Spectral Filtrations via Generalized Morphisms

SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS MOHAMED BARAKA T Abstra t. This pap er in tro dues a reform ulation of the lassial on v ergene theorem for sp etral sequenes of ltered omplexes whi h pro vides an algorithm to ee tively ompute the indued ltration on the total (o)homology , as so on as the omplex is of nite t yp e, its ltration is nite, and the underlying ring is omputable. So-alled gener alize d maps pla y a deisiv e role in simplifying and streamlining all in v olv ed algorithms. Contents 1. In tro dution 1 2. A generalit y on sub ob jet latties 5 3. Long exat sequenes as sp etral sequenes 6 4. Generalized maps 12 5. Sp etral sequenes of ltered omplexes 20 6. Sp etral sequenes of biomplexes 23 7. The Car t an-Eilenber g resolution of a omplex 29 8. Gr othendiek 's sp etral sequenes 30 9. Appliations 31 9.1. The double- Ext sp etral sequene and the ltration of T or 32 9.2. The T or - Ext sp etral sequene and the ltration of Ext 35 App endix A. The triangulation algorithm 37 App endix B. Examples with GAP 's homalg 39 Referenes 55 1. Intr odution The motiv ation b ehind this w ork w as the need for algorithms to expliitly onstrut sev eral natural ltrations of mo dules. It is already kno wn that all these ltrations an b e desrib ed in a unied w a y using sp etral sequenes of ltered omplexes, whi h in turn suggests a unied algorithm to onstrut all of them. Desribing this algorithm is the main ob jetiv e of the presen t pap er. Sine Verdier it b eame more and more apparen t that one should b e studying om- plexes of mo dules rather than single mo dules. A single mo dule is then represen ted b y one of its resolutions, all quasi-isomorphi to ea h other. The idea is no w v ery simple: 1 2 MOHAMED BARAKA T If there is no diret w a y to onstrut a ertain natural ltration on a mo dule M , it migh t b e simpler to expliitly realize M as one of the (o)homologies H n ( C ) of some omplex C with some easy onstrutible (natural) ltration, su h that the ltration indued on H n ( C ) (b y the one on C ) maps b y the expliit isomorphism H n ( C ) ∼ = M on to the lo ok ed-for ltration on M . In this w ork it will b e sho wn ho w to ompute the indued ltration on H n ( C ) using sp etral sequenes of ltered omplexes, enri hed with some extra data. This pro vides a unied approa h for onstruting n umerous imp ortan t ltrations of mo dules and shea v es of mo dules (f. [ W ei94 , Chap. 5℄ and [ Rot79 , Chap. 11℄). Sine w e are in terested in ee- tiv e omputations w e restrit ourself for simpliit y to nite typ e omplexes arrying nite ltrations. When talking ab out D -mo dules the ring D is assumed asso iativ e with one. Denition 1.1 (Filtered mo dule) . Let M b e a D -mo dule. (a) A  hain of submo dules ( F p M ) p ∈ Z of the mo dule M is alled an asending ltra- tion if F p − 1 M ≤ F p M . The p -th graded part is the subfator mo dule dened b y gr p M := F p M /F p − 1 M . (d) A  hain of submo dules ( F p M ) p ∈ Z of the mo dule M is alled a desending ltra- tion if F p M ≥ F p +1 M . The p -th graded part is the subfator mo dule dened b y gr p M := F p M /F p +1 M . All ltrations of mo dules will b e assumed exhaustiv e (i.e. S p F p M = M ), Hausdor (i.e. T p F p M = 0 ), and will ha v e nite length m (i.e. the dierene b et w een the highest and the lo w est stable index is at most m ). Su h ltrations are alled m -step ltrations. W e start with t w o examples that will b e pursued in Setion 9 : (d) Let M and N b e righ t D -mo dules and M ∗ := Hom D ( M , D ) the dual (left) D - mo dule of M . The map ϕ :  N ⊗ D M ∗ → Hom D ( M , N ) n ⊗ α 7→ ( m 7→ nα ( m )) is in general neither injetiv e nor surjetiv e. In fat, im ϕ is the last (graded) part of a d esending ltration of Hom( M , N ) . • Hom( M , N ) • • • N ⊗ M ∗ / / •        cok er ϕ • ϕ / / coim ϕ  •  im ϕ • k er ϕ  (a) Dually , let M b e a left mo dule, L a righ t mo dule, and ε : M → M ∗∗ := Hom(Hom( M , D ) , D ) SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 3 the ev aluation map . The omp osition ψ L ⊗ D M ψ 2 2 id ⊗ ε / / L ⊗ M ∗∗ ϕ / / Hom D ( M ∗ , L ) is in general neither injetiv e nor surjetiv e. It will turn out that its oimage coim ψ is the last graded part of an a sending ltration of L ⊗ M . • Hom( M ∗ , L ) • L ⊗ M / / •  cok er ψ • ψ / / coim ψ  •  im ψ • k er ψ      • • Example (a) has a geometri in terpretation. (a') Let D b e a omm utativ e Noether ian ring with 1 . Reall that the Kr ull di- mension dim D is dened to b e the length d of a maximal  hain of prime ideals D > p 0 > · · · > p d . F or example, the Kr ull dimension of a eld k is zero, dim Z = 1 , and dim D [ x 1 , . . . , x n ] = dim D + n . The denition of the Kr ull dimension is then extended to non trivial D -mo dules using dim M := dim D Ann D ( M ) . Dene the o dimension of a non trivial mo dule M as co dim M := dim D − dim M and set the o dimension of the zero mo dule to b e ∞ . If for example D is a (om- m utativ e) prinipal ideal domain whi h is not a eld, then the nitely generated D -mo dules of o dimension 1 are preisely the nitely generated torsion mo dules. Denition 1.2 (Purit y ltration) . Let D b e a omm utativ e Noether ian ring with 1 and M a D -mo dule. Dene the submo dule t − c M as the biggest submo dule of M of o dimension ≥ c . The as ending ltration · · · ≤ t − ( c +1) M ≤ t − c M ≤ · · · ≤ t − 1 M ≤ t 0 M := M is alled the purit y ltration of M [ HL97 , Def. 1.1.4℄. The graded part M c := t − c / t − ( c +1) is pure of o dimension c , i.e. an y non trivial submo dule of M c has o dimension c . t − 1 M is nothing but the torsion submo dule t( M ) . This suggests alling t − c M the c -th (higher) torsion submo dule of M . Early referenes to the purit y ltration are J.-E. R oos 's pioneering pap er [ Ro o62 ℄ where he in tro dued the bidualizing omplex , M. Kashiw ara 's master thesis (Deem b er 1970) [ Kas95 , Theorem 3.2.5℄ on algebrai D -mo dules, and J.-E. Björk's 4 MOHAMED BARAKA T standard referene [ Bjö79 , Chap. 2, Thm. 4.15℄. All these referenes address the onstrution of this ltration from a homologial 1 p oin t of view, where the assump- tion of omm utativit y of the ring D an b e dropp ed. Under some mild onditions on the not neessarily omm utativ e ring D one an  haraterize the purit y ltration in the follo wing w a y: There exist so-alled higher ev aluation maps ε c , generalizing the standard ev aluation map, su h that the sequene 0 − → t − ( c +1) M − → t − c M ε c − → Ex t c D (Ext c D ( M , D ) , D ) is exat (f. [ AB69 , Qua01 ℄). ε c an th us b e view ed as a natural transforma- tion b et w een the c -th torsion funtor t − c and the c -th bidualizing funtor Ext c (Ext c ( − , D ) , D ) . In Subsetion 9.1.3 it will b e sho wn ho w to use sp etral se- quenes of ltered omplexes to onstrut all the higher ev aluation maps ε c . More generally it is eviden t that sp etral sequenes are natural birthplaes for man y natural transformations. No w to see the onnetion to the previous example (a) set L = D as a righ t D -mo dule. ψ then b eomes the ev aluation map ε . There still exists a misunderstanding onerning sp etral sequenes of ltered omplexes and it migh t b e appropriate to address it here. Let C b e a ltered omplex (f. Def. 3.1 and Remark 4.6 ). (*) W e ev en assume C of nite typ e and the ltration nite . The ltration on C indu es a ltration on its (o)homologies H n ( C ) . It is sometimes b eliev ed that the sp etral sequene E r pq asso iated to the ltered omplex C annot b e used to determine the indued ltration on H n ( C ) , but an only b e used to determine its graded parts gr p H n ( C ) . One migh t b e easily led to this onlusion sine the last page of the sp etral sequene onsists of preisely these graded parts E ∞ pq = gr p H p + q ( C ) , and omputing the last page is traditionally regarded as the last step in determining the sp etral sequene. It is lear that ev en the kno wledge of the total (o)homology H n ( C ) as a whole (along with the kno wledge of the graded parts gr p H n ( C ) ) is in general not enough to determine the ltration. Another reason migh t b e the use of the phrase omputing a sp etral sequene. V ery often this means a suessful attempt to gure out the morphisms on some of the pages of the sp etral sequene, or ev en b etter, w orking skillfully around determining most or ev en all of these morphisms and nev ertheless deduing enough or ev en all information ab out of the last page E ∞ . This often mak es use of ingen uous argumen ts only v alid in the example or family of examples under onsideration. F or this reason w e add the w ord eetiv e to the ab o v e phrase, and b y eetiv ely omputing the sp etral sequene w e mean expliitly determining al l morphisms on al l pages of the sp etral sequene. Indeed, the denition one nds in standard textb o oks lik e [ W ei94 , Setion 5.4℄ of the sp etral sequene asso iated to a omplex of nite typ e arrying a nite ltration is  onstrutive in the sense that it an b e implemen ted on a omputer (see [ Bar09 ℄). The message of this w ork is the follo wing: 1 Kashiw ara did not use sp etral sequenes: Instead of using sp etral sequenes, Sato devised [...℄ a metho d using asso iated ohomology, [ Kas95 , Setion 3.2℄. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 5 If the sp etral sequene of a ltered omplex is eetiv ely omputable, then, with some extra w ork, the indued ltration on the total (o)homology is eetiv ely omputable as w ell. By denition, the ob jets E r pq of the sp etral sequene asso iated to the ltered omplex C are subfators of the total ob jet C p + q (see Setions 3 and 5 ). In Setion 4 w e in tro due the notion of a generalized em b edding to k eep tra k of this information. The en tral idea of this w ork is to use the generalized em b eddings E ∞ pq → C p + q to lter the total (o)homology H p + q ( C )  also a subfator of C p + q . This is the on ten t of Theorem 5.1 . Eetiv ely omputing the indued ltration is not a main stream appliation of sp etral sequenes. V ery often, esp eially in top ology , the total ltered omplex is not ompletely kno wn, or is of innite t yp e, although the (total) (o)homology is kno wn to b e of nite t yp e. But from some page on, the ob jets of the sp etral sequene b eome intrinsi and of nite typ e . Pushing the sp etral sequene to on v ergene and determining the isomorphism t yp e of the lo w degree total (o)homologies is already highly non trivial. The reader is referred to [ RS02 ℄ and the impressiv e program Kenzo [ RSS ℄. In its urren t stage, Kenzo is able to ompute A ∞ -strutures on ohomology . The goal here is nev ertheless of dieren t nature, namely to eetiv ely ompute the indued ltration on the a priori known (o)homology . The shap e of the sp etral sequene starting from the intrinsi page will also b e used to dene new n umerial in v arian ts of mo dules and shea v es of mo dules (f. Subsetion 9.1.5 ). The approa h fa v ored here mak es extensiv e use of generalized maps , a onept moti- v ated in Setion 3 , in tro dued in Setion 4 , and put in to ation starting from Setion 5 . Generalized maps an b e view ed as a data strutur e that allo ws r e or ganizing man y algorithms in homologial algebra as lose d formulas . Although the whole theoretial on ten t of this w ork an b e done o v er an abstrat ab elian ategory , it is sometimes on v enien t to b e able to refer to elemen ts. The disussion in [ Har77 , p. 203℄ explains wh y this an b e assumed without loss of generalit y . 2. A generality on subobjet la tties The follo wing situation will b e rep eatedly enoun tered in the sequel. Let C b e an ob jet in an ab elian ategory , Z , B , and A sub ob jets with B ≤ Z . Then the sub ob jet lattie 2 of C is at most a degeneration of the one in Figure 1 . This lattie mak es no statemen t ab out the size of B or Z ompared to A , sine, in general, neither B nor Z is in a ≤ -relation with A . The seond 3 isomorphism theorem an b e applied ten times within this lattie, t w o for ea h of the v e parallelograms. The sub ob jet A leads to the in termediate sub ob jet A ′ := ( A + B ) ∩ Z sitting b et w een B and Z , whi h in general neither oinides with Z nor with B . Hene, a 2 -step ltration 0 ≤ A ≤ C leads to a 2 -step ltration 0 ≤ A ′ /B ≤ Z /B . 2 I learned dra wing these pitures from Prof. Jo a him Neubüser . He made in tensiv e use of subgroup latties in his ourses on nite group theory to visualize argumen ts and ev en mak e pro ofs. 3 Here w e follo w the n um b ering in Emmy Noether 's fundamen tal pap er [ No e27 ℄. 6 MOHAMED BARAKA T PSfrag replaemen ts C A B Z A ′ Figure 1. A general lattie with sub ob jets B ≤ Z and A Arguing in terms of sub ob jet latties is a manifestation of the isomorphism theorems, all b eing immediate orollaries of the homomorphism theorem (f. [ No e27 ℄). 3. Long exa t sequenes as spetral sequenes Long exat sequenes are in a preise sense a preursor of sp etral sequenes of ltered omplexes. They ha v e the adv an tage of b eing a lot easier to omprehend. The ore idea around whi h this w ork is built an already b e illustrated using long exat sequenes, whi h is the aim of this setion. Long exat sequenes often o ur as the sequene onneting the homologies · · · ← H n − 1 ( A ) ∂ ∗ ← − H n ( R ) ν ∗ ← − H n ( C ) ι ∗ ← − H n ( A ) ∂ ∗ ← − H n +1 ( R ) ← − · · · of a short exat sequene of omplexes 0 ← − R ν ← − C ι ← − A ← − 0 . If one views ( A, ∂ A ) as a sub omplex of ( C , ∂ ) , then ( R, ∂ R ) an b e iden tied with the quotien t omplex C / A . Moreo v er ∂ A is then ∂ | A and ∂ R is b oundary op erator indued b y ∂ on the quotien t R . The natural maps ∂ ∗ app earing in the long exat sequene are the so-alled onneting homomorphisms and are, lik e ∂ A and ∂ R , indued b y the b oundary op erator ∂ of the total omplex C . T o see in whi h sense a long exat sequene is a sp eial ase of a sp etral sequene of a ltered omplex w e rst reall the denition of a ltered omplex. Denition 3.1 (Filtered omplex) . W e distinguish b et w een  hain and o  hain omplexes: (a) A  hain of sub omplexes ( F p C ) p ∈ Z (i.e. ∂ ( F p C n ) ≤ F p C n − 1 for all n ) of the  hain omplex ( C • , ∂ ) is alled an asending ltration if F p − 1 C ≤ F p C . The p -th graded part is the subfator  hain omplex dened b y gr p C := F p C /F p − 1 C . (d) A  hain of sub omplexes ( F p C n ) p ∈ Z (i.e. ∂ ( F p C n ) ≤ F p C n +1 for all n ) of the  o  hain omplex ( C • , ∂ ) is alled a desending ltration if F p C ≥ F p +1 C . The p - th graded part is the subfator o  hain omplex dened b y gr p C := F p C /F p +1 C . Lik e for mo dules all ltrations of omplexes will b e exhaustiv e (i.e. S p F p C = C ), Haus- dor (i.e. T p F p C = 0 ), and will ha v e nite length m (i.e. the dierene b et w een the SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 7 highest and the lo w est stable index is at most m ). Su h ltrations are alled m -step ltrations in the sequel. Con v en tion: F or the purp ose of this w ork ltrations on  hain omplexes are automatially asending whereas on  o  hain omplexes desending. Remark 3.2. Before on tin uing with the previous disussion it is imp ortan t to note that (a) The ltration ( F p C n ) of C n indu es an asending ltration on the homology H n ( C ) . Its p -th graded part is denoted b y gr p H n ( C ) . (d) The ltration ( F p C n ) of C n indu es a desending ltration on the ohomology H n ( C ) . Its p -th graded parts is denoted b y gr p H n ( C ) . More preisely , F p H n ( C ) is the image of the morphism H n ( F p C ) → H n ( C ) . A short exat sequene of (o) hain omplexes 0 ← − R ν ← − C ι ← − A ← − 0 an b e view ed as a 2 -step ltration 0 ≤ A ≤ C of the omplex C with graded parts A and R . F ollo wing the ab o v e on v en tion the ltration is asending or desending dep ending on whether C is a  hain or o  hain omplex. The main idea b ehind long exat sequenes is to relate the homologies of the total  hain omplex C with the homologies of its graded parts A and R . This preisely is also the idea b ehind sp etral sequenes of ltered omplexes but generalized to m -step ltrations, where m ma y no w b e larger than 2 . Roughly sp eaking, the sp etral sequene of a ltered omplex measures ho w far the graded part gr p H n ( C ) of the ltered n -th homology H n ( C ) of the total ltered omplex C is a w a y from simply b eing the homology H n (gr p C ) of the p -th graded part of C . This w ould for example happ en if the ltration F p C is indued b y its o wn grading 4 , i.e. F p C = L p ′ ≤ p gr ′ p C , sine then the homologies of C will simply b e the diret sum of the homologies of the graded parts gr p C . In general, gr p H n ( C ) will only b e a subfator of H n (gr p C ) . Long exat sequenes do not ha v e a diret generalization to m -step ltrations, m > 2 . The language of sp etral sequenes oers in this resp et a b etter alternativ e. In order to mak e the transition to the language of sp etral sequenes notie that the graded parts cok er( ι ∗ ) and k er( ν ∗ ) of the ltered total homology H n ( C ) indiated in the diagram b elo w (1) H n − 1 ( A ) H n ( R ) ∂ ∗ o o H n ( C ) ν ∗ o o H n ( A ) ι ∗ o o H n +1 ( R ) ∂ ∗ o o • • • o o • • o o ∂ ∗ • o o • • o o ν ∗ ι ∗ } cok er( ι ∗ ) k er( ν ∗ )  • o o • • o o • o o • • o o ∂ ∗ • 4 In the on text of long exat sequenes this w ould mean that the short exat sequene of omplexes 0 ← − Q ν ← − C ι ← − T ← − 0 splits. 8 MOHAMED BARAKA T b oth ha v e an alternativ e desription in terms of the onneting homomorphisms: (2) cok er( ι ∗ ) ∼ = k er ( ∂ ∗ ) and k er( ν ∗ ) ∼ = cok er ( ∂ ∗ ) . These natural isomorphisms are nothing but the statemen t of the homomorphism theorem applied to ι ∗ and ν ∗ . Belo w w e will giv e the denition of a sp etral sequene and in Setion 5 w e will reall ho w to asso iate a sp etral sequene to a ltered omplex. But b efore doing so let us desrib e in simple w ords the rough piture, v alid for general sp etral sequenes (ev en for those not asso iated to a ltered omplex). A sp etral sequene an b e view ed as a b o ok with sev eral pages E a , E a +1 , E a +2 , . . . starting at some in teger a . Ea h page on tains a double arra y E r pq of ob jets, arranged in an arra y of omplexes. The pattern of arranging the ob jets in su h an arra y of omplexes dep ends only on the in teger a and is xed b y a ommon on v en tion one and for all. The ob jets on page r + 1 are the homologies of the omplexes on page r . It follo ws that the ob jet E r pq on page r are subfators of the ob jets E t pq on al l the previous pages t < r . No w w e turn to the morphisms of the omplexes. F rom what w e ha v e just b een sa ying w e kno w that at least the soure and the target of a morphism on page r + 1 are ompletely determined b y page r . This an b e regarded as a sort of restrition on the morphism, and indeed, in the ase when zero is the only morphism from the giv en soure to the giv en target, the morphism then b eomes uniquely determined. This happ ens for example whenev er either the soure or the target v anishes, but ma y happ en of ourse in other situations ( Hom Z ( Z / 2 Z , Z / 3 Z ) = 0 ). So no w it is natural to ask whether page r or an y of its previous pages imp ose further restritions on the morphisms on page r + 1 , apart from determining their soures and targets. The answ er is, in general, no. This will b eome lear as so on as w e onstrut the sp etral sequene asso iated to a 2 -step ltered omplex b elo w (or more generally for an m -step ltration in Setion 5 ) and understand the nature of data on ea h page. Summing up: T aking homology only determines the ob jets of the omplexes on page r + 1 , but not their morphisms. Cho osing these morphisms not only ompletes the ( r + 1) -st page, but again determines the ob jets on the ( r + 2) -nd page. Iterating this pro ess nally denes a sp etral sequene. T ypially , in appliations of sp etral sequenes there exists a natural  hoie of the mor- phisms on the suessiv e pages. This is illustrated in the follo wing example, where w e asso iate a sp etral sequene to a 2 -ltered omplex. But rst w e reall the denition of a sp etral sequene. Denition 3.3 (Homologial sp etral sequene) . A homologial sp etral sequene (starting at r 0 ) in an ab elian ategory A onsists of (1) Ob jets E r pq ∈ A , for p, q , r ∈ Z and r ≥ r 0 ∈ Z ; arranged as a sequene (indexed b y r ) of latties (indexed b y p, q ); (2) Morphisms ∂ r pq : E r pq → E r p − r,q + r − 1 with ∂ r ∂ r = 0 , i.e. the sequenes of slop e − r +1 r in E r form a  hain omplex; SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 9 (3) Isomorphisms b et w een E r +1 pq and the homology k er ∂ r pq / im ∂ r p + r,q − r +1 of E r at the sp ot ( p, q ) . E r is alled the r -th sheet (or page , or term ) of the sp etral sequene. Note that E r +1 pq is b y denition (isomorphi to) a subfator of E r pq . p is alled the ltration degree and q the omplemen tary degree . The sum n = p + q is alled the total degree . A morphism with soure of total degree n , i.e. on the n -th diagonal, has target of degree n − 1 , i.e. on the ( n − 1) -st diagonal. So the total degree is de r e ase d b y one. q E 2 02 E 2 12 E 2 22 E 2 01 E 2 11 E 2 21 ∂ h h Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q E 2 00 E 2 10 E 2 20 ∂ h h Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q O O / / p Figure 2. E 2 Denition 3.4 (Cohomologial sp etral sequene) . A ohomologial sp etral seq- uene (starting at r 0 ) in an ab elian ategory A onsists of (1) Ob jets E pq r ∈ A , for p, q , r ∈ Z and r ≥ r 0 ∈ Z ; arranged as a sequene (indexed b y r ) of latties (indexes b y p, q ); (2) Morphisms d pq r : E pq r → E p + r,q − r +1 r with d r d r = 0 , i.e. the sequenes of slop e − r +1 r in E r form a o  hain omplex; (3) Isomorphisms b et w een E pq r +1 and the ohomology of E r at the sp ot ( p, q ) . E r is alled the r -th sheet of the sp etral sequene. Here the total degree n = p + q is inr e ase d b y one. Reeting a ohomologial sp etral sequene at the origin ( p, q ) = (0 , 0) , for example, denes a homologial one E r pq = E − p, − q r , and vie v ersa. F or more details and terminology ( b oundedness , on v ergene , b er terms , base terms , edge homomorphisms , ollapsing , E ∞ term , regularit y ) see [ W ei94 , Setion 5.2℄. 10 MOHAMED BARAKA T P art of the data w e ha v e in the on text of long exat sequenes an b e put together to onstrut a sp etral sequene with three pages E 0 , E 1 , and E 2 : E 0 pq : A n R n +1 A n − 1 R n A n − 2 R n − 1 add the arro ws / / /o /o /o E 0 pq : A n ∂ A   R n +1 ∂ R   A n − 1 ∂ A   R n ∂ R   A n − 2 R n − 1 homology tak e   ? ? ? ? E 1 pq : H n ( A ) H n +1 ( R ) H n − 1 ( A ) H n ( R ) H n − 2 ( A ) H n − 1 ( R ) add the arro ws / / /o /o /o E 1 pq : H n ( A ) H n +1 ( R ) ∂ ∗ o o H n − 1 ( A ) H n ( R ) ∂ ∗ o o H n − 2 ( A ) H n − 1 ( R ) ∂ ∗ o o homology tak e   ? ? ? ? E 2 pq : cok er ( ∂ ∗ ) k er ( ∂ ∗ ) cok er ( ∂ ∗ ) k er ( ∂ ∗ ) cok er ( ∂ ∗ ) k er ( ∂ ∗ ) no arro ws to add / / /o /o /o E 2 pq : cok er ( ∂ ∗ ) k er ( ∂ ∗ ) cok er ( ∂ ∗ ) k er ( ∂ ∗ ) cok er ( ∂ ∗ ) k er ( ∂ ∗ ) with p, q ∈ Z , n = p + q . T aking the t w o olumns o v er p = 0 and p = 1 , for example, is equiv alen t to setting F − 1 C := 0 , F 0 C := A , and F 1 C := C . Sev eral remarks are in order. First note that all the arro ws in the ab o v e sp etral sequene are indued b y ∂ , the b oundary op erator of the total omplex C . Sine ∂ resp ets the ltration, i.e. ∂ ( F p C ) ≤ F p C , the indued map ¯ ∂ : F p C → C /F p C v anishes. So resp eting the ltration means that ∂ annot arry things up in the ltration. But sine ∂ do es not neessarily resp et the grading indued b y the ltration it ma y v ery w ell arry things do wn one or more lev els. No w w e an in terpret the pages: E 0 onsists of the graded parts gr p C with b oundary op erators ∂ A and ∂ Q  hopping o all what ∂ arries do wn in the ltration. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 11 E 1 desrib es what ∂ arries do wn exatly one lev el. This in terpretation of the onneting homomorphisms ∂ ∗ puts them on the same oneptual lev el as ∂ A and ∂ Q . Finally , E 2 desrib es what ∂ arries exatly t w o lev els do wn, but sine a 2 -step ltration has t w o lev els it should no w b e lear wh y E 2 do es not ha v e arro ws. Seond, as w e ha v e seen in ( 2 ) using the homomorphism theorem, the ob jets of the last page E 2 an b e naturally iden tied with the graded parts gr p H n ( C ) of the ltered total homology H n ( C ) . And sine the ob jets on ea h page are subfators of the ob jets on the previous pages one an view the ab o v e sp etral sequene as a pro ess suessiv ely appro ximating the graded parts gr p H n ( C ) of the ltered total homology H n ( C ) : ( A n , R n ) ❀ ( H n ( A ) , H n ( R ) ) ❀ ( cok er ( ∂ ∗ ) , k er ( ∂ ∗ )) . The appro ximation is a hiev ed b y suessiv ely taking deep er in ter-lev el in teration in to aoun t. Finally one an ask if the sp etral sequene ab o v e aptured all the information in the long exat sequene. The answ er is no . The long exat sequene additionally on tains the short exat sequene (3) 0 ← − k er ( ∂ ∗ ) ν ∗ ← − H n ( C ) ι ∗ ← − cok er ( ∂ ∗ ) ← − 0 , expliitly desribing the total homology H n ( C ) as an extension of its graded parts cok er ( ∂ ∗ ) and k er ( ∂ ∗ ) . Lo oking to what happ ens inside the sub ob jet lattie of C n during the appro ximation pro ess will help understanding ho w to remedy this defet. PSfrag replaemen ts C n Z n ( R ) B n ( R ) A n Z n ( A ) B n ( A ) Z n ( C ) B n ( C ) H n ( C ) H n ( C ) ∼ = k er ( ∂ ∗ ) ∼ = cok er ( ∂ ∗ ) Figure 3. The 2 -step ltration 0 ≤ A ≤ C and the indued 2 -step ltration on H ∗ ( C ) Figure 3 sho ws the n -th ob jet C n in the  hain omplex together with the sub ob jets that dene the dieren t homologies: H n ( R ) := Z n ( R ) / B n ( R ) , H n ( A ) := Z n ( A ) / B n ( A ) , 12 MOHAMED BARAKA T PSfrag replaemen ts C n E 0 1 ,n − 1 = R n E 0 0 ,n = A n H n ( C ) Figure 4. E 0 ❀ PSfrag replaemen ts C n A n E 1 1 ,n − 1 = H n ( R ) E 1 0 ,n = H n ( A ) H n ( C ) Figure 5. E 1 ❀ PSfrag replaemen ts C n A n E 2 1 ,n − 1 = k er ( ∂ ∗ ) E 2 0 ,n = cok er ( ∂ ∗ ) H n ( C ) Figure 6. E 2 = E ∞ The appro ximation pro ess of the graded parts of H n ( C ) and H n ( C ) := Z n ( C ) / B n ( C ) . Here w e replaed Z n ( R ) and B n ( R ) b y their full preimages in C n under the anonial epimorphism C n ν − → R n := C n / A n . Figures 4 - 6 sho w ho w the graded parts of H n ( C ) get suessiv ely appro ximated b y the ob jets in the sp etral sequene E r pq , naturally iden tied with ertain subfators of C n for n = p + q . Figure 6 pro v es that the seond isomorphism theorem pro vides  anoni-  al isomorphisms b et w een the graded parts of the total homology H n ( C ) and the ob jets E ∞ 1 ,n − 1 = E 2 1 ,n − 1 and E ∞ 0 ,n = E 2 0 ,n of the stable sheet. And mo dulo these natural isomor- phisms Figure 6 further suggests that kno wing ho w to iden tify E ∞ 1 ,n − 1 and E ∞ 0 ,n with the indiated subfators of C n will sue to expliitly onstrut the extension ( 3 ) in the form (4) 0 ← − E ∞ 1 ,n − 1 ← − H n ( C ) ← − E ∞ 0 ,n ← − 0 . But sine w e annot use maps to iden tify ob jets with subfators of other ob jets w e are lead to in tro due the notion of generalized maps in the next Setion. Roughly sp eaking, this notion enables us to in terpret the pairs of horizon tal arro ws in Figure 7 as generalized em b eddings . 4. Generalized maps A morphism b et w een t w o ob jets (mo dules, omplexes, . . . ) indues a map b et w een their lattie of sub ob jets, and the homomorphism theorem implies that this map giv es rise to a bijetiv e orresp ondene b et w een the sub ob jets of the target lying in the image and those sub ob jets of the soure on taining the k ernel. This motiv ates the visualization in Figure 8 of a morphism T ϕ ← − S with soure S and target T . The homomorphism theorem states that the morphism ϕ , indiated b y the horizon tal pair of arro ws in Figure 8 , maps SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 13 PSfrag replaemen ts C n A n E ∞ 1 ,n − 1 E ∞ 0 ,n H n ( C ) Figure 7. The generalized em b eddings S/ k er( ϕ ) on to the sub obje t im( ϕ ) in a struture-preserving w a y . In this sense, the exat ladder of morphisms in ( 1 ) visualizes part of the long exat homology sequene. PSfrag replaemen ts T S ϕ im ϕ k er ϕ Figure 8. The homomorphism theorem The simplest motiv ation for the notion of a generalized morphism T ψ ← − S is the desire to giv e sense to the piture in Figure 9 mapping a quotien t of S on to a subfator of T . Denition 4.1 (Generalized morphism) . Let S and T b e t w o ob jets in an ab elian ategory (of mo dules o v er some ring). A generalized morphism ψ with soure S and target T is a pair of morphisms ( ¯ ψ , ı ) , where ı is a morphism from some third ob jet F to T and ¯ ψ is a morphism from S to cok er ı = T / im ( ı ) . W e all ¯ ψ the morphism asso iated to ψ and ı the morphism aid of ψ and denote it b y Aid ψ . F urther w e all L := im ı ≤ T the morphism aid sub ob jet . T w o generalized morphisms ( ¯ ψ , ı ) and ( ¯ ϕ,  ) with ( im ı = im  and) ¯ ψ = ¯ ϕ will b e iden tied. Philosophially sp eaking, this denition frees one from the onserv ativ e standp oin t of viewing ψ as morphism to the quotien t T / im ı . Instead it allo ws one to view ψ as a 14 MOHAMED BARAKA T PSfrag replaemen ts T S L ψ im ψ Im ψ k er ψ Figure 9. A generalized morphism morphism to the full ob jet T b y diretly inorp orating ı in the v ery denition of ψ . The in tuition b ehind the notion morphism aid (resp. morphism aid sub ob jet) is that ı (resp. L = im ı ) aids ψ to b eome a (w ell-dened) morphism. Figure 10 visualizes the generalized morphism ψ as a pair ( ¯ ψ , ı ) . PSfrag replaemen ts F T S T / im ı ¯ ψ π ı π ı ı im ψ im ¯ ψ L = im ı k er ¯ ψ π − 1 ı (im ¯ ψ ) =: Im ψ Figure 10. The morphism aid ı and the asso iated morphism ¯ ψ Note that replaing ı b y a morphism with the same image do es not alter the generalized morphism. W e will therefore often write ( ¯ ψ , L ) for the generalized morphism ( ¯ ψ , ı ) , where ı is an y morphism with im ı = L ≤ T . The most natural  hoie w ould b e the em b edding ı : L → T . Figure 9 visualizes the generalized morphism ψ as a pair ( ¯ ψ , L ) . It also reets the idea b ehind the denition more than the expanded Figure 10 do es. If L = im ı v anishes, then ψ is nothing but the (ordinary) morphism ¯ ψ . Con v ersely , an y morphism an b e view ed as a generalized morphism with trivial morphism aid sub ob jet L = 0 . Denition 4.2 (T erminology for generalized morphisms) . Let ψ = ( ¯ ψ , ı ) : S → T b e a generalized morphism. Dene the k ernel k er( ψ ) := k er ¯ ψ , the k ernel of the asso iated SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 15 map. If π ı denotes the natural epimorphism T → T / im ı , then dene the om bined image Im ψ to b e the submo dule π − 1 ı (im ¯ ψ ) of T . In general it diers from the image im ψ whi h is dened as the subfator Im ψ / im ı of T (f. Figure 10 ). W e all ψ a generalized monomorphism (resp. generalized epimorphism , generalized isomorphism ) if the asso iated map ¯ ψ is a monomorphism (resp. epimorphism, isomorphism). Sometimes w e use the terminology generalized map instead of generalized morphism and generalized em b edding instead of generalized monomorphism, esp eially when the ab elian ategory is a ategory of mo dules (or omplexes of mo dules, et.). As a rst appliation of the notion of generalized em b eddings w e state the follo wing denition, whi h is en tral for this w ork. Denition 4.3 (Filtration system) . Let I = ( p 0 , . . . , p m − 1 ) b e a nite in terv al in Z , i.e. p i +1 = p i + 1 . A nite sequene of generalized em b eddings ψ p = ( ¯ ψ p , L p ) , p ∈ I with ommon target M is alled an asending m -ltration system of M if (1) ψ p 0 is an ordinary monomorphism, i.e. L p 0 v anishes; (2) L p = Im ψ p − 1 , for p = p 1 , . . . , p m − 1 ; (3) ψ p m − 1 is a generalized isomorphism, i.e. Im ψ p m − 1 = M . PSfrag replaemen ts ψ p 0 ψ p 1 ψ p m − 2 ψ p m − 1 L p 1 L p 2 L p m − 2 L p m − 1 M Figure 11. An asending m -ltration system A nite sequene of generalized em b eddings ψ p = ( ¯ ψ p , L p ) , p ∈ I with ommon target M is alled a desending m -ltration system of M if (1) ψ p 0 is a generalized isomorphism, i.e. Im ψ p 0 = M ; (2) L p = Im ψ p +1 , for p = p 0 , . . . , p m − 2 ; (3) ψ p m − 1 is an ordinary monomorphism, i.e. L p m − 1 v anishes. W e sa y ( ψ p ) omputes a giv en ltration ( F p M ) if Im ψ p = F p M for all p . 16 MOHAMED BARAKA T No w w e ome to the denition of the basi op erations for generalized morphisms. T w o generalized maps ψ = ( ¯ ψ , ı ) and ϕ = ( ¯ ϕ,  ) are summable only if im ı = im  and w e set ψ ± ϕ := ( ¯ ψ ± ¯ ϕ, ı ) . The follo wing notational on v en tion will pro v e useful: It will often happ en that one w an ts to alter a generalized morphism ψ = ( ¯ ψ , L ψ ) with target T b y replaing L ψ with a larger sub ob jet L , i.e. a sub ob jet L ≤ T on taining L ψ . W e will sloppily write e ψ = ( ¯ ψ , L ) , where ¯ ψ no w stands for the omp osition of ¯ ψ with the natural epimorphism T /L ψ → T /L . W e will sa y that ψ w as oarsened to e ψ to refer to the passage from ψ = ( ¯ ψ , L ψ ) to e ψ = ( ¯ ψ , L ) with L ψ ≤ L ≤ T . As Figure 12 sho ws, oarsening ψ migh t v ery w ell enlarge its om bined image Im ψ . The w ord oarse refers to the fat that the image im e ψ is naturally isomorphi to a quotient of im ψ , and Figure 12 sho ws that this natural isomorphism is giv en b y the seond isomorphism theorem. W e sa y that the oarsening e ψ = ( ¯ ψ , L ) of ψ = ( ¯ ψ , L ψ ) is eetiv e , if Im ψ ∩ L = L ψ . Figure 12 sho ws that in this ase the images im ψ and im e ψ are naturally isomorphi . PSfrag replaemen ts T S S L L ψ ψ ψ e ψ Im ψ Im e ψ im ψ im e ψ k er ψ k er e ψ Figure 12. Coarsening the generalized map ψ = ( ¯ ψ , K ) to e ψ = ( ¯ ψ , L ) F or the omp osition ψ ◦ ϕ of S ϕ ϕ − → T ϕ = S ψ ψ − → T ψ follo w the lled area in Figure 13 from left to righ t. F ormally , rst oarsen ϕ = ( ¯ ϕ,  ) → e ϕ = ( ¯ ϕ, K ) , where K := im  + k er ψ ≤ T ϕ . Then oarsen ψ = ( ¯ ψ , ı ) → e ψ = ( ¯ ψ , L ) , where L := π − 1 ı (im( ¯ ψ ◦  )) = π − 1 ı ( ¯ ψ ( K )) ≤ T ψ and π ı as ab o v e. No w set ψ ◦ ϕ := ( ¯ ψ ◦ ¯ ϕ, L ) . SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 17 PSfrag replaemen ts S ϕ T ϕ = S ψ T ψ T ψ ϕ ϕ k er ϕ k er ψ ◦ ϕ = k er e ϕ Im ϕ im ϕ im  K ψ ψ ψ k er ψ Im ψ im ψ ◦ ϕ Im ψ ◦ ϕ L := π − 1 ı (im( ¯ ψ ◦  )) im ı Figure 13. The omp osition ψ ◦ ϕ Note that k er ψ ◦ ϕ = k er e ϕ . Finally w e dene the division β − 1 ◦ γ of t w o generalized maps S γ γ − → T β ← − S β under the onditions of the next denition. Denition 4.4 (The lifting ondition) . Let γ = ( ¯ γ , L γ ) and β = ( ¯ β , L β ) b e t w o generalized morphisms with the same target N . M ′ γ ! ! C C C C C C C C N ′ β / / N . Consider the ommon oarsening of the generalized maps β and γ , i.e. the generalized maps e β := ( ¯ β , L ) and e γ := ( ¯ γ , L ) , where L = L γ + L β ≤ N . W e sa y β lifts γ (or divides γ ) if the follo wing t w o onditions are satised: (im) The om bined image of e β on tains the om bined image of e γ : Im e γ ≤ Im e β . (e ) The oarsening γ → e γ is eetiv e, i.e. Im γ ∩ L = L γ . W e will refer to e γ as the eetiv e oarsening of γ with resp et to β . The follo wing lemma justies this denition. Both the denition and the lemma are visualized in Fig- ure 14 . T o state the lemma one last notion is needed: Dene t w o generalized morphisms ψ = ( ¯ ψ , L ψ ) and ϕ = ( ¯ ϕ, L ϕ ) to b e equal up to eetiv e ommon oarsening or 18 MOHAMED BARAKA T quasi-equal if their ommon oarsenings e ψ := ( ψ , L ) and e ϕ := ( ϕ, L ) oinide and are b oth eetiv e. W e write ψ , ϕ . PSfrag replaemen ts N M ′ N ′ L Im α im α L α L β L γ Im e β Im e γ β β β k er β Im β γ k er γ Im γ e ψ I m ψ I m e ψ im ψ im e ψ k e r ψ k e r e ψ Figure 14. The lifting ondition and the lifting lemma Lemma 4.5 (The lifting lemma) . L et γ = ( ¯ γ , L γ ) and β = ( ¯ β , L β ) b e two gener alize d morphisms with the same tar get N . Supp ose that β lifts γ . Then ther e exists a gener alize d morphism α : M ′ → N ′ with β ◦ α , γ , M ′ γ ! ! C C C C C C C C α   N ′ β / / N . i.e. β ◦ α is e qual to γ up to ee tive  ommon  o arsening. α is  al le d a lift of γ along β . F urther let e γ := ( ¯ γ , L e γ ) b e the ee tive  o arsening of γ with r esp e t to β , i.e. L e γ = L = L γ + L β . Then ther e exists a unique lift α = ( ¯ α, L α ) satisfying (a) Im α = ¯ β − 1 (Im e γ ) and (b) L α = ¯ β − 1 ( L e γ ) . This α is  al le d the lift of γ along β , or the quotient of γ by β and is denote d by β − 1 ◦ γ or by γ /β . Pr o of. The sub ob jet lattie(s) in Figure 14 desrib es the most general setup imp osed b y onditions (im) and (e ), in the sense that all other sub ob jet latties of ongurations satisfying these t w o onditions are at most degenerations of the one in Figure 14 . No w to onstrut the unique α simply follo w the lled area from righ t to left.  The reader ma y ha v e already notied that the  hoie of the sym b ol , for quasi-equalit y w as motiv ated b y Figure 14 , with L at the tip of the p yramid. The pro of mak es it lear that the lifting lemma is y et another inarnation of the homomorphism theorem. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 19 Remark 4.6 (Eetiv e omputabilit y) . Note that the lift α = ( ¯ α, L α ) sees from N ′ only its subfator N ′ /L α . Replaing N ′ b y its subfator N ′ /L α turns β in to a generalized em- b edding, whi h w e again denote b y β . No w γ and this β ha v e eetiv e ommon oarsenings e γ = ( ¯ γ , L ) and e β = ( ¯ β , L ) , whi h see from N only N/ L , where L = L γ + L β . And mo d- ulo L the generalized morphism e γ b eomes a morphism and the generalized em b edding e β b eomes an (ordinary) em b edding. So from the p oin t of view of eetiv e omputations the setup an b e redued to the follo wing situation: γ : M ′ → N is a morphism and β : N ′ → N is a monomorphism . When M ′ , N ′ , and N are nitely presen ted mo dules o v er a omputable ring (f. Def. A.1 ) it w as sho wn in [ BR08 , Subsetion 3.1.1℄ that in this ase the unique morphism α : M ′ → N is eetiv ely omputable. With the notion of a generalized em b edding at our disp osal w e an nally giv e the horizon tal arro ws in Figure 7 a meaning. No w onsider the three generalized em b eddings ι : H n ( C ) → C n , ι 0 : E ∞ 0 ,n → C n , and ι 1 : E ∞ 1 ,n − 1 → C n in Figure 15 . ι p is alled the total em b edding of E ∞ p,n − p . PSfrag replaemen ts C n A n E ∞ 1 ,n − 1 E ∞ 0 ,n H n ( C ) H n ( C ) ι ι ι 0 ι 1 Figure 15. ι lifts ι 0 and ι 1 Corollary 4.7. The gener alize d emb e dding ι in Figur e 15 lifts b oth total emb e ddings ι 0 and ι 1 . Thus the two lifts ǫ 0 := ι 0 /ι and ǫ 1 := ι 1 /ι ar e gener alize d emb e ddings that form a ltr ation system of H n ( C ) , visualize d in Figur e 16 . Mor e pr e isely, ǫ 0 is an (or dinary) emb e dding and ǫ 1 is a gener alize d isomorphism. Pr o of. There are t w o ob vious degenerations of the sub ob jet lattie(s) in Figure 14 , b oth leading to a sublattie of the lattie in Figure 15 , one for the pair ( β , γ ) = ( ι, ι 0 ) and the other for ( β , γ ) = ( ι, ι 1 ) . In other w ords: F ollo wing the t w o lled areas from righ t to left onstruts ǫ 0 := ι − 1 ◦ ι 0 and ǫ 1 := ι − 1 ◦ ι 1 .  20 MOHAMED BARAKA T PSfrag replaemen ts E ∞ 1 ,n − 1 E ∞ 0 ,n H n ( C ) ǫ 0 = ι 0 /ι ǫ 1 = ι 1 /ι Figure 16. The ltration of H n ( C ) giv en b y the 2 -ltration system ǫ 0 , ǫ 1 Corollary 4.8 (Generalized in v erse) . L et ψ : S → T b e a gener alize d epimorphism. Then ther e exists a unique gener alize d epimorphism ψ − 1 : T → S , suh that ψ − 1 ◦ ψ = (id S , ker ψ ) and ψ ◦ ψ − 1 = (id T , Aid ψ ) . ψ − 1 is  al le d the gener alize d inverse of ψ . In p artiular, if ψ is an (or dinary) epimorphism, then ψ − 1 is a gener alize d isomorphism, and vi e versa. Pr o of. Sine ψ lifts id T dene ψ − 1 := id T /ψ .  Rephrasing short exat sequenes (also alled 1 -extensions) in terms of 2 -ltration sys- tems is no w an easy appliation of this orollary . In partiular, the information in the short exat sequene ( 4 ) is fully aptured b y the 2 -ltration system in Figure 16 . This is last step of remedying the defet men tioned while in tro duing the short exat sequene ( 3 ) in Setion 3 . 5. Spetral sequenes of fil tered omplexes Ev erything substan tial already happ ened in Setions 3 and 4 . Here w e only sho w ho w the ideas already dev elop ed for 2 -ltrations and their 2 -step sp etral sequenes easily generalize to m -ltrations and their m -step sp etral sequenes. W e start b y realling the onstrution of the sp etral sequene asso iated to a ltered omplex . The exp osition till Theorem 5.1 losely follo ws [ W ei94 , Setion 5.4℄. W e also remain lo y al to our use of sub ob jet latties as they are able to sum up a onsiderable amoun t of relations in one piture. Consider a  hain omplex C with (an asending) ltration F p C . The omplemen tary degree q and the total degree n are dropp ed for b etter readabilit y . Dene the natural pro jetion F p C → F p C /F p − 1 C =: E 0 p . It is elemen tary to  he k that the sub ob jets of r -appro ximate yles A r p := k er( F p C → F p C /F p − r C ) = { c ∈ F p C | ∂ c ∈ F p − r C } satisfy the relations of Figure 17 , with Z r p := A r p + F p − 1 C , B r p := ∂ A r − 1 p +( r − 1) + F p − 1 C , and E r p := Z r p /B r p . These denitions deviate a bit from those in [ W ei94 , Setion 5.4℄. Here Z r p and B r p sit b et w een F p C and F p − 1 C . His Z r p and B r p are the pro jetions under η p on to E 0 p := F p C /F p − 1 C of the ones here, and hene sit in the ob jets of the 0 -th sheet E 0 p . The sub ob jet lattie in Figure 17 should b y no w b e onsidered an old friend as it is ubiquitous throughout all our argumen ts. Setting Z ∞ p := ∩ ∞ r =0 Z r p and B ∞ p := ∪ ∞ r =0 B r p ompletes the to w er of sub ob jets F p − 1 C = B 0 p ≤ B 1 p ≤ · · · ≤ B r p ≤ · · · ≤ B ∞ p ≤ Z ∞ p ≤ · · · ≤ Z r p ≤ · · · ≤ Z 1 p ≤ Z 0 p = F p C SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 21 PSfrag replaemen ts F p C F p − 1 C E r p Z r p B r p A r p A r − 1 p − 1 ∂ A r − 1 p +( r − 1) ∂ A r p − 1+( r ) Figure 17. The fundamen tal sub ob jet lattie b et w een F p − 1 C and F p C . F rom Figure 17 it is immediate that E r p := Z r p B r p ∼ = A r p ∂ A r − 1 p +( r − 1) + A r − 1 p − 1 . It is no w routine to v erify that the total b oundary op erator ∂ indues morphisms ∂ r p : E r p → E r p − r . And as men tioned in Setion 3 these morphisms derease the ltration degree b y r . They omplete the denition of the r -th sheet. F rom the p oin t of view of eetiv e omputations the ab o v e denition of ∂ r p is  onstrutive , as long as all in v olv ed ob jets are of nite typ e . In fat, it an easily b e turned in to an algorithm using generalized maps. But sine the ltered omplexes relev an t to our appliations are total omplexes of biomplexes, the desription of this algorithm is deferred to Setion 6 , where the biomplex struture will b e exploited. T o see that ( E r ) indeed denes a sp etral sequene it remains to sho w the taking homol- ogy in E r repro dues the ob jets of E r +1 up to (natural) isomorphisms. F or this purp ose one uses the statemen ts eno ded in Figure 17 to dedue that (a) Z r p /Z r +1 p ∼ = B r +1 p − r /B r p − r , (b) k er ∂ r p ∼ = Z r +1 p /B r p , () im ∂ r p + r ∼ = B r +1 p /B r p , and nally (d) E r +1 p ∼ = k er ∂ r p / im ∂ r p + r . () follo ws from (a) and (b) sine they state that ∂ r p deomp oses as E r p := Z r p /B r p (b) − → Z r p /Z r +1 p (a) − → B r +1 p − r /B r p − r ֒ → Z r p − r /B r p − r =: E r p − r , 22 MOHAMED BARAKA T sho wing that im ∂ r p ∼ = B r +1 p /B r p . No w replae p b y p + r . (d) is the rst isomorphism theorem applied to E r +1 p := Z r +1 p /B r +1 p using (b) and (). F or (a) and (b) see [ W ei94 , Lemma 5.4.7 and the subsequen t disussion℄. Before stating the main theorem w e mak e some remarks ab out on v ergene. Reall that all our ltrations are assumed nite of length m . This means that E m runs out of arro ws and th us stabilizes, i.e. E m = E m +1 = · · · . W e already sa w this for m = 2 in Setion 3 . As ustomary , the stable sheet is denoted b y E ∞ . The stable form of Figure 17 is Figure 18 , where A ∞ p := ∪ ∞ r =0 A r p and A ∞ p + ∞ := ∪ ∞ r =0 A r p + r . PSfrag replaemen ts F p C F p − 1 C E ∞ p ι p Z ∞ p B ∞ p A ∞ p A ∞ p − 1 ∂ A ∞ p + ∞ ∂ A ∞ p − 1+ ∞ Figure 18. The stable fundamen tal sub ob jet lattie The iden tities (5) A ∞ p = k er ∂ | F p C = { c ∈ F p C | ∂ c = 0 } and (6) ∂ A ∞ p + ∞ = im ∂ | F p C = ∂ C ∩ F p C are diret onsequenes of the resp etiv e denitions. Theorem 5.1 (Bey ond E ∞ ) . L et C b e a hain  omplex with an as ending m -step ltr ation. The gener alize d emb e dding ι : H ( C ) → C divides al l gener alize d emb e ddings ι p : E ∞ p → C ,  al le d the total emb e dding of E ∞ p . The quotients ǫ p := ι p /ι form an m -ltr ation system whih  omputes the indu e d ltr ation on H ( C ) . Pr o of. W e only need to v erify the t w o lifting onditions for the pairs ( ι, ι p ) . Ev erything else is immediate. F or the morphism aid sub ob jets of ι p and ι w e ha v e L ι p = ∂ A ∞ p + ∞ + F p − 1 C (see Figure 18 ) and L ι = ∂ C. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 23 Dene L := L ι p + L ι = ( ∂ A ∞ p + ∞ + F p − 1 C ) + ∂ C = ∂ C + F p − 1 C . Condition (im) : Sine Im ι p = A ∞ p + F p − 1 C and Im ι = k er ∂ w e obtain Im e ι p ≤ Im e ι ⇐ ⇒ ( A ∞ p + F p − 1 C ) + L ≤ k er ∂ + L ⇐ ⇒ A ∞ p + ∂ C + F p − 1 C ≤ k er ∂ + F p − 1 C . No w ∂ C ≤ k er ∂ sine ∂ is a b oundary op erator, and A ∞ p ≤ k er ∂ b y ( 5 ). Condition (e ) : Im ι p ∩ L = ( ∂ C + F p − 1 C ) ∩ ( A ∞ p + F p − 1 C ) ( 5 ) = ( ∂ C ∩ F p C ) + F p − 1 C ( 6 ) = ∂ A ∞ p + ∞ + F p − 1 C = L ι p . The lifting lemma 4.5 is no w appliable, yielding the generalized em b eddings ǫ p := ι p /ι .  Corollary 4.7 is the sp eial ase m = 2 . In ligh t of Remark 4.6 the theorem th us states that the indued ltration on the total (o)homology is eetiv ely omputable, as long as the generalized em b eddings ι and ι p are eetiv ely omputable for all p . Hene, it an b e view ed as a (more) onstrutiv e v ersion of the lassial on v ergene theorem of sp etral sequenes of ltered omplexes, a v ersion that mak es use of generalized em b eddings: Theorem 5.2 (Classial on v ergene theorem [ W ei94 , Thm. 5.5.1℄) . L et C b e hain  om- plex with a nite ltr ation ( F p C ) . Then the asso iate d sp e tr al se quen e  onver ges to H ∗ ( C ) : E 0 pq := F p C p + q /F p − 1 C p + q = ⇒ H p + q ( C ) . Ev erything in this setion an b e reform ulated for  o  hain omplexes and ohomologial sp etral sequenes. 6. Spetral sequenes of biomplexes Biomplexes are one of the main soures for ltered omplexes in algebra. They are less often enoun tered in top ology . A homologial biomplex is a lattie B = ( B pq ) ( p, q ∈ Z ) of ob jets onneted with v ertial morphisms ∂ v p oin ting down and horizon tal 24 MOHAMED BARAKA T morphisms ∂ h p oin ting left , su h that ∂ v ∂ h + ∂ h ∂ v = 0 . q B 02 ∂ v   B 12 ∂ v   ∂ h o o B 22 ∂ v   ∂ h o o B 01 ∂ v   B 11 ∂ v   ∂ h o o B 21 ∂ v   ∂ h o o B 00 B 10 ∂ h o o B 20 ∂ h o o O O / / p The sign tri k ˆ ∂ pq := ( − 1) p ∂ v pq on v erts the an tiomm utativ e squares in to omm uta- tiv e ones, and hene turns the biomplex in to a omplex of omplexes onneted with  hain maps as morphisms, and vie v ersa. The diret sum of ob jets T ot( B ) n := L p + q = n B pq together with the total b oundary op erator ∂ n := P p + q = n ∂ v pq + ∂ h pq form a  hain omplex alled the the total omplex asso iated to the biomplex B . ∂ ∂ = 0 is a diret onsequene of the an tiomm utativit y . The v ertial morphisms d v of a ohomologial biomplex ( B pq ) p oin t up and the horizon tal d h p oin t right . W e assume all biomplexes b ounded, i.e. only nitely man y ob jets B pq are dieren t from zero. There exists a natural so-alled olumn ltration of the total omplex T ot( B ) su h that the 0 -th page E 0 = ( E 0 pq ) = ( B pq ) of the sp etral sequene asso iated to this ltration onsists of the v ertial arro ws of B and the 1 -st page E 1 on tains morphisms indued b y the v ertial ones. Its asso iated sp etral sequene is alled the rst sp etral sequene of the biomplex B and is often denoted b y I E . F or a formal denition see [ W ei94 , Def. 5.6.1℄. The seond sp etral sequene is the (rst) sp etral sequene of the transp osed biomplex tr B = ( tr B pq ) := ( B q p ) . It is denoted b y II E . Note that T ot( B ) = T o t( tr B ) , only the t w o orresp onding ltrations and their indued ltrations on the total ohomology H ∗ (T ot( B )) dier in general. So the short notation I E a pq = ⇒ H p + q (T ot( B )) ⇐ = II E a pq refers in general to t w o dieren t ltrations of H p + q (T ot( B )) . Here is an algorithm using generalized maps to ompute the arro ws ∂ r pq : E r pq → E r p − r,q + r − 1 of the r -th term of the homologial (rst) sp etral sequene E r . Again, ev erything an b e easily adapted for the ohomologial ase. Denote b y α S : E r pq → B pq resp. α T : E r p − r,q + r − 1 → B p − r,q + r − 1 the generalized em b edding of the s oure resp. t arget of ∂ r pq in to the ob jet B pq = E 0 pq ≤ T ot( B ) p + q resp. B p − r,q + r − 1 ≤ T ot( B ) p + q − 1 . These so-alled absolute em b eddings are SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 25 the suessiv e omp ositions of the relativ e em b eddings E r pq → E r − 1 pq . F or the sak e of ompleteness w e also men tion the total em b eddings ι S : E r pq → T ot( B ) p + q resp. ι T : E r p − r,q + r − 1 → T ot( B ) p + q − 1 , the omp ositions of α S resp. α T with the gener alize d em b eddings 5 B pq → T ot( B ) p + q resp. B p − r,q + r − 1 → T ot( B ) p + q − 1 . PSfrag replaemen ts E ∞ pq E ∞ pq E r pq E 0 pq C p + q = T ot( B ) p + q α pq ι pq · · · Figure 19. The relativ e, absolute, and total em b eddings F or r > 1 let h r pq : B pq → r − 1 M i =1 B p − i,q + i − 1 and v r p − r +1 ,q + r − 1 : B p − r +1 ,q + r − 1 → r − 1 M i =1 B p − i,q + i − 1 b e the restritions of the total b oundary op erator ∂ p + q to the sp eied soures and targets. Similarly , for r > 2 let l r pq : r − 2 M i =1 B p − i,q + i → r − 1 M i =1 B p − i,q + i − 1 , 5 It iden ties B pq with the subfator of T o t( B ) p + q ditated b y the ltration. 26 MOHAMED BARAKA T again the restrition of the total b oundary op erator ∂ p + q to the sp eied soure and target. E r p − r,q + r − 1 _  α T   B p − r,q + r − 1 B p − r +1 ,q + r − 1 ∂ h o o ∂ v   B p − r +1 ,q + r − 2 B p − r +2 ,q + r − 2 o o   O O O O O O . . . B p − 1 ,q +1 o o   B p − 1 ,q B pq ∂ h o o E pq ?  α S O O W e distinguish four ases r = 0 , 1 , 2 , and r > 2 . r = 0 : ∂ 0 pq := ∂ v pq . Note that E 0 pq := B pq . r = 1 : ∂ 1 pq := α − 1 T ◦ ( ∂ h pq ◦ α S ) . r = 2 : ∂ 2 pq := α − 1 T ◦ ( ∂ h p − 1 ,q +1 ◦ ( − β − 1 ◦ ( h 2 pq ◦ α S ))) , where β := v 2 p − 1 ,q +1 . Note that h 2 pq = ∂ h pq and v 2 p − 1 ,q +1 = ∂ v p − 1 ,q +1 . r > 2 : ∂ r pq := α − 1 T ◦ ( ∂ h p − r +1 ,q + r − 1 ◦ ( − β − 1 ◦ ( h r pq ◦ α S ))) , with β := ( v r p − r +1 ,q + r − 1 , l r pq ) , the oarsening of v r p − r +1 ,q + r − 1 with aid l pq . W e sa y: v r p − r +1 ,q + r − 1 aided b y l r pq lifts h r pq ◦ α S . W e announed an algorithm and pro vided losed form ulas. This is the true v alue of generalized maps men tioned in the In tro dution. As an easy exerise, the reader migh t try to rephrase the diagram  hasing of the snak e lemma as a losed form ula in terms of generalized maps. The onept of a generalized map ev olv ed during the implemen tation of the homalg pa k age in GAP [ Bar09 ℄. It follo ws from Remark 4.6 that the sp etral sequene of a nite t yp e b ounded biomplex (in fat, of a nite t yp e omplex with nite ltration) o v er a omputable ring is eetiv ely omputable (f. Def. A.1 ). The homalg pa k age [ Bar09 ℄ on tains routines to ompute sp etral sequenes of biomplexes. W e end this setion with a simple example from linear algebra. Let k b e a eld and λ ∈ k a eld elemen t. The Jord an -form matrix J ( λ ) =   λ 1 · · λ 1 · · λ   ∈ k 3 × 3 SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 27 turns V := k 1 × 3 in to a left k [ x ] -mo dule (of nite length), where x ats via J ( λ ) , i.e. xv := J ( λ ) v for all v ∈ V . The k [ x ] -mo dule V is ltered and the ltrations stems from a biomplex: Example 6.1 ( Sp etrum of an endomorphism) . Let k b e a eld and λ ∈ k . Consider the seond quadran t biomplex B λ B − 2 , 3 ( x − λ )   B − 2 , 2 B − 1 , 2 ( − 1 ) o o − ( x − λ )   B − 1 , 1 B 0 , 1 ( − 1 ) o o ( x − λ )   B 0 , 0 with B 0 , 0 = B 0 , 1 = B − 1 , 1 = B − 1 , 2 = B − 2 , 2 = B − 2 , 3 = k [ x ] , all other sp ots b eing zero. The total omplex on tains exatly t w o non trivial k [ x ] -mo dules at degrees 0 and 1 and a single non trivial morphism ∂ 1 ( λ ) : T ot( B ) 1 = k [ x ] 1 × 3 / / k [ x ] 1 × 3 = T ot( B ) 0 with matrix x Id − J ( λ ) =   x − λ − 1 · · x − λ − 1 · · x − λ   . The rst sp etral sequenes I E liv es in the seond quadran t and stabilizes already at I E 1 =: I E ∞ · · · I E 1 − 2 , − 2 · · · I E 1 − 1 , − 1 · · · I E 1 0 , 0 with I E ∞ 0 , 0 = I E ∞ − 1 , − 1 = I E ∞ − 2 , − 2 = k [ x ] / h x − λ i . 28 MOHAMED BARAKA T The seond sp etral sequenes II E liv es in the fourth quadran t, has only zero arro ws at lev els 1 and 2 II E 1 0 , 0 · · · · · · · · · · II E 1 3 , − 2 II E 2 0 , 0 · · · · · · · · · · II E 2 3 , − 2 with II E 1 0 , 0 = II E 1 3 , − 2 = k [ x ] , and hene II E 2 0 , 0 = II E 2 3 , − 2 = k [ x ] = II E 3 0 , 0 = II E 3 3 , − 2 . A t lev el 3 there exists a single nonzero arro w ∂ 3 3 , − 2 with matrix  ( x − λ ) 3  : II E 3 0 , 0 · · · · · · · · · · II E 3 3 , − 2 ∂ 3 3 , − 2 J J J J J J J d d J J J J J J J II E nally ollapses to its p -axes at II E 4 =: II E ∞ II E 4 0 , 0 · · · · · · · · · · · with II E ∞ 0 , 0 = k [ x ] / h ( x − λ ) 3 i , pro viding a sp etral sequene pro of for the elemen tary fat that cok er ∂ 1 ( λ ) ∼ = k [ x ] / h ( x − λ ) 3 i . Con v ersely , this isomorphism implies that the matrix of the morphism ∂ 3 3 , − 2 is equal to  ( x − λ ) 3  , up to a unit a ∈ k × . SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 29 7. The Car t an-Eilenber g resolution of a omplex The Car t an-Eilenber g resolution generalizes the horse sho e lemma in the fol- lo wing sense: The horse sho e lemma pro dues a sim ultaneous pro jetiv e resolution 6 0   0   · · · 0   0 M ′ o o   P ′ 0 o o   · · · o o P ′ d o o   0 o o 0 M o o   P 0 o o   · · · o o P d o o   0 o o 0 M ′′ o o   P ′′ 0 o o   · · · o o P ′′ d o o   0 o o 0 0 · · · 0 of a short exat sequene 0 ← − M ′′ ← − M ← − M ′ ← − 0 , where sim ultaneous means that ea h ro w is a pro jetiv e resolution and all olumns are exat. No w let us lo ok at this threefold resolution in the follo wing w a y: The short exat sequene denes a 2 -step ltration of the ob jet M with graded parts M ′ and M ′′ and the horse sho e lemma states that an y resolutions of the graded parts an b e put together to a resolution of the total ob jet M . In fat, as P ′′ i is pro jetiv e, it follo ws that the total ob jet P i m ust ev en b e the diret sum of the graded parts P ′ i and P ′′ i . The non-trivialit y of the ltration on M is reeted in the fat that the morphisms of the total resolution P ∗ are in general not merely the diret sum of the morphisms in the resolutions P ′ ∗ and P ′′ ∗ of the graded parts M ′ and M ′′ . This statemen t an no w b e generalized to m -step ltrations simply b y applying the ( 2 -step) horse sho e lemma indutiv ely . No w onsider a omplex ( C , ∂ ) , whi h is not neessarily exat. On ea h ob jet C n the omplex struture indues a 3 -step ltration 0 ≤ B n ≤ Z n ≤ C n , with b oundaries B n := im ∂ n +1 and yles Z n := k er ∂ n . The ab o v e disussion no w applies to the three graded parts B n , H n := Z n /B n and C n / Z n and an y three resolution thereof an b e put together to a resolution of the total ob jet C n . If one tak es in to aoun t the fat that ∂ n +1 indues an isomorphism b et w een C n +1 / Z n +1 and B n (for all n , b y the homomorphism theorem), then all total resolutions of all the C n 's an b e onstruted in a ompatible w a y so that they t together in one omplex of omplexes. This omplex is alled the Car t an-Eilenber g resolution of the omplex C . A formal v ersion of the ab o v e disussion an b e found in [ HS97 , Lemma 9.4℄ or [ W ei94 , Lemma 5.7.2℄. Sine the pro jetiv e horse sho e lemma is onstrutiv e, the pro jetiv e Car t an-Eilenber g resolution is so as w ell. 6 W e will only refer to pro jetiv e resolutions as they are more relev an t to eetiv e omputations. 