Higher Extension Modules and the Yoneda Product

HIGHER EXTENSION MODULES AND THE YONED A PR ODUCT MOHAMED BARAKA T & BARBARA BREMER Abstract. A chain of c submo dules E =: E 0 ≥ E 1 ≥ · · · ≥ E c ≥ E c +1 := 0 gives rise to c comp osable 1-co cycles in Ext 1 ( E i − 1 /E i , E i /E i +1 ), i = 1 , . . . , c . In this pap er we follow the converse question: When are c comp osable 1-co cycles induced by a module E together with a chain of submo dules as ab o ve? W e ca ll such mo dules c -extension mo dules. The c ase c = 1 is the classica l corresp ondence b et ween 1- extensions and 1- co cycles. F or c = 2 w e prov e an e x istence theor em stating that a 2-ex tension mo dule exists for tw o compo sable 1-co cycles η M L ∈ Ext 1 ( M , L ) and η L N ∈ Ext 1 ( L, N ), if and only if their Yoneda pr o duct η M L ◦ η L N ∈ Ext 2 ( M , N ) v anishes. W e further prov e a mo delling theorem for c = 2: In case the set of all s uch 2-ex tension mo dules is non-empt y it is an affine space mo delled ov er the ab elian g roup that w e call the fir s t ex tension gro up of 1-co cycles, Ext 1 ( η M L , η L N ) := Ext 1 ( M , N ) / (Ext 0 ( M , L ) ◦ η L N + η M L ◦ Ext 0 ( L, N )). 1. Intr oduction Let D b e a ring with one. W e will not restrict this condition fur t her in the pap er, only for the examples w e need D to b e a ring o v er whic h one can effectiv ely compute extension groups. W e are in terested in the computational problem of (re)constructing a (left) D - module from g iven subfactor mo dules: A chain of c submo dules E =: E 0 ≥ E 1 ≥ · · · ≥ E c ≥ E c +1 := 0 giv es r ise to c + 1 subfactor mo dules E i /E i +1 , f o r i = 0 , . . . , c and we w an t to describ e the additional data necessary to reconstruct E out of t he se c + 1 subfactors. Since suc h a c hain describes a m ultiple extension pro ces s, w e call the mo dule E together with its c hain of submo dules a c -extension mo dule . This definition app ears in Section 4 together with t he adequate notio n of equiv alence. It is ob vious that the da t a needed to reconstruct a c hain of length c con tains the corre- sp onding data of all its sub c hains. F or example, eac h pair of success ive subfactors leads to the subchain E i /E i +2 ≥ E i +1 /E i +2 ≥ 0, whic h corresp onds to the short exact sequence 0 ← E i /E i +1 ← E i /E i +2 ← E i +1 /E i +2 ← 0. Th us, one is lead to consider the problem for c = 1 first. So let E ≥ N ≥ 0 b e a 1-extension mo dule with corresp onding short exact sequence 0 ← M ← E ← N ← 0 for M := E / N . Suc h short exact sequence s with fixed factor mo dule M and submo dule N are classified b y the first extension group 1 Ext 1 ( M , N ). Its elemen ts are called 1 -extension c o cycles . In par t icular, for the case c = 1, the 1-co cycles in Ext 1 ( M , N ) are precisely t he data describing how to put M on top of Date : 2 008. 1 F or simplicity we will ommit the subscript D in E xt c D ( M , N ). 1 2 MOHAMED BAR AKA T & BAR B AR A BREMER N to reconstruct E . This is part of the classical Yoneda equiv alