ ✲ ✲ ✲ ✲ ✲ · · · · ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ X A B C D E F G H J K L M N P Q Y (38) One ﬁnds that this double complex leads to a system of four linked “half-long” exact sequence s. T o the left and to the righ t, the diagram lo oks lik e the n = 0 case, (35 ), but there is a p eculiar “splicing” in the middle: ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ A X · B · B C · C · H H · D · C · H · H D · J · J E · K · K F F · L · G G · M · H H · N · M · H · H N · N · P P · Y · Q (39) Here is the same diag ram, redrawn more smo othly . F ∼ = A X · F · B · L · G ∼ = B C · G · C · M · H H · H · H D · N · J · N · P ∼ = J E · P · K · Y · Q ∼ = K ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ❫ ✣ ✣ ❫ ✼ ✇ ✇ ✼ ❘ ✒ ❘ ✒ (40) 19 The exact sequences are those c hains of arrows whic h can b e follo w ed without making sharp turns. W e remark that the ﬁrst step in ve rifying the exactness o f the sequenc es in (39), equiv alen tly , (40) is to che c k that the following isomorphisms follow from Corollaries 2.1 and 3.1. X · ∼ = X ∼ = B , C · ∼ = C , C · ∼ = C , D · ∼ = D ∼ = J , P ∼ = J , Q ∼ = K , E · ∼ = E ∼ = K , L · ∼ = L ∼ = F , F ∼ = A , Y · ∼ = Y ∼ = P , N · ∼ = N , N · ∼ = N , M · ∼ = M ∼ = G , G ∼ = B . (41) Using these, the v eriﬁcation of the exactness conditions is immediate. (Note that the part of (4 0) b et w een “ G ∼ = B ” and “ P ∼ = J ” is, up to lab eling, just a cop y of ( 2 7), with the to p and left no des o f (27) iden tiﬁed, and lik ewise the b ottom and righ t no des, and with substitutions from (4 1 ) made where appropria te. So all the exactness conditions in that part of the dia g ram are immediate.) The in terp olation of some exact rows b et w een the three nonexact rows of (38) do es not aﬀect the resulting system of exact sequences (39), (40) except b y a shift of indices. General v a lues of m and n in (36) yie ld systems that, for most o f their length, consis t of m + n − 2 intert wining exact sequences (cf. (33)), but hav e ﬁnitely man y “splicings”; essen tially , o ne f o r eac h ob ject of the giv en double complex whic h lies a t the inte rsection of a nonexact row and a nonexact column, and do es not hav e, either to its upp er righ t or lo w er left, a region where (due to exactness and weak b oundedness ) all donors and receptors are zero. Th us, t he one splice in (39) and (40) comes fro m the ob ject H of (38). If m = n = 2 , a s in P U Q V A B C D E F H J K L M N S X T Y ❄ ❄ ❄ ❄ ❄ ❄ ❄ · · · · ❄ ❄ ❄ ❄ ❄ ❄ ❄ · · · · ❄ ❄ ❄ ❄ ❄ ❄ ❄ · · · · ❄ ❄ ❄ ❄ ❄ ❄ ❄ · · · · ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ (42) 20 then the tw o ob jects C and L lead to tw o “splicings”: Q Q · V · C C · D · D · L · L E · M · M V · C · C D · D · L L · E · B B · J · C C · K · K · L · L S · X · X J · C · C K · K · L L · S · ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ (43) The same dia g ram in “smo oth” fo r mat (and carried o ne step further at eac h end) is A ∼ = P P · A · U · H · Q ∼ = B B · Q · J · V · C C · C · C D · K · D · K · L L · L · L S · E · X · M · M ∼ = X F · T · N · Y · N ∼ = Y ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ✣ ❫ ❫ ✣ ❘ ✒ ❫ ✣ ✣ ❫ ✣ ❫ ❫ ✣ ✣ ❫ ❫ ✣ ❘ ✒ ❘ ✒ ❘ ✒ ❘ ✒ ✣ ❫ ❘ ✒ ❫ ✣ (44) If exact row s or columns are introduced b etw een the giv en nonexact ones, the splicings mo v e farther apart ( as the “staircases” on whic h C and L lie mov e apart), with a “normal” stretc h b etw een them. 6. T ota l homo logy It is probably fo olhar dy , at ve ry least, for someone who do es not kno w t he theory of sp ectral sequences to attempt to sa y something ab out the total homolog y of a do uble complex. How eve r, I shall no te here some connections b et w een that sub ject and the constructions A a nd A w e hav e b een w orking with. Let us b e giv en a double complex with ob jects A i,r , v ertical arrows δ 1 : A i,r → A i +1 ,r , and horizon tal arrow s δ 2 : A i,r → A i,r +1 , ( i, r ∈ Z ) . (45) In particular, at eac h ob ject A i,r , we ha v e δ 1 δ 2 = δ 2 δ 1 , δ 1 δ 1 = 0 , δ 2 δ 2 = 0 . (46) A t this p oin t, one usually deﬁnes the total c omplex induced by t his double complex to hav e for ob jects the direct sums A n = L i + r = n A i,r , assuming the coun table direct sum cons truction to b e deﬁned and exact in our ab elian category A . But w e ma y as w ell b e mor e general. Let A Z denote the ab elian category of all Z -tuples X = ( X i ) i ∈ Z of ob jects of A , a nd let P : A Z → B b e any exact functor into an ab elian categor y B 21 whic h comm utes with shift, i.e., has the prop erty that for ( X i ) ∈ A Z w e ha v e a functorial isomorphism P i X i ∼ = P i X i +1 . (47) F or instance, supp ose A = B = the category of all R -mo dules for R a ﬁxed ring. (As noted in § 1, “mo dule” can mean either left or r ig h t mo dule.) Then w e migh t tak e P to b e: (i) the op erator of direct sum, or (ii) the op erator of direct pro duct, o r (iii) or (iv) the righ t- or left-truncated pro duct op erators, giv en by P X i = ( Q i< 0 X i ) × ( L i ≥ 0 X i ) , resp ectiv ely , P X i = ( L i< 0 X i ) × ( Q i ≥ 0 X i ) (these migh t be called “f ormal Lauren t sum” op erations; the reader should ch ec k that they indeed satisfy ( 4 7)), or ev en some v ery “un-sum-lik e” constructions, such a s (v) P X i = ( Q X i ) / ( L X i ) , or more generally (vi) the r educed pro duct of the R -modules X i ( i ∈ Z ) with resp ect to any translation-inv arian t ﬁlter on Z . Before sa ying what w e will do with these f unctors, let me digress and p o in t out that in any ab elian category A with countable copro ducts, i.e., countable direct sums, the functor L i ∈ Z : A Z → A satisﬁes ( 4 7) and is right exact (since it is a left adjoin t, and therefore resp ects co equalizers); and, dually , when the countable direct pro duct functor Q i ∈ Z exists, it satisﬁes (47) a nd is left exact. F or A the catego ry of all R -mo dules, it is easy t o c hec k that these constructions are exact on b oth sides; but there are ab elian categories in which coun table pro ducts or copro ducts exist but are no t exact. F or instance, in the category A of torsion ab elian groups, inﬁnite pro ducts are g iv en b y the tor sion subgroup of the direct pro duct as groups [10, Exercise I.8]. It is easy to see that t he direct pro duct in this category of the fa mily of short exact sequenc es 0 → Z / ( p i ) → Z / ( p i +1 ) → Z / ( p ) → 0 , for p a ﬁxed prime a nd i ranging ov er the natural n um b ers, loses exactness on the righ t: no elemen t of t he pro duct o f the middle terms maps to the elemen t (1 , 1 , 1 , . . . ) o f the pro duct of the right-hand terms. (As describ ed, this example is a family of short exac t seque nces indexed by N ; but by asso ciat ing copies of the zero short exact sequence to indices i < 0 , we get a n example of non-rig h t-exactness of pro ducts indexed b y Z . ) Applying P on try agin dualit y [1 4, Theorem 1.7.2] to this example, w e get non-left-exactness of countable c opr o ducts in the catego ry of totally disc onnected compact Hausdorﬀ a b elian gro ups (though it seems harder to describ e the elemen ts in v olv ed). Th us, these t w o functors, and others lik e them, are exclude d as candidates f o r the P w e are considering. Returning to where w e left oﬀ, supp ose w e are giv en an exact functor P : A Z → B satisfying (4 7), and a double complex (45) in A . Then w e deﬁne A n = P i ∈ Z A i,n − i , for eac h n ∈ Z . (48) The families o f maps δ 1 and δ 2 of (45) induce maps whic h w e shall denote by the same sym b o ls, δ 1 , δ 2 : A n → A n +1 , for each n ∈ Z . (49) 22 (Remark: the isomorphism (47) is needed in the deﬁnition of δ 1 , but no t in that of δ 2 ; essen tially b ecause we decided arbitrarily that the i of the op erator P i in (48) w ould index t he ﬁrst subscript of A i,n − i . ) These maps will aga in clearly satisfy (46). Since δ 1 and δ 2 no w represen t maps whic h can simultaneous ly ha v e the same range and the same domain (as in (49)), we can add a nd subtract them, and (46) immediately yields ( δ 1 + δ 2 ) ( δ 1 − δ 2 ) = 0 = ( δ 1 − δ 2 ) ( δ 1 + δ 2 ) . (50) Th us, if for each n we let δ = δ 2 + ( − 1) n δ 1 : A n − → A n +1 , (51) w e get a complex · · · ✲ δ A n − 1 ✲ δ A n ✲ δ A n +1 ✲ δ · · · . (52) W e shall call (52) the total c omplex (with respect to the functor P ) of our giv en double complex. Since the ma ps δ come from maps going do wn w ard and to the rig h t on o ur original double complex, we shall denote the homology ob jects of the ab ov e complex by A n · = Ker( A n δ → A n +1 ) / Im ( A n − 1 δ → A n ) . (53) So far, this is no thing new. W e no w bring o ur donor a nd receptor ob jects in to the picture. Let us deﬁne A n = P i A i,n − i , A n = P i A i,n − i . (54) F rom the exactness and shift-inv ariance of P , it follows that within eac h ob ject A n , sub ob jects suc h as Ker( δ 1 : A n → A n +1 ) and Im( δ 1 : A n − 1 → A n ) will b e giv en by the corresp onding “sums”, P i Ker( δ 1 : A i,n − i → A i +1 ,n − i ) and P i Im( δ 1 : A i − 1 ,n − i → A i,n − i ) , and similarly for more complicated expressions. One deduces that A n = Ker ( δ 1 δ 2 ) / (Im δ 1 + Im δ 2 ) , A n = (Ker δ 1 ∩ Ker δ 2 ) / Im( δ 1 δ 2 ) . (55) (Here, in the n umerators, the sym bo ls δ 1 , δ 2 , δ 1 δ 2 , denote the maps so named hav ing domain A n , and in the denominators, the maps with co domain A n . ) In view of (55), the iden tit y morphism o f A n induces in tramural maps A n − → A n · − → A n . (56) (W e could hav e deﬁned A n · and A n · analogously to ( 5 4), noted characterizations of them analogo us to (55), and got t en a comm uting diagram A n A n · A n · A n · ; A n ✠ ❄ ❘ ❘ ❄ ✠ (57) 23 but w e shall not need these additio na l ob jects.) Finally , the t w o sets o f extr a m ural maps constructed from our original double complex in § 1, combine d with (54), yield, for each n, tw o maps whic h we shall call A n ✲ δ 1 ✲ δ 2 A n +1 . (58) In terms o f the description (55) of A n and A n +1 , we see that δ 1 and δ 2 are induced in B b y δ 1 , δ 2 : A n → A n +1 . Let us now write (a na logous to (51)) , δ = δ 2 + ( − 1) n δ 1 : A n − → A n +1 . (59) W e ﬁnd that the comp osite of this map with the ﬁrst in tram ural map of (56), A n − 1 ✲ δ A n ✲ A n · (60) is zero, since the ﬁrst arrow maps into t he denominator of (53). This sa ys that the tw o comp osites A n − 1 ✲ δ 1 ✲ δ 2 A n ✲ A n · (61) agree up to sign. Hence, b elo w, w e shall just refer to the comp osite inv olving δ 1 . The same commen ts apply to the comp osites A n · ✲ A n ✲ δ 1 ✲ δ 2 A n +1 . (62) W e can now state a v ersion o f the Salamander Lemma for total homology . 6.1. Lemma . In the a b ove c ontext, for e ach n the 6 -term se quenc e of intr amur al and extr amur al m a ps and their c omp osites A n − 1 ✲ δ 1 A n cf. (61) · A n · ✲ A n ✲ A n +1 ✲ A n +1 · ✲ A n +1 δ 1 cf. (62 ) · A n +2 (63) is exact. Proo f . Rather than re- pro ving this, let us deduc e it from t he Sa la mander Lemma (Lemma 1.7) b y a trick . W e deﬁne a double complex ( B i,r ) in B in whic h B i,r = A i + r , the horizontal ma ps are the maps δ , and the v ertical maps are the δ 1 . Th us, letting n = i + r, w e hav e B i − 1 ,r B i,r B i,r +1 B i +1 ,r +1 ✲ δ ✲ δ ✲ δ ❄ δ 1 ❄ δ 1 ❄ δ 1 ❄ δ 1 ❄ δ 1 ❄ δ 1 ❄ δ 1 ❄ δ 1 A n − 1 A n A n +1 A n +2 ✲ δ 2 ∓ δ 1 ✲ δ 2 ± δ 1 ✲ δ 2 ∓ δ 1 (64) 24 F rom ( 4 6) and (51), w e see that δ 1 δ = δ 1 δ 2 = δ 2 δ 1 = δ δ 1 , and that in each ob ject, Im δ 1 + Im δ = Im δ 1 + Im δ 2 and Ker δ 1 ∩ Ker δ = Ker δ 1 ∩ Ker δ 2 . It easily follows that B i,r · = A n · , B i,r = A n , B i,r = A n , and that (63) is just the 6 -term exact sequence asso ciated with the a b o v e horizontal arrow of this double complex. Lik ewise, Corollary 4.2, applied to an y ro w of the complex (64) giv es: 6.2. Lemma . If our original double c omplex has ex a ct c olumns, then one ha s a long exact se quenc e in B : · · · ✲ δ A n ✲ A n · ✲ A n ✲ δ A n +1 ✲ · · · . (65) (Note the curious prop ert y of this seque nce: t hat the ob jects joined b y the connecting morphisms δ are isomo rphic under a diﬀerent map, δ 1 , by Corollary 2.1.) 6.3. Cor ollar y . If our original do uble c omplex has exact ro ws and columns , and is w eakly b ounded ( e.g., if A i,r = 0 whenever i or r is ne gative ) , then al l total homolo gy obje cts A n · ar e zer o. Proo f . In the original double complex ( A i,r ) , the donor and receptor ob jects form iso- morphic c hains going up w ard to the righ t and do wn w ard to the left by Coro llary 2.1 and our exactness assumptions; hence, b y w eak b oundedness, they are all zero. Th us, for all n, A n = P A i,n − i = P 0 = 0 , and similarly A n = 0 . So by (65), A n · = 0 . There is no more to b e said ab out total homology under this h yp othesis, so to ﬁnish this section, let us return to the weak er h yp othesis of Lemma 6.2, and examine t he b eha vior of the exact sequence ( 6 5) for v arious c hoices of A and P . 6.4. Convention . F or the r emainder of this se c tion , we sha l l assume as in L emm a 6.2 that our give n double c omplex ( A i,r ) i,r ∈ Z has exac t c olumns. Consider now the pa ir o f ob jects A n and A n +1 of the exact sequence (65 ). W e ha v e t w o maps, δ 1 and δ 2 , b etw een them, as in ( 5 8), of whic h δ 1 is now an isomorphism b ecause of o ur a ssumption of exact columns (Corollary 2.1 ). Let us think of the o b jects A n and A n +1 as the com binations, under the functor P , o f the donor ob jects, respec- tiv ely the receptor ob jects, in the ch ain o f ob jects and maps in A lying on one of the “staircases” in (32). Flatt ened out, such a staircase lo oks like · · · ✲ δ 2 A i +1 ,n − i ∼ = δ − 1 1 A i,n − i ✲ δ 2 A i,n − i +1 ∼ = δ − 1 1 A i − 1 ,n − i +1 ✲ δ 2 · · · . (66) Here the maps δ 1 , δ 2 : A n → A n +1 b et w een our “total” dono r and receptor ob j ects arise under P from the ab o v e maps of (6 6). If we comp ose eac h arrow δ 2 of (66) with t he preceding (or the f ollo wing) isomorphism δ − 1 1 , we ma y regard (6 6) as a directed system in A . This suggests that we lo ok a t the direct limit Lim − → i →−∞ A i,n − i = Lim − → i →−∞ A i,n − i +1 of that system, if this exists in A . 25 If in fact A has coun table copro ducts (direct sums), then it has general countable colimits (b y [3, Prop osition 7.6.6], and the fa ct that, b eing an ab elian category , it has co equalizers) and so, in par t icular, coun table direct limits. Examining ho w this is pro v ed, w e see that in this situation, the direct limit of (66) can b e constructed as the cokerne l of the map δ 1 − δ 2 : L i A i,n − i − → L i A i,n − i +1 . (67) No w supp o se coun table copro ducts are exact in A , and tak e B = A and P = L . Then (up to a p ossible change of sign in δ 1 , whic h clearly w on’t change the direct limit of (66)) we see that (67) is just δ : A n − → A n +1 . (68) In summary: if P = L , then the cok ernel of the step (68) in the exac t sequen ce (65) is giv en b y the direct limit of (6 6). The kernel of (68 ) do es not in g eneral hav e suc h a natural description for P = L . But if A is the category of R -mo dules ( R any ring), that kerne l will b e zero! Indeed, consider any nonzero x ∈ A n = L i A i,n − i . Let i b e the largest inte ger such that 0 6 = x i ∈ A i,n − i (corresp onding diagra mmatically to the lowe s t p osition where A n has nonzero component on the staircase where it liv es). Examining the A i +1 ,n − i comp onen t of δ ( x ) , w e see that this is δ 1 ( x i ) , whic h is nonzero b ecause δ 1 is an isomorphism. So δ ( x ) is injectiv e, as claimed. Applying these observ a tions to the exact sequenc e (6 5), w e get: 6.5. Cor ollar y . If R is a ring, and ( A i,r ) a double c omplex of R -mo dules with exact c olumns, then its total hom o lo gy with r esp e ct to L is describ e d by A n · ∼ = Cok( δ : A n − 1 → A n ) ∼ = Lim − → i →−∞ A i,n − i − 1 ∼ = Lim − → i →−∞ A i,n − i , (69) wher e the dir e ct limits ar e over the system (66) . Dualizing the observ ation following (68 ) , w e see that if A an ab elian category with coun table direct pro ducts and these are exact, and w e tak e P = Q , then the k ernel of (68) is the inverse limit of (66). The c okernel is no w hard to describe, ev en for A the category of R -modules; but I shall dev elop the description in that case (with n − 1 in place of n, for con v enience) b elow . W e b egin by noting that for general A and P , the exactness of (65) at A n and A n · tells us that Cok( δ : A n − 1 → A n ) ∼ = Im( A n → A n · ) = Ker( A n · → A n ) . (70) (where the arrow s in the last t w o expressions are in tram ural maps). No w let A again b e the category of R -mo dules, let ( A i,r ) still b e a double complex in A with exact columns, and let its total complex, its total ho mo lo gy , and our related 26 ob jects b e deﬁned using the functor P = Q on A . Supp o se an elemen t x ∈ A n · has the prop ert y t ha t for ev ery ﬁnite subset I ⊆ Z , x can b e represen ted b y a cycle in A n whic h has zero comp o nen t in A i,n − i for all i ∈ I . Then I will call x a “p eek ab o o elemen t” (b ecause wherev er you loo k, it isn’t there!) The set of these elemen ts forms a submo dule of A n · , whic h I shall denote PB( A n · ) . I claim that (70), as a submo dule of A n · , is precisely PB( A n · ) . Let us ﬁrst sho w that PB( A n · ) is contained in (70), lo ok ed at as Ker( A n · → A n ) . Supp ose x ∈ PB( A n · ) , and let x b e represen ted by a cycle ( x i ) ∈ Q i A i,n − i . (71) Since x is a p eek abo o elemen t, for an y j ∈ Z we can mo dify ( x i ) by a boundary δ ( y i ) to get a n elemen t havin g j - th co ordinate 0 . But doing t his c hanges x j b y δ 2 ( y j ) ± δ 1 ( y j − 1 ) , so if it can send x j to zero, w e m ust ha v e x j ∈ δ 2 ( A j,n − j − 1 ) + δ 1 ( A j − 1 ,n − j ) , whic h means that x j has zero ima g e in A j,n − j . Since w e hav e prov ed this for arbitrar y j, the image of x in A n = Q A i,n − i is zero, i.e., x ∈ Ker( A n · → A n ) , as claimed. Con v ersely , supp ose x ∈ A n · lies in (70), whic h w e no w lo ok at as Im( A n → A n · ) . This sa ys that x can b e represen ted by a cycle ( x i ) as in (71) suc h that eac h co ordinate x i is annihilated by b oth δ 1 and δ 2 . W e wish to sho w that for an y ﬁnite subset I ⊆ Z , our cycle ( x i ) can b e mo diﬁed by a b oundary so that it b ecomes zero at eac h co ordinate in I . Supp ose inductiv ely that we hav e b een able to ac hiev e zero en tries at all co o rdinates in I − { j } , where j = min ( I ) (corresp onding to the high e st p oint w e are in teres ted in o n our staircase ), in the process preserving the condition that eac h coordinat e is annihilated b y δ 1 and δ 2 . F or notational simplicity , let us again call this mo diﬁed elemen t ( x i ) . In particular, the co ordinate x j is annihilated b y δ 1 ; so b y exactness of the columns of our double complex, w e can write x j = δ 1 ( z j − 1 ) for some z j − 1 ∈ A j − 1 ,n − j . If w e let ( z i ) ∈ A n − 1 b e the elemen t with j − 1-st co ordinate z j − 1 and all other co ordinates 0 , it is easy to v erify tha t ( x i ) + ( − 1) n δ (( z i )) has zero co ordinates at all indices in I . (The t w o co ordinates that diﬀer from those of ( x i ) are the j -th, whic h has b een bro ugh t to zero, and the j − 1-st, whic h w e don’t care ab out b ecause j − 1 / ∈ I . ) T o see that all co ordinates of this ele men t are still annihilated by δ 1 and δ 2 , note that it suﬃces to pro v e tha t δ (( z i )) has this pro p ert y , for whic h, b y ( 4 6) and (51), it will suﬃce to show that δ 2 δ 1 ( z j − 1 ) = 0 . But δ 2 δ 1 ( z j − 1 ) = δ 2 ( x j ) = 0 b y c hoice of ( x i ) . This completes the pro o f that (70) is given b y PB ( A n · ) . Inserting into (65) our earlier description of the k ernel of δ , and this description of its cok ernel, we get 6.6. Cor ollar y . If R is a ring, and ( A i,r ) a double c omplex of R -mo dules with exact c olumns, and we fo rm its total homolo gy obje cts A n · with r esp e ct to Q , then for e ach n we have a sh o rt exact se quenc e 0 − → PB( A n · ) − → A n · − → (Lim ← − i →∞ A i,n − i ∼ = Lim ← − i →∞ A i,n − i +1 ) − → 0 , (72) wher e the inverse limits ar e over the system (66) . 27 F or an example in whic h the term PB( A n · ) of (72) is nonzero, let p b e any prime, and consider the double complex of a b elian gr o ups, ❄ ✲ Z ✲ p ❄ ✲ Z ✲ p ❄ ✲ Z ✲ p ❄ ✲ Z ✲ p ❄ ✲ 0 Z ❄ ✲ ❄ ✲ 0 Z ❄ ✲ ❄ ✲ 0 Z ❄ ✲ ❄ ✲ 0 0 ❄ ✲ ❄ ✲ 0 0 ❄ ✲ ❄ ✲ 0 0 ❄ ✲ , (73) where the arrows lab eled “ p ” represen t m ultiplication by p, and the v ertical equals-signs denote the identit y map. Let n b e the common v alue of the sum of the subsc ripts on the ob jects Z in the lo w er diagonal string of such ob jects in (73), so that A n is the direct pro duct of these ob jects. Then A n +1 = 0 , so all mem bers o f A n are cycles. F or the same reason, the second inv erse limit in (72) is zero; so (72) shows that all elemen ts of A n · are p eek abo o elemen ts. T o establish the existenc e of no nzer o p eek abo o elemen ts, we must therefore sho w tha t A n · 6 = 0 . T o this end, consider the map σ from A n to the group Z p of p -adic inte gers , given b y σ (( x i ) i ∈ Z ) = P ∞ i =0 ( − 1) in p i x i ∈ Z p . (74) (Note the “cut-oﬀ ”: all x i with i < 0 are ignored.) If ( x i ) ∈ A n is a b oundary , ( x i ) = δ (( y i )) , then f or eac h i w e hav e x i = δ 2 ( y i ) + ( − 1) n − 1 δ 1 ( y i − 1 ) = p y i − ( − 1) n y i − 1 . This mak es the computation ( 7 4) of σ (( x i )) a “telescoping sum”, where all terms cancel except − ( − 1) n y − 1 . Th us, for ev ery b ound- ary ( x i ) = δ ( ( y i )) , the elemen t σ (( x i )) ∈ Z p b elongs to Z . On the other hand, w e can clearly c ho ose elemen ts (hence, cycles) ( x i ) ∈ A n for whic h σ (( x i )) is an arbitra ry mem b er of Z p . Th us, there ar e cycles in A n whic h are not b oundaries, hence nonzero p eek ab o o elemen ts in A n · . It is not hard to show , con v ersely , tha t eve ry ( x i ) ∈ A n suc h that σ ( x i ) ∈ Z is a b oundary , and to conclude that A n · ∼ = Z p / Z . (Inciden tally , in (73) we could replace Z by a p olynomial ring k [ x ] , for k a ﬁeld, and the elemen t p b y x. In place of the Z p / Z in t he ﬁnal result w e w ould then get k [[ x ]] /k [ x ] . Regarding this example as a double complex of k -ve ctor-sp ac es, we see that the exis tence of nonzero p eek ab o o elemen ts in A n · do es not r equire any noncompleteness prop erty of the base ring.) W e now come to our last choic e of P on the category A of R -mo dules, the r ig h t truncated pro duct (or “left formal La uren t sum”) functor. Thus w e let A n = ( Q i< 0 A i,n − i ) × ( L i ≥ 0 A i,n − i ) . (75) 28 (This w as example (iii) in the sen tence follo wing (47). I use “rig h t” with reference to the subscript i in P X i = ( Q i< 0 X i ) × ( L i ≥ 0 X i ) , thinking of the indices as written in increasing o rder from left to righ t. Unfort unat ely , since in our double complex the ﬁrst subscript indexes the row , and increases in the do wn w ard direction in our diagrams, “ right truncated” means, for these diagrams, “truncated in the downw a r d left direction along eac h diago na l”.) W e shall ﬁnd that this yields the simplest, i.e., most trivial, c haracterization of the total homology . Note that if in δ = δ 2 ± δ 1 : A n − 1 → A n , we lo ok at the eﬀects of the resp ectiv e op erators δ 1 and δ 2 on the ﬁrst subscript of A i,n − 1 − i , eac h op erator adds to this sub- script a constan t, namely 1 , resp ectiv ely 0 . W e can think of this as sa ying our o p erators are each “homogeneous”; but they are homogeneous of distinct degrees, and the op era t or of higher degree, δ 1 , is in v ertible. It follo ws that δ will b e inv ertible! Indeed, let us write δ = (1 − ε )( ± δ 1 ) , where ε = ± δ 2 δ − 1 1 , noting that ε is homogeneous of degree − 1 . Then w e see tha t the formal inv erse δ − 1 = ± δ − 1 1 (1 + ε + ε 2 + . . . ) (76) con v erges on our righ t t runcated pro duct mo dules (75), a nd thus giv es a g en uine in v erse to δ . Lemma 6 .2 no w immediately gives : 6.7. Cor ollar y . Every double c omplex of R -mo dules with exact c olumns has trivial total homolo gy w ith r esp e ct to the right trunc ate d pr o duct ( left fo rmal L aur ent sum ) functor Q i< 0 × L i ≥ 0 . F or the left truncated pro duct functor L i< 0 × Q i ≥ 0 (the right formal La uren t sum), the k ernel a nd cok ernel of δ seem muc h more diﬃcult to describ e in terms of the directed system (66), and I will not try to do so. (Of course, if, rev ersing Conv en tion 6.4, we ta k e ro ws rather than columns exact in our double comple x of R -mo dules, the b eha viors of left and right truncated pro ducts are rev ersed.) In a diﬀeren t direction, one ﬁnds that for eac h of the ab ov e four constructions P , o n any ab elian category A where it is deﬁned and exact, w e akly b ounde d double complexes with exact columns alwa ys ha v e trivial to tal homology: 6.8. Cor ollar y . L et A b e any ab elian c ate gory w ith c ountable c opr o ducts, r esp e ctively c ountable pr o ducts, r esp e ctively b oth, wh ich ar e exact functors; let ( A i,r ) b e a w e akly b ounde d double c omplex in A with e x act c olumns, and supp ose we form its total c omplex with r esp e ct to the c opr o duct functor, r esp e ctively, the pr o duct functor, r esp e ctively, the right- or left-trunc ate d pr o duct functor. Then in e ach c ase, the in duc e d m a ps δ : A n → A n +1 ( se e (59)) wil l b e isomorphisms , and he n c e the total homolo gy wil l b e zer o. Sketch of Pro of . The w eak b oundedness hypothesis has the eﬀect that for eac h n, the string of ob jects and maps · · · ✲ δ 2 A i +1 ,n − i ✛ δ 1 ∼ = A i,n − i ✲ δ 2 A i,n − i +1 ✛ δ 1 ∼ = A i − 1 ,n − i +1 ✲ δ 2 · · · (77) 29 breaks up in to (generally inﬁnitely man y) ﬁnite substrings, separated b y zero ob jects. Th us, the map δ : A n → A n +1 b ecomes a P -sum of maps fro m ﬁnite direct sums of consecutiv e donor ob jects to ﬁnite direct sums of consecutiv e receptor ob jects. (One v eriﬁes this by examining how eac h of our four functors P b eha v es with resp ect to decom- p ositions of its domain into ﬁnite subfamilies.) O n eac h of these pairs of ﬁnite sums, one v eriﬁes that the restriction of δ is inv ertible, b y a ﬁnite v ersion of the computation ( 7 6). A P -sum of inv ertible maps is inv ertible, completing the pro o f. I ha v e not inv e stigated the b ehavior of to t a l ho mology with resp ect to any other functors P . In particular, I do not kno w of an y examples of exact functors P : A Z → B satisfying (47) for whic h the analog of Corollary 6.8 fails. Cf. Corollary 6.3, whic h say s that any suc h functor do es giv e trivial homology on double complexes hav ing b oth exact ro ws and exact columns. 7. F urther notes 7.1. A formall y simpler appro a ch . A more sophisticated fo rm ulation of the basic ideas of this note (sa y , of Deﬁnition 1.1 through Lemma 1.7, plus Corollary 4.3 and Lemmas 6.1 and 6.2) w ould refer to a single o b ject A of an ab elian catego ry , p ossibly graded, with t w o comm uting (or an ticomm uting) square-zero endomorphisms δ 1 and δ 2 . The situation w e hav e b een studying w ould b e the particular case where the category is the bigraded additiv e catego ry o f double complexes in our A . (Big raded b ecause w e w ould a llo w subscript-shifting as w ell a s subscript-preserving morphisms.) F or instance, in the formulation of Coro lla ry 4 .3, the v ertical exactness assumption w ould simply tak e the form Im( δ 1 ) = Ker( δ 1 ) , and the diagram obtained, (65), w ould r educe to what is called in [6 ] an exact couple: A ✲ A ✢ ❪ A · ; (78) alongside whic h w e w ould ha v e an isomorphism A ∼ = A . But I will not attempt t o dev elop any of the material in this form. 7.2. Triple complexes . A t ev ery o b ject A of an ordinary complex, one has a single homology ob ject, while ab ov e w e hav e asso ciat ed to eac h A in a double complex fo ur homology ob jects, A , A · , A · and A. What w ould b e the a nalogous constructions in a triple complex? T o answ er this, let us examine ho w the four constructions w e asso ciate to a double complex arise. Let us start with a picture , represen ting an ob ject A of our double complex, together with the three other ob jects o f the complex from whic h the double complex structure giv es us p ossibly nontrivial maps in to A, and the three into which it 30 giv es us p ossibly non trivial maps from A (cf. (26)); and let us mark with dots those of these ob jects o ccurring in the deﬁnition of eac h of our homology ob jects: Diagrams • • • • • • • • • • Homology ob j ects A ✠ ❘ A · A · ✠ ❘ A (79) F or each of the ab ov e diagrams, the associated homo lo gy ob jec t is the quotien t of t he in tersection of the ke rnels of the maps from A to the mark ed ob jects in the low er square, b y the sum of the images in A of the marked ob jects in the upp er square. Th e dots in the t w o sq uares are in eac h case lo cated so that the maps fro m the mark ed ob j ects of the upp er square in to the mark ed ob jects in the lo w er square via A are all zero, i.e., so that the image in question is indeed con tained in the k ernel named. On the other hand, the dots in the upp er square are in eac h case as low and as f a r to the rig h t as they can b e without violating this condition, give n the p ositions of the dots in the low er square, and the dots in the low er square are a s high and as far t o the left as they can b e, giv en the p ositions of the dots in the upp er square. In fact, if w e partially order the v ertices of our diagrams by considering the arrow s of our double complex to go from higher to low er elemen ts, and we supplemen t our dots in the low er square with all the dots b elo w them under this ordering, a nd those in the upp er square with a ll the do t s ab ov e them (noting that so doing do es not change the resulting sums of imag es, or intersec tions o f k ernels), • • • • • • • • • • • • • • • • (80) then w e see that the arrays of dot s in the lo w er squares of the ab o v e four diagr a ms are precisely the four prop er nonempt y down-se ts ( sets closed under ≤ ) of that partia lly or- dered set, and the array s of dots in t he upp er squares are (if we momen tarily sup erimp o se the upp er and low er squares) the c omplementary up-sets. Kno wing this, w e can see what the analo g should b e for triple complexes. One ﬁnds that the set of v ertices of a cub e has 18 prop er nonempty dow n-sets, each with its com- plemen tary up-set. Under p erm utation of the three co ordinat es, thes e 18 complemen tary pairs form 8 equiv alence classes. F or simplic ity , w e sho w on the rig h t b elo w only one r ep- resen tativ e of eac h equiv alence class, a nd we again sho w b y dots only maximal elemen ts of eac h do wn-set and minimal elemen ts of eac h up- set, 31 since these corresp ond to the ob jects actually needed to compute our homology ob jects. W e mark each dia g ram × 1 or × 3 , to indicate the size of its orbit under p erm utations of co ordinates. On t he left, w e sho w the full partially ordered system of these ob jects. The lines showin g the order-relations in that partially ordered set induce in tram ural maps among our homology ob jects. • • • • × 1 • • • × 3 • • • • × 3 • • × 3 • • • • • • × 1 • • • • × 3 • • • × 3 • • • • × 1 · · · · · · · · · · · · · · · · · · · (81) The reader familiar with lattice theory will recognize the diagr a m at the left as the free distributiv e lattice on three generators [5, Figure 19, p. 84]. This is b ecause the distributiv e lattice of prop er nonempt y do wn-sets o f the set of v ertices o f our cub e, under union and in tersection, is freely generated by t he three dow n-sets • • • • , • • • • and • • • • . (These fr ee generators are the three “outer” elemen ts at the middle lev el of the diagram on the left ab o v e, corr esp o nding, in the diagram on the righ t, to the picture at that lev el mark ed × 3 . As constructions on o ur triple complex, they represen t the classical ho molo gy constructions corresp onding to the three a xial directions in that complex.) I ha v e not in v estigated what extram ural ma ps and exact sequences relate these 18 constructions. I will not pro p ose iconic notations for them lik e those w e hav e used in studying double complexes ; pro bably the b est notation w ould, rather, in v olv e indexing them b y express ions for t he elemen ts of a free distributiv e lat t ice; e.g., h ( x 1 ∧ x 2 ) ∨ x 3 ( A ij k ) or h (( x ∧ y ) ∨ z ; A ij k ) , where x 1 , x 2 , x 3 , or x, y , z , denote the free generators. F or informal purp oses, t hough, something like h ( • • , A ij k ) migh t o ccasionally b e conv en ien t (as long a s w e don’t go b ey ond triple complexes). Are these ob jects likely to b e of use? I don’t kno w! 32 7.3. Kernel and image ra tio s . J. Lambek [8] (cf. [6, Lemma I I I.3.1]) asso ciates to an y comm uting square, P ✲ a ❘ f R ❄ b ❄ c Q ✲ d S where ca = f = db (82) t w o ob jects, whic h he calls the ke rn e l r atio and the image r atio of t he square. T o br ing out the similarity to t he concepts of this pap er, let me name them P ∗ = Ker f / (Ker a + Ker b ) , ∗ S = (Im c ∩ Im d ) / Im f . (83) In fa ct, if the giv en square is em b edded in a double complex whic h is ve rtically and horizon tally exact at P , resp ectiv ely at S, then w e see that P ∗ = P , respective ly , ∗ S = S . No w supp ose w e ha v e a comm uting diagram with exact r ows, P ✲ R ✲ T ❄ ❄ c ❄ Q ✲ S ✲ U . (84) Then w e can extend this diagram, b y putting in the kerne l and cok erne l of c, to a double complex exact in b oth directions at R and S. Hence Coro llary 2.1, applied to c, giv es us R ∗ ∼ = R ∼ = S ∼ = ∗ S. (85) The isomorphism R ∗ ∼ = ∗ S is prov ed b y Lam b ek [8 ] and used to get other results, as I use Corollary 2 .1 in § 2 ab o v e. The constructions ( ) ∗ and ∗ ( ) ha v e the adv an tage of b eing deﬁnable with reference to a smaller diagram than m y ( ) and ( ) . They share with these constructions the prop erty of v a nishing on an y do ubly ex a ct double complex with ﬁnite suppor t . But one do esn’t seem to b e able to do anything with them without some exactness assumptions. With suc h assumptions, one gets extram ural iso mor phisms as in Corollary 2.1, but without them, one do es not ha v e analo gs of the extramu ral homo mo r phisms o f Lemma 1.7. 7.4. Non-abelian gr oups . Lam b ek prov es the results referred to ab o v e for not nec- essarily ab elian gro ups, through he applies them in ab elian situations. Note that for non-ab elian groups, the exactness of the top ro w o f (84), or something similar, is needed to conclude that ∗ S w ill b e a group, i.e., that the denominator in the deﬁnition will b e a n ormal subgroup of the n umerator. Without that condition, ∗ S is a “homogeneous space”. I lik ewise noticed when ﬁrst w orking out the Salamander Lemma that a v ersion 33 could b e stated for not-necessarily-ab elian g r o ups, but there w ere ev en worse diﬃculties – e.g., in D eﬁnition 1.1, A w ould just b e a p ointed set, the quotien t of the g roup Ker q b y the left action of the subgroup Im c and the right action of the subgroup Im d. Ho w- ev er, it w ould certainly b e nice to ha v e a to ol lik e the Salamander Lemma fo r proving noncomm utativ e v ersions of basic diagram-c hasing lemmas, when these hold. Leic h t [11], Kopy lov [7], and others ha v e give n mo r e general conditions on a category under whic h Lam b ek’s result holds. In [9], Lam b ek gets related results for v arieties of algebras in the sense of univ ersal algebra satisfying an a ppropriate Mal’cev-type condition. 8. Two exer cises I ha v e left man y k ey calculations in this note to the r eader, including the v eriﬁcation of the Salama nder Lemma (Lemma 1.7) itself. My “ske tc hes of pro ofs” can like wise b e regarded “exercises with hin ts”. Here are t w o further interes ting exercises. 8.1. Building up finite exact double complexes . Let A b e an ab elian catego r y , and let A # denote the ab elian category of double complexes in A , where a morphism f : A → B is a family of morphisms f i,r : A i,r → B i,r comm uting with δ 1 and δ 2 . (W e are not admitting subscript-shifting morphisms here.) Let FX ⊆ Ob( A # ) (standing for “ﬁnite exact”) b e the class of double complexes with only ﬁnitely many nonzero ob jects, and all ro ws a nd columns exact, and EX ⊆ FX (“elemen tary exact”) b e the class of double complexes of the fo rm 0 0 0 A A 0 0 A A 0 0 0 ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ (86) placed at a r bitrary lo cat io ns in the grid, where the maps among all copies of A are the iden tit y . (a) Show that FX is the least sub class o f Ob( A # ) containing EX and closed therein under extensions. Hin t: Giv en an ob ject of F X , prov e that o ne can map it epimorphically to a n ob ject 34 of EX as suggested b elow . 0 C D 0 A B ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ 0 0 0 D D 0 0 D D 0 0 0 ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ . (87) T o get epimorphicit y at A, sho w tha t D = 0 . (b) In con trast, sho w by example that for a double complex whic h is not assumed to b e ﬁnite, but nonetheless has exact ro ws and columns, and has an ob ject D with zero ob jects immediately b elow it and to its right, the map of (87) ma y fail to b e an epimorphism. I exp ect that an epimorphicit y result analog o us to (87) should hold for ﬁnite exact triple (and higher) complexes. If so, o ne could get t he analog of (a) ab ov e, and deduce from this t ha t the 18 constructions of § 7.2 (corresp onding to the diagrams of (81 )) all giv e zero at ob j ects of a ﬁnite triple complex exact in all three directions. 8.2. Complexes with a twist . (a) Suppo se w e a re give n a commu tative diagram 0 0 0 0 0 0 0 0 0 0 0 0 ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ✲ A B C L M N E P Q R U V W F X Y Z H J K (88) in whic h the columns b eginning and ending with 0 are short exact sequenc es, while the three “r ows” (the tw o ordinary ones, and the one that mak es a detour from the top to the b ott o m) are merely assumed to b e complexes. 