30 MOHAMED BARAKA T 8. Gr othendiek's spetral sequenes Let C F ← − B G ← − A b e omp osable funtors of ab elian ategories. The so-alled Gr o- thendiek sp etral sequene relates, under mild assumptions, the omp osition of the deriv ations of F and G with the deriv ation of their omp osition F ◦ G . There are 16 v er- sions of the Gr othendiek sp etral sequene, dep ending on whether F resp. G is o- or on tra v arian t, and whether F resp. G is b eing left or righ t deriv ed. F our of them do not use injetiv e resolutions and are therefore rather diretly aessible to a omputer. In this se- tion t w o v ersions out of the four are review ed: The ltrations of L ⊗ D M and Hom D ( M , N ) men tioned in the In tro dution are reo v ered in the next setion as the sp etral ltrations indued b y these t w o Gr othendiek sp etral sequenes, after appropriately  ho osing the funtors F and G . Theorem 8.1 ( Gr othendiek sp etral sequene, [ Rot79 , Thm. 11.41℄) . L et F and G b e  ontr avariant funtors and let every obje t in A and B has a nite pr oje tive r esolution. Under the assumptions that (1) G maps pr oje tive obje ts to F -ayli obje ts and that (2) F is left exat, then ther e exists a se  ond quadr ant homolo gi al sp e tr al se quen e with E 2 pq = R − p F ◦ R q G = ⇒ L p + q ( F ◦ G ) . Pr o of. Let M b e an ob jet in A and P • = ( P p ) a nite pro jetiv e resolution of M . De- note b y C E = ( C E p,q ) the pro jetiv e Car t an-Eilenber g resolution of the o omplex ( Q p ) := ( G ( P p )) . It exists sine B has enough pro jetiv es. Note that q ≤ 0 sine C E is a ohomologial biomplex. Dene the homologial biomplex B = ( B p,q ) := ( F ( C E p,q )) . W e all B the Grothendie k biomplex asso iated to M , F , and G . It liv es in the fourth quadran t and is b ounded in b oth diretions. The rst sp etral sequene I E : F or xed p the v ertial o omplex C E p, • is, b y onstrution, a pro jetiv e resolution of G ( P p ) . Hene I E 1 pq = R − q F ( G ( P p )) . But sine G ( P p ) is F -ayli b y assumption (1), the rst sheet ollapses to the 0 -th ro w. The left exatness of F implies that R 0 F = F and hene I E 1 p 0 = ( F ◦ G )( P p ) . I.e. the 0 -th ro w of I E 1 is nothing but the o v arian t funtor F ◦ G applied to the pro jetiv e resolution ( P p ) of M . The rst sp etral sequenes of B th us stabilizes at lev el 2 with the single ro w I E 2 n, 0 = L n ( F ◦ G )( M ) . The seond sp etral sequene II E : The seond sp etral sequene of the biomplex B is b y denition the sp etral sequene of its transp osed ( tr B pq ) := ( B q p ) , a seond quadran t biomplex. Ob viously tr B = F ( tr C E ) . By denition, the q -th ro w II E 1 • ,q := H v ert • ,q ( tr B ) = H v ert • ,q ( F ( tr C E )) = F ( H • ,q v ert ( tr C E )) , where the last equalit y follo ws from the prop erties of the Car t an-Eilenber g resolution and the additivit y of F . No w reall that the v ertial ohomologies H • ,q v ert ( tr C E ) are for xed q , again b y onstrution, pro jetiv e resolutions of the ohomology H q ( G ( P • )) =: R q G ( M ) . Hene II E 2 pq = R − p F (R q G ( M )) .  SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 31 The pro of sho ws that assumptions (1) and (2) only in v olv e the rst sp etral sequene. Assumption (1) guaran teed the ollapse of the rst sp etral sequene at the rst lev el, while (2) ensures that the natural transformation F → R 0 F is an equiv alene. In other w ords, dropping (2) means replaing L p + q ( F ◦ G ) b y L p + q (R 0 F ◦ G ) . Theorem 8.2 ( Gr othendiek sp etral sequene) . L et F b e a  ovariant and G a  on- tr avariant funtor and let every obje t in A and B has a nite pr oje tive r esolution. Under the assumptions that (1) G maps pr oje tive obje ts to F -ayli obje ts and that (2) F is right exat, then ther e exists a se  ond quadr ant  ohomolo gi al sp e tr al se quen e with E 2 pq = L − p F ◦ R q G = ⇒ R p + q ( F ◦ G ) . Pr o of. Again the rst sp etral sequene is a fourth quadran t sp etral sequene while the seond liv es in the seond quadran t. Assumption (2) ensures that the natural transfor- mation L 0 F → F is an equiv alene. The ab o v e pro of and the subsequen t remark an b e opied with the ob vious mo diations.  Remark 8.3 (One sided b oundedness) . The existene of nite pro jetiv e resolutions in A and B led the sp etral sequenes to b e b ounded in b oth diretions. In order to a v oid on v ergene subtleties it w ould sue to assume b oundedness in just one diretion b y requiring that either A or B allo ws nite pro jetiv e resolutions while the other has enough pro jetiv es. The assumption of the existene of nite pro jetiv e resp. injetiv e resolutions an b e dropp ed when dealing with the v ersions of the Gr othendiek sp etral sequenes that liv e in the rst resp. third quadran t. 9. Applia tions This setion realls ho w the natural ltrations men tioned in examples (a), (a'), and (d) of the In tro dution an b e reo v ered as sp etral ltrations . Theorems 8.1 and 8.2 admit an ob vious generalization. The omp osed funtor F ◦ G an b e replaed b y a funtor H that oinides with F ◦ G on pro jetiv es (for other v ersions of the Gr othendiek sp etral sequene the pro jetiv es has to b e replaed b y injetiv es). As usual, D is an asso iativ e ring with 1 . Ext n D and T or D n are abbreviated as Ext n and T or n . Assumption: In this setion the left or righ t global dimension 7 of D is assumed nite. The in v olv ed sp etral sequenes will then b e b ounded in (at least) one diretion (see Remark 8.3 ). 7 Reall, the left global (homologial) dimension is the suprem um o v er all pro jetiv e dimensions of left D -mo dules (see Subsetion 9.1.5 ). If D is left Noether ian, then the left global dimension of D oinides with the w eak global (homologial) dimension , whi h is the largest in teger µ su h that T o r D µ ( M , N ) 6 = 0 for some righ t mo dule M and left mo dule N , otherwise innit y (f. [ MR01 , 7.1.9℄). This last denition is ob viously left-righ t symmetri. The same is v alid if left is replaed b y righ t. 32 MOHAMED BARAKA T 9.1. The double- Ext sp etral sequene and the ltration of T or . Corollary 9.1 (The double- Ext sp etral sequene) . L et M b e a left D -mo dule and L a right D -mo dule. Then ther e exists a se  ond quadr ant homolo gi al sp e tr al se quen e with E 2 pq = Ext − p (Ext q ( M , D ) , L ) = ⇒ T or p + q ( L, M ) . In p artiular, ther e exists an as ending ltr ation of T or p + q ( L, M ) with gr p T or p + q ( L, M ) natur al ly isomorphi to a subfator of Ext − p (Ext q ( M , D ) , L ) , p ≤ 0 . The sp eial ase p + q = 0 reo v ers the ltration of L ⊗ M men tioned in Example (a) of the In tro dution via the natural isomorphism L ⊗ M ∼ = T or 0 ( L, M ) . 9.1.1. Using the Gr othendiek bi omplex. Corollary 9.1 is a onsequene of Theorem 8.1 for F := Hom D ( − , L ) and G := Hom D ( − , D ) , sine F ◦ G oinides with L ⊗ D − on pro jetiv es. T o b e able to eetiv ely ompute double- Ext (groups in) the Gr othendiek biomplex the ring D m ust b e omputable in the sense that two sided inhomogeneous linear systems A 1 X 1 + X 2 A 2 = B m ust b e eetiv ely solv able, where A 1 , A 2 , and B are matries o v er D (see [ BR08 , Subsetion 6.2.4℄). This is immediate for omputable omm utativ e rings (f. Def. A.1 ). In B.2 an example o v er a omm utativ e ring is treated. 9.1.2. Using the bi omplex I L ⊗ P M . The bifuntorialit y of ⊗ leads to the follo wing homologial biomplex B := I L ⊗ P M ∼ = Hom(Hom( P M , D ) , I L ) , where P M is an injetiv e resolution of M and I L is an injetiv e resolution of L . Start- ing from r = 2 the rst and seond sp etral sequene of B oinide with those of the Gr othendiek biomplex asso iated to M , F := Hom D ( − , L ) , and G := Hom D ( − , D ) . In on trast to the Gr othendiek biomplex the biomplex B is o v er most of the in terest- ing rings in general highly nononstrutiv e as an injetiv e resolution en ters its denition. In [ HL97 , Lemma 1.1.8℄ a sheaf v arian t of this biomplex w as used to ompute the purit y ltration (see b elo w). 9.1.3. The bidualizing  omplex. T aking L = D as a righ t D -mo dule in Corollary 9.1 reo v- ers the bidualizing sp etral sequene of J.-E. R oos [ Ro o62 ℄. E 2 pq = Ext − p (Ext q ( M , D ) , D ) = ⇒  M for p + q = 0 , 0 otherwise. The Gr othendiek biomplex is then kno wn as the bidualizing omplex . The ase p + q = 0 denes the purit y ltration 8 (t − c M ) of M , whi h w as motiv ated in Example (a') of the In tro dution. F or more details f. [ Bjö79 , Chap. 2, 5,7℄. The mo dule M c = E ∞ − c,c is for c = 0 and c = 1 a submo dule of Ext c (Ext c ( M , D ) , D ) = E 2 − c,c and for c ≥ 2 in general only a subfator. All this is ob vious from the shap e of the bidualizing sp etral sequene. 8 Unlik e [ Bjö79 , Chap. 2, Subsetion 4.15℄, w e only mak e the w eak er assumption stated at the b eginning of the setion. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 33 Sine M c = t − c M / t − ( c +1) M it follo ws that the higher ev aluations maps ε c 0 − → t − ( c +1) M − → t − c M ε c − → Ex t c D (Ext c D ( M , D ) , D ) men tioned in the In tro dution are only a dieren t w a y of writing the generalized em b ed- dings ¯ ε c : M c → Ex t c (Ext c ( M , D ) , D ) . So without further assumptions ε c (resp. ¯ ε c ) is kno wn to b e an ordinary morphism (resp. em b edding) only for c = 0 and c = 1 . No w assuming that E 2 pq := Ext − p (Ext q ( M , D ) , D ) v anishes 9 for p + q = 1 , then all arro ws ending at total degree p + q = 0 v anish (as they all start at total degree p + q = 1 ). It follo ws that for all c the mo dule M c is not merely a subfator of Ext c (Ext c ( M , D ) , D ) but a submo dule, or, equiv alen tly , ε c (resp. ¯ ε c ) is an ordinary morphism (resp. em b edding). In an y ase the mo dule Ext c (Ext c ( M , D ) , D ) is alled the reexiv e h ull of the pure subfator M c . Denition 9.2 (Pure, reexiv ely pure) . A mo dule M is alled pure if it onsists of exatly one non trivial pure subfator M c or is zero. A non trivial mo dule M is alled reexiv ely pure if it is pure and if the generalized em b edding M = M c → Ex t c (Ext c ( M , D ) , D ) is an isomorphism. Dene the zero mo dule to b e reexiv ely pure. If M is a nitely generated D -mo dule, then all ingredien ts of the bidualizing omplex are again nitely generated (pro jetiv e) D -mo dules, ev en if the ring D is non omm utativ e. It follo ws that the purit y ltration o v er a omputable ring D is eetiv ely omputable. A omm utativ e and a nonomm utativ e example are giv en in B.3 and B.4 resp etiv ely . The latter demonstrates ho w the purit y ltration (as a ltration that alw a ys exists) an b e used to transform a linear system of PDEs in to a triangular form where no w a asade in tegration strategy an b e used to obtain exat solutions. The idea of viewing a linear system of PDEs as a mo dule o v er an appropriate ring of dieren tial op erators w as emphasized b y B. Malgrange in the late 1960's and aording to him go es ba k to Emmy Noether . 9.1.4. Criterions for r eexive purity. This subsetion lists some simple riterions for re- exiv e purit y of mo dules. First note that the existene of the bidualizing sp etral sequene immediately implies that the set c ( M ) := { c ≥ 0 | Ext c D ( M , D ) 6 = 0 } is empt y only if M = 0 . Reall that if c ( M ) is nonempt y , then its minim um is alled the grade or o dimension of M and denoted b y j ( M ) or co dim M . The o dimension of the zero mo dule is set to b e ∞ . F urther dene ¯ q ( M ) := sup c ( M ) in ase c ( M ) 6 = ∅ , and ∞ otherwise. All of the follo wing argumen ts mak e use of the shap e of the bidualizing sp etral sequene in the resp etiv e situation. • If c ( M ) on tains a single elemen t, i.e. if co dim M = ¯ q ( M ) =: ¯ q < ∞ , then M = M ¯ q is reexiv ely pure of o dimension ¯ q , giving a simple sp etral sequene pro of of [ Qua01 , Thm. 7℄. 9 This ondition is satised for an A uslander regular ring D : Ext − p (Ext q ( M , D ) , D ) = 0 for all p + q > 0 and all D -mo dules M . See [ Bjö79 , Chap. 2: Cor. 5.18, Cor. 7.5℄. 