35 Obtain a long exact sequence of (mostly horizontal) homology ob jects · · · → V · → C · → N · → W · → E · → E · → F · → F · → P · → X · → H · → Q · → · · · . (89) (Suggestion: Square oﬀ the curv ed arrows in (8 8) b y inserting the ke rnel D and the cok ernel G of the a rro w E → F , and a dd zero es ab ov e, resp ectiv ely , b elo w these, getting an ordinary double complex with all columns exact. Apply to this the idea used in deriving Lemma 4.4.) (b) Consider an y double complex (90), in whic h we ha v e lab eled some o f the ob jects to the upp er left and low er right of one arr o w, F → G : · · · M · · · · · K L · D E F G H J · B C · · · · · A · · · ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ . (90) Obtain fro m this a diagram 0 0 0 0 0 0 0 0 ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ D ✲ E ✲ A ⊕ B ⊕ D ✲ C ⊕ E ✲ F ✲ K ✲ L ⊕ M ✲ ✲ A ⊕ B ✲ C ✲ G ✲ H ⊕ K ✲ J ⊕ L ⊕ M ✲ H ✲ J ✲ (91) taking the arrow s to b e the sums of “all av ailable” morphisms. (E.g., the arrow E → C ⊕ E is just the inclusion, but the arro w C ⊕ E → F is the sum of the give n arro ws C → F 36 and E → F . ) V erify that, a f ter a bit of sign-tw eakin g, (91) satisﬁes the h yp otheses of part (a ) . W rite out, for this dia gram, the middle six terms o f the exact sequence (89), then one mo r e on either side. Th us we see that w e can extend the 6 -term “ sala ma nders” of Lemma 1.7 to longer exact sequences , if we are willing to deﬁne more complicated auxiliary ob j ects. (c) Do these new constructions still hav e the prop erty of b eing zero on b ounde d e x act double complexes? 9. Ac knowledgemen ts, and a ﬁnal remar k My paren ts, L ester and Sylvia Bergman, w ork ed tog ether in scien tiﬁc photogr a ph y and illustration; and t ho ugh they nev er instructed me in the latter, I acquired from them m y lo v e o f an eﬀectiv e visual displa y . (But the resp onsibility for the of t en inconsisten t T E X co ding underlying the diagrams of this note is m y own.) Most of this work w as done, and a rough draft written, in the F all of 1972, when I w as supp orted b y an Alfred P . Sloan Researc h F ello wship, and was a guest at the Univ ersit y of Leeds’ stim ulating Ring Theory Y ear. In 2007 An ton G erasc henk o digitized, and, with m y p ermission, put online, a cop y of that draft. His en th usiasm for the material help ed spur me to create this b etter v ersion. I am also indebted t o the referee for sev eral though tful suggestions, though shortage of time has prev en ted me from f ollo wing many of them. I ha v e not attempted to mak e this not e a “deﬁnitiv e” dev elopmen t of the sub ject: that should b e left to t ho se who actually work with double complexes, and can judge ho w b est to dev e lop the material. Hence the v ariation, in the presen t note, b et w een sections whe re ob jects are denoted b y arbitrary letters and those where they ar e distinguishe d b y double subscripts (as seemed to giv e the most r eada ble presen tation of one o r a nother topic), the lac k of an y general notation for intram ural and extram ural maps, and the c hoice of v ertical arr o ws that p oint down w a rd (as in most familiar lemmas pro v ed b y diag ram c hasing) rather than up w ard (as migh t b e preferable based on systematic considerations). References [1] Hyman Bass, Algebr aic K-the ory , W. A. Benjamin, 1968. MR 40 #2736. [2] G eorge M. Berg ma n, A n o te on ab elian c ate gories – tr an slating elem ent-chasing pr o ofs, and ex a ct emb e d d ing in ab elian gr oups, unpublished, 7 pp., 1974, readable at http://math.be rkeley.edu/ ~ gbergman/pa pers/unpub/ . [3] G eorge M. Bergman, A n Invitation to Gener al Algebr a and Universal Constructions, 422 pp. Readable at http://math.ber keley.edu/ ~ gbergman/24 5 . MR 99h :18001. (Up dated ev ery few ye ars. The MR review is o f the 1998 v ersion.) [4] T emple H. F ay , Keith A. Har die, and P eter J. Hilton, The two- squar e lemma, Publicacions Matem` atiques 33 (1989) 13 3–137. 37 http://www. raco.cat/index. php/ PublicacionsMatematiques/article/view/3757 6 . MR 90h :18011. [5] G . G r¨ atzer, L attic e The ory: F oundation, Birkh¨ auser V erlag, Basel, 2011. ISBN:978 - 3-0348- 0017-4. MR 27685 81. [6] P . J. Hilton and U. Stamm bac h, A Course in Homolo gic al Algebr a, 2nd ed., Gra duate T exts in Mathematics, v. 4, Springer, 1997 . MR 97k :180 0 1. [7] Y aroslav Kop ylo v, On the L amb ek invarian ts of c ommutative squar es in a quasi- ab elian c ate gory , Sci. Ser. A Math. Sci. (N.S.) 11 (200 5) 57-67. MR 2007d :180 07. [8] Joa chim Lam b ek, Goursat’s the or em and homolo gic a l algebr a, Canad. Math. Bull. 7 (1964) 597–6 0 8. MR 30 #4813a. [9] J. Lambek, The butterﬂy a nd the serp ent, in L o gic and Algebr a ( Pontignano, 199 4 ), Lecture Notes in Pure and Applied Mathematics, 180, Dekk er, 1996. MR 97k :08006 . [10] Serge Lang, Algebr a, Addison-W esley , third edition, 19 9 3, reprin ted as Graduate T exts in Mathematics, v.211, Springer, 2002. MR 2003e :0000 3 . [11] J. B. Leic h t, Axiomatic pr o of of J. L amb ek’s homo lo gic al the or em, Canad. Math. Bull. 7 (1964) 6 09–613. MR 30 #4813b. [12] Sa unders Mac Lane, Homolo gy, Grundlehren der mathematisc hen Wissensc haften, Bd. 114, Academic Press a nd Springer, 1963. MR 28 #122 . [13] Sa unders Mac Lane, Cate gories for the Working Mathematician, Gradua t e T exts in Mathematics, v.5, Springer, 197 1. MR 50 #7275. [14] W alter Rudin, F ourier Analysis on Gr oups, In terscie nce T racts in Pure and Applied Mathematics, No. 12 , In terscience Publishers, 1 9 62. MR 27 #2808. Dep artment of Mathematics, University of California Berkeley, CA 9 4 720-3840, USA Email: gbergman@math.berk eley.edu

On diagram-chasing in double complexes

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