34 MOHAMED BARAKA T F or the remaining riterions assume that Ext − p (Ext q ( M , D ) , D ) = 0 for p + q = 1 : • If ¯ q := ¯ q ( M ) is nite, then E 2 − ¯ q, ¯ q = E ∞ − ¯ q, ¯ q , i.e. M ¯ q is reexiv ely pure (p ossibly zero). This generalizes the ab o v e riterion (under the assumption just made). • No w if M is a left (resp. righ t) D -mo dule, then assume further that the righ t (resp. left) global dimension d of the ring D is nite. It follo ws that E 2 − c,c = E ∞ − c,c for c = d and c = d − 1 . This means that under the ab o v e assumptions the subfators M d and M d − 1 are alw a ys reexiv ely pure 10 . 9.1.5. Co de gr e e of purity. As a Gr othendiek sp etral sequene the bidualizing sp etral sequene b eomes in trinsi at lev el 2 . Ea h E 2 − c,c starts to shrink un til it stabilizes at E ∞ − c,c = M c . Motiv ated b y this dene the o degree of purit y cp M of a mo dule M as follo ws: Set cp M to ∞ if M is not pure. Otherwise cp M is a tuple of nonnegativ e in tegers, the length of whi h is one plus the n um b er of times E a − c,c shrinks (non trivially 11 ) for a ≥ 2 un til it stabilizes at M c . The en tries of this tuple are the n um b ers of pages b et w een the drops, i.e. the width of the steps in the stairase of ob jets ( E a − c,c ) c ≥ 2 . It follo ws that the sum o v er the en tries of cp M is the n um b er of pages it tak es for E 2 − c,c un til it rea hes M c . In partiular, a mo dule is r eexively pur e if and only if cp M = (0) . The o degree of purit y app ears in Examples B.3 and B.4 . In Example B.7 the o degree of purit y is ompared with t w o other lassial homologial in v arian ts: Reall, the pro jetiv e dimension of a mo dule M is dened to b e the length d of the shortest pro jetiv e resolution 0 ← − M ← − P 0 ← − · · · ← − P d ← − 0 . A uslander 's degree of torsion-freeness of a mo dule M is dened follo wing [ AB69 , Def. on p. 2 & Def. 2.15(b)℄ to b e the smallest nonne gative in teger i , su h that Ext i +1 (A( M ) , D ) 6 = 0 , otherwise ∞ , where A( M ) is the so-alled A uslander dual of M (see also [ Qua01 , Def. 5℄, [ CQR05 , Def. 19℄). T o onstrut A( M ) tak e a pro jetiv e presen tation 0 ← − M ← − P 0 d 1 ← − P 1 of M and set A( M ) := cok er( P ∗ 0 d ∗ 1 − → P ∗ 1 ) , where d ∗ 1 := Hom( d 1 , D ) (f. [ AB69 , p. 1 & Def. 2.5℄). Lik e the syzygies mo dules, it is pro v ed in [ AB69 , Prop. 2.6(b)℄ that A( M ) is uniquely determined b y M up to pro jetiv e equiv alene (see also [ Qua99 ℄ and [ PQ00 , Thm. 2℄). In partiular, the degree of torsion- freeness is w ell-dened. The fundamen tal exat sequene [ AB69 , (0.1) & Prop. 2.6(a)℄ 0 − → Ext 1 D (A( M ) , − ) − → M ⊗ D − − → Hom D ( M ∗ , − ) − → Ext 2 D (A( M ) , − ) − → 0 , applied to D ,  haraterizes torsion-freeness and reexivit y of the mo dule M (see also [ HS97 , Exerise IV.7.3℄, [ CQR05 , Thm. 6℄). F or a  haraterization of pro jetivit y using the degree of torsion-freeness see [ CQR05 , Thm. 7℄. The o degree of purit y an b e dened for quasi-oheren t shea v es of mo dules replaing D b y the struture sheaf O X or b y the dualizing sheaf 12 if it exists. It is imp ortan t to note 10 In ase D = A n , the n -th Weyl algebra o v er a eld, this sa ys that holonomi and subholonomi mo dules are reexiv ely pure. See [ Bjö79 , Chap. 2, 7℄. 11 i.e. passes to a true subfator. 12 It ma y ev en b e dened for ob jets in an ab elian ategory with a dualizing ob jet. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 35 that the o degree of purit y of a oheren t sheaf F of O X -mo dules on a pro jetiv e s heme X = Pro j( S ) ma y dier from the o degree of purit y of a graded S -mo dule M used to represen t F = f M = Pro j M . This is mainly due to the fat that F = f M v anishes for Ar tin ian mo dules M . There are sev eral ob vious w a ys ho w one an rene the o degree of purit y to get sharp er in v arian ts. The o degree of purit y is an example of what an b e alled a sp etral in v ari- an t . 9.2. The T or - Ext sp etral sequene and the ltration of Ext . Corollary 9.3 (The T or − Ext sp etral sequene) . L et M and N b e left D -mo dules. Then ther e exists a se  ond quadr ant  ohomolo gi al sp e tr al se quen e with E pq 2 = T or − p (Ext q ( M , D ) , N ) = ⇒ Ext p + q ( M , N ) . In p artiular, ther e exists a des ending ltr ation of Ext p + q ( M , N ) with gr p Ext p + q ( M , N ) natur al ly isomorphi to a subfator of T or − p (Ext q ( M , D ) , N ) , p ≤ 0 The sp eial ase p + q = 0 reo v ers the ltration of Hom( M , N ) men tioned in Example (d) of the In tro dution via the natural isomorphism Hom( M , N ) ∼ = Ext 0 ( M , N ) . F or holonomi mo dules M o v er the Weyl k -algebra D := A n the sp eial ase form ula Hom( M , N ) ∼ = T or n (Ext n ( M , D ) , N ) (f. [ Bjö79 , Chap. 2, Thm. 7.15℄) w as used b y H. Tsai and U. W al ther in the ase when also N is holonomi to ompute the nite dimensional k -v etor spae of homomorphisms [ TW01 ℄. The indued ltration on Ext 1 ( M , N ) an b e used to atta h a n umerial in v arian t to ea h extension of M with submo dule N . This giv es another example of a sp etral in v arian t . 9.2.1. Using the Gr othendiek bi omplex. Corollary 9.3 is a onsequene of Theorem 8.2 for F := − ⊗ D N and G := Hom D ( − , D ) sine F ◦ G oinides with Hom D ( − , N ) on pro jetiv es. See Example B.5 . 9.2.2. Using the bi omplex Hom( P M , P N ) . The bifuntorialit y of Hom leads to the fol- lo wing ohomologial biomplex B := Hom( P M , P N ) ∼ = Hom( P M , D ) ⊗ P N , where P L denotes a pro jetiv e resolution of the mo dule L . It is an easy exerise (f. [ Bjö79 , Chap. 2, 4.14℄) to sho w that starting from r = 2 the rst and seond sp etral sequene of B oinide with those of the Gr othendiek biomplex asso iated to M , F := − ⊗ D N and G := Hom D ( − , D ) . Both biomplexes are onstrutiv e as only pro jetiv e resolutions en ter their denitions. The biomplex B has the omputational adv an tage of a v oiding the rather exp ensiv e Car t an-Eilenber g resolution used to dene the Gr othendiek bi- omplex. See Example B.6 . Compare the output of the ommand homalgRingStatistis in Example B.6 with orresp onding output in Example B.5 . 36 MOHAMED BARAKA T Sine the rst sp etral sequene of the biomplex B := Hom( P M , P N ) ollapses, a small part of it is often used to ompute Hom( M , N ) o v er a  ommutative ring D , as then all arro ws of B are again morphisms of D -mo dules. See [ GP02 , p. 104℄ and [ BR08 , Subsetion 6.2.3℄. If the ring D is not omm utativ e, then the ab o v e biomplex and the Gr othendiek biomplex in the previous subsetion fail to b e D -biomplexes (unless when M or N is a D -bimo dule). The biomplexes are ev en in a lot of appliations of in terest not of nite t yp e o v er their natural domain of denition. In ertain situations there nev ertheless exist quasi- isomorphi subfator (bi)omplexes whi h an b e used to p erform eetiv e omputations. In [ TW01 ℄, ited ab o v e, and in the pioneering w ork [ OT01 ℄ Kashiw ara 's so-alled V - ltration is used to extrat su h subfators. SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 37 Appendix A. The triangula tion algorithm Denition A.1 (Computable ring [ BR08 , Subsetion 1.2℄) . A left and righ t no etherian ring is alled omputable if there exists an algorithm whi h solv es one sided inhomogeneous linear systems X A = B and AX = B , where A and B are matries with en tries in D . The w ord solv es means: The algorithm an deide if a solution exists, and, if solv able, is able to ompute a partiular solution as w ell as a nite generating set of solutions of the orresp onding homogeneous system. F rom no w on the ring D is assumed omputable. Let M b e a nitely generated left D -mo dule. Then M is nitely presen ted, i.e. there exists a matrix M ∈ D p × q , view ed as a morphism M : D 1 × p → D 1 × q , su h that cok er M ∼ = M . M is alled a matrix of relations or a presen tation matrix for M . It forms the b eginning of a free resolution 0 ← − M ← − D 1 × q d 1 = M ← − − − D 1 × p d 2 ← − D 1 × p 2 d 3 ← − · · · . d i is alled the i -th syzygies matrix of M and K i := cok er d i +1 the i -th syzygies mo dule. K i is uniquely determined b y M up to pro jetiv e equiv alene . No w supp ose that M is endo w ed with an m -ltration F = ( F p M ) . W e will sk et h an algorithm that, starting from a presen tation matrix M ∈ D p × q for M and presen tation matries M p for the graded parts M p := gr p M , omputes another upp er triangular presen tation matrix M F of the form 13 M F =       M p m − 1 ∗ · · · · · · ∗ M p m − 2 ∗ · · · ∗ . . . . . . . . . M p 1 ∗ M p 0       ∈ D p ′ × q ′ and an isomorphism cok er M F ∼ = − → cok er M giv en b y a matrix T ∈ D q ′ × q : Let ( ψ p ) b e an asending m -ltration system omputing F (f. Def. 4.3 ). T o start the indution tak e p to b e the highest degree p m − 1 in the ltration and set F p M := M . Sine µ p := ψ p : M p = c ok er M p → cok er F p M is a generalized isomorphism, its unique generalized in v erse exists and is an epimorphism (f. Cor. 4.8 ), whi h w e denote b y π p : F p M → M p . (Note: cok er F p M = F p M = M for p = p m − 1 .) Sine D is omputable w e are able to determine (a matrix of ) an injetiv e morphism ι p mapping on to the k ernel of π p . The soure of ι p is a mo dule isomorphi to F p − 1 M , whi h w e also denote b y F p − 1 M . No onfusion an o ur sine w e will only refer 13 Note that  ho osing a generating system of M adapted to the ltration F is not enough to pro due a triangular presen tation matrix, as  hanging the set of generators only orresp onds to olumn op erations on M . 38 MOHAMED BARAKA T to the latter. All maps in the exat-ro ws diagram 0 M p o o P 0 η 0   ν o o K 1 η   M p o o 0 o o 0 M p o o F p M π p o o F p − 1 M ι p o o 0 o o are omputable, where P 0 is a free D -mo dule and K 1 is the 1 -st syzygies mo dule of M p : η 0 is omputable sine P 0 is free and η is omputable sine ι p is injetiv e (see [ BR08 , Subsetion 3.1℄). This yields the short exat sequene 0 − → K 1 κ := “ M p η ” − − − − − − − → P 0 ⊕ F p − 1 M ρ := 0 @ − η 0 ι p 1 A − − − − − − − → F p M − → 0 . Hene, the ok ernel of κ :=  M p η  is isomorphi to F p M whi h therefore admits a pre- sen tation matrix of the form M p F =  M p η 0 F p − 1 M  , where F p − 1 M is a presen tation matrix of F p − 1 M (for more details see [ BB , Subsetion 7.1℄). If χ : P 0 ⊕ F p − 1 M − → cok er κ = coker M p F denotes the natural epimorphism and ρ :=  − η 0 ι p  , then the matrix T p of the morphism T p := ρ ◦ χ − 1 is an isomorphism b et w een cok er M p F and cok er F p M . By the indution h yp othesis w e ha v e e M p +1 F :=  stable p η p 0 F p M  =   stable p +1 ∗ ∗ 0 M p +1 ∗ 0 0 F p M   =  stable p +1 ∗ ∗ 0 M p +1 F  with cok er e M p +1 F ∼ = cok er M . (Sine p w as dereased b y one the old F p − 1 M is no w addressed as F p M , et.). Before pro eeding indutiv ely on the submatrix F p M of e M p +1 F tak e the quotien t µ p := ( ι p m − 1 ◦ · · · ◦ ι p +1 ) − 1 ◦ ψ p : M p = cok er M p → cok er F p M , whi h is lik e µ p +1 again a generalized isomorphism. Note that matrix T p of the morphism T p := ρ ◦ χ − 1 pro viding the isomorphism b et w een cok er M p F and cok er F p M no w has to b e m ultiplied from the righ t to the submatrix η p of e M p +1 F whi h lies ab o v e F p M . This ompletes the indution. The algorithm terminates with M F := e M p 0 F and T is the omp osition of all the suessiv e olumn op erations on M .  The ab o v e algorithm is implemen ted in homalg pa k age [ Bar09 ℄ under the name Isomor- phismOfFiltration . It tak es an m -ltration system ( ψ p ) of M = cok er M as its input and returns an isomorphism τ : coke r M F → cok er M with a triangular presen tation matrix M F , as desrib ed ab o v e. IsomorphismOfFiltration will b e extensiv ely used in the examples in App endix B . SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 39 Appendix B. Examples with GAP 's homalg The pa k ages homalg , IO_ForHomalg , and RingsForHomalg are assumed loaded: gap> LoadPakage( "RingsForHomalg" ); true F or details see the homalg pro jet [ h t09 ℄. Example B.1 ( LeftPresentation) . Dene a left mo dule W o v er the p olynomial ring D := Q [ x, y , z ] . Also dene its righ t mirror Y . gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";; gap> wmat := HomalgMatrix( "[ \ x*y, y*z, z, 0, 0, \ x^3*z,x^2*z^2,0, x*z^2, -z^2, \ x^4, x^3*z, 0, x^2*z, -x*z, \ 0, 0, x*y, -y^2, x^2-1,\ 0, 0, x^2*z, -x*y*z, y*z, \ 0, 0, x^2*y-x^2,-x*y^2+x*y,y^2-y \ ℄", 6, 5, Qxyz ); gap> W := LeftPresentation( wmat ); gap> Y := Hom( Qxyz, W ); Example B.2 (Homologial GrothendiekSpetralSequen e ) . Example B.1 on tin ued. Compute the double- Ext sp etral sequene for F := Hom( − , Y ) , G := Hom( − , D ) , and the D -mo dule W . This is an example for Subsetion 9.1.1 . gap> F := InsertObjetInMultiFuntor( Funtor_Hom, 2, Y, "TensorY" ); gap> G := LeftDualizingFuntor( Qxyz );; gap> II_E := GrothendiekSpetralSequene( F, G, W ); gap> Display( II_E ); The assoiated transposed spetral sequene: a homologial spetral sequene at bidegrees [ [ 0 .. 3 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * * * * * 40 MOHAMED BARAKA T . * * * . . * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spetral sequene of the biomplex: a homologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 3 ℄ ℄ --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * * * * * . * * * . . . * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 41 * s s s . . s * . . . * --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpetralSequene( II_E, 0 ); -1: -2: -3: of > gap> ByASmallerPresentation( filt ); -1: -2: -3: of > gap> m := IsomorphismOfFiltration( filt ); Example B.3 ( PurityFiltration) . Example B.1 on tin ued. This is an example for Subsetions 9.1.3 and 9.1.5 . gap> filt := PurityFiltration( W ); -1: -2: -3: of > gap> W; 42 MOHAMED BARAKA T gap> m := IsomorphismOfFiltration( filt ); gap> IsIdentialObj( Range( m ), W ); true gap> Soure( m ); gap> Display( last ); 0, 0, x, -y,0,1, 0, 0, 0, x*y,-y*z,-z,0, 0,0, 0, 0, 0, x^2,-x*z,0, -z,1,0, 0, 0, 0, 0, 0, 0, 0, y,-z,0, 0, 0, 0, 0, 0, 0, x,0, -z, 0, 1, 0, 0, 0, 0, 0,x, -y, -1, 0, 0, 0, 0, 0, 0,-y,x^2-1,0, 0, 0, 0, 0, 0, 0,0, 0, z, 0, 0, 0, 0, 0, 0,0, 0, y-1,0, 0, 0, 0, 0, 0,0, 0, 0, z, 0, 0, 0, 0, 0,0, 0, 0, y, 0, 0, 0, 0, 0,0, 0, 0, x Cokernel of the map Q[x,y,z℄^(1x12) --> Q[x,y,z℄^(1x9), urrently represented by the above matrix gap> Display( filt ); Degree 0: 0, 0, x, -y, x*y,-y*z,-z,0, x^2,-x*z,0, -z Cokernel of the map Q[x,y,z℄^(1x3) --> Q[x,y,z℄^(1x4), urrently represented by the above matrix ---------- Degree -1: y,-z,0, x,0, -z, 0,x, -y, 0,-y,x^2-1 Cokernel of the map SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 43 Q[x,y,z℄^(1x4) --> Q[x,y,z℄^(1x3), urrently represented by the above matrix ---------- Degree -2: Q[x,y,z℄/< z, y-1 > ---------- Degree -3: Q[x,y,z℄/< z, y, x > gap> Display( m ); 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, -x^2,-x*z, 0, -z, 0, 0, 0, x, -y, 0, 0, 0, 0, 0, -1, 0, 0, x^2,-x*y,y, x^3, x^2*z,0, x*z, -z the map is urrently represented by the above 9 x 5 matrix Example B.4 ( PurityFiltration , non omm utativ e) . This is a non omm utativ e exam- ple for Subsetions 9.1.3 and 9.1.5 . Let A 3 := Q [ x, y , z ] h D x , D y , D z i b e the 3 -dimensional Weyl algebra. gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );; gap> nmat := HomalgMatrix( "[ \ 3*Dy*Dz-Dz^2+Dx+3*Dy-Dz, 3*Dy*Dz-Dz^2, \ Dx*Dz+Dz^2+Dz, Dx*Dz+Dz^2, \ Dx*Dy, 0, \ Dz^2-Dx+Dz, 3*Dx*Dy+Dz^2, \ Dx^2, 0, \ -Dz^2+Dx-Dz, 3*Dx^2-Dz^2, \ Dz^3-Dx*Dz+Dz^2, Dz^3, \ 2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\ ℄", 8, 2, A3 ); gap> N := LeftPresentation( nmat ); gap> filt := PurityFiltration( N ); 44 MOHAMED BARAKA T -1: -2: -3: of > gap> II_E := SpetralSequene( filt ); gap> Display( II_E ); The assoiated transposed spetral sequene: a homologial spetral sequene at bidegrees [ [ 0 .. 3 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * . * * * . . * * . . . * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s . . . . . . . . . . . . . . . Now the spetral sequene of the biomplex: a homologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 3 ℄ ℄ --------- SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 45 Level 0: * * * * . * * * . . * * . . . * --------- Level 1: * * * * . * * * . . * * . . . . --------- Level 2: s . . . . s . . . . s . . . . . gap> m := IsomorphismOfFiltration( filt ); gap> IsIdentialObj( Range( m ), N ); true gap> Soure( m ); gap> Display( last ); Dx,-1/3,-2/9*x, 0, Dy, -1/3, 0, Dx, 1, 0, 0, Dz, 0, 0, Dy, 0, 0, Dx Cokernel of the map R^(1x6) --> R^(1x3), ( for R := Q[x,y,z℄ ) urrently represented by the above matrix gap> Display( filt ); Degree 0: 0 ---------- Degree -1: 46 MOHAMED BARAKA T Q[x,y,z℄/< Dx > ---------- Degree -2: Q[x,y,z℄/< Dy, Dx > ---------- Degree -3: Q[x,y,z℄/< Dz, Dy, Dx > gap> Display( m ); 1, 1, -3*Dz-3, -3*Dz, -3*Dz^2+3*Dx-3*Dz,-3*Dz^2 the map is urrently represented by the above 3 x 2 matrix Example B.5 (Cohomologial GrothendiekSpetralSequen e ) . Example B.1 on tin- ued. Compute the T or - Ext sp etral sequene for the triple F := − ⊗ W , G := Hom( − , D ) , and the D -mo dule W . This is an example for Subsetion 9.2.1 . gap> F := InsertObjetInMultiFuntor( Funtor_TensorProdut, 2, W, "TensorW" ); gap> G := LeftDualizingFuntor( Qxyz );; gap> II_E := GrothendiekSpetralSequene( F, G, W ); gap> homalgRingStatistis(Qxyz); re( BasisOfRowModule := 110, BasisOfColumnModule := 16, BasisOfRowsCoeff := 50, BasisOfColumnsCoeff := 60, DeideZeroRows := 241, DeideZeroColumns := 31, DeideZeroRowsEffetively := 51, DeideZeroColumnsEffetively := 63, SyzygiesGeneratorsOfRows := 184, SyzygiesGeneratorsOfColumns := 63 ) gap> Display( II_E ); The assoiated transposed spetral sequene: a ohomologial spetral sequene at bidegrees [ [ 0 .. 3 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * * * * * . * * * . . * * SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 47 --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spetral sequene of the biomplex: a ohomologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 3 ℄ ℄ --------- Level 0: * * * * * * * * . * * * . . * * --------- Level 1: * * * * * * * * . * * * . . . * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s . s s s . . s * 48 MOHAMED BARAKA T . . . s --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpetralSequene( II_E, 0 ); -2: -1: 0: of > gap> ByASmallerPresentation( filt ); -2: -1: 0: of > gap> m := IsomorphismOfFiltration( filt ); Example B.6 ( T or - Ext sp etral sequene) . Here w e ompute the T or - Ext sp etral se- quene of the biomplex B := Hom( P W , D ) ⊗ P W . This is an example for Subsetion 9.2.2 . gap> P := Resolution( W ); gap> GP := Hom( P ); gap> FGP := GP * P; gap> BC := HomalgBiomplex( FGP ); gap> p_degrees := ObjetDegreesOfBiomplex( BC )[1℄; [ 0 .. 3 ℄ gap> II_E := SeondSpetralSequeneWithFiltration( BC, p_degrees ); SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 49 gap> homalgRingStatistis(Qxyz); re( BasisOfRowModule := 109, BasisOfColumnModule := 1, BasisOfRowsCoeff := 48, BasisOfColumnsCoeff := 0, DeideZeroRows := 190, DeideZeroColumns := 1, DeideZeroRowsEffetively := 49, DeideZeroColumnsEffetively := 0, SyzygiesGeneratorsOfRows := 166, SyzygiesGeneratorsOfColumns := 2 ) gap> Display( II_E ); The assoiated transposed spetral sequene: a ohomologial spetral sequene at bidegrees [ [ 0 .. 3 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * * * * * * * * * * * * * --------- Level 1: * * * * . . . . . . . . . . . . --------- Level 2: s s s s . . . . . . . . . . . . Now the spetral sequene of the biomplex: a ohomologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 3 ℄ ℄ --------- Level 0: * * * * * * * * * * * * 50 MOHAMED BARAKA T * * * * --------- Level 1: * * * * * * * * * * * * * * * * --------- Level 2: * * s s * * * * . * * * . . . * --------- Level 3: * s s s . s s s . . s * . . . s --------- Level 4: s s s s . s s s . . s s . . . s gap> filt := FiltrationBySpetralSequene( II_E, 0 ); -2: -1: 0: of > gap> ByASmallerPresentation( filt ); -2: -1: 0: SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 51 of > gap> m := IsomorphismOfFiltration( filt ); Example B.7 ( CodegreeOfPurity) . F or t w o torsion-free D -mo dules V and W of rank 2 ompute the three homologial in v arian ts • pro jetiv e dimension, • A uslander 's degree of torsion-freeness, and • o degree of purit y men tioned in Subsetion 9.1.5 are omputed. The o degree of purit y is able to distinguish the t w o mo dules. gap> vmat := HomalgMatrix( "[ \ 0, 0, x,-z, \ x*z,z^2,y,0, \ x^2,x*z,0,y \ ℄", 3, 4, Qxyz ); gap> V := LeftPresentation( vmat ); gap> wmat := HomalgMatrix( "[ \ 0, 0, x,-y, \ x*y,y*z,z,0, \ x^2,x*z,0,z \ ℄", 3, 4, Qxyz ); gap> W := LeftPresentation( wmat ); gap> Rank( V ); 2 gap> Rank( W ); 2 gap> ProjetiveDimension( V ); 2 gap> ProjetiveDimension( W ); 2 gap> DegreeOfTorsionFreeness( V ); 1 gap> DegreeOfTorsionFreeness( W ); 1 52 MOHAMED BARAKA T gap> CodegreeOfPurity( V ); [ 2 ℄ gap> CodegreeOfPurity( W ); [ 1, 1 ℄ gap> filtV := PurityFiltration( V ); -1: -2: of > gap> filtW := PurityFiltration( W ); -1: -2: of > gap> II_EV := SpetralSequene( filtV ); gap> Display( II_EV ); The assoiated transposed spetral sequene: a homologial spetral sequene at bidegrees [ [ 0 .. 2 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * * * * * * . * * --------- Level 1: * * * . . . . . . . . . SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 53 --------- Level 2: s . . . . . . . . . . . Now the spetral sequene of the biomplex: a homologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 2 ℄ ℄ --------- Level 0: * * * * * * * * . * * * --------- Level 1: * * * * * * * * . . * * --------- Level 2: * . . . * . . . . . * * --------- Level 3: * . . . . . . . . . . * --------- Level 4: . . . . . . . . . . . s gap> II_EW := SpetralSequene( filtW ); 54 MOHAMED BARAKA T gap> Display( II_EW ); The assoiated transposed spetral sequene: a homologial spetral sequene at bidegrees [ [ 0 .. 2 ℄, [ -3 .. 0 ℄ ℄ --------- Level 0: * * * * * * . * * . . * --------- Level 1: * * * . . . . . . . . . --------- Level 2: s . . . . . . . . . . . Now the spetral sequene of the biomplex: a homologial spetral sequene at bidegrees [ [ -3 .. 0 ℄, [ 0 .. 2 ℄ ℄ --------- Level 0: * * * * . * * * . . * * --------- Level 1: * * * * . * * * . . . * --------- Level 2: SPECTRAL FIL TRA TIONS VIA GENERALIZED MORPHISMS 55 * . . . . * . . . . . * --------- Level 3: * . . . . . . . . . . * --------- Level 4: . . . . . . . . . . . s An alternativ e title for this w ork ould ha v e b een "Squeezing sp etral sequenes". Referenes [AB69℄ Maurie Auslander and Mark Bridger, Stable mo dule the ory , Memoirs of the Amerian Mathe- matial So iet y , No. 94, Amerian Mathematial So iet y , Pro videne, R.I., 1969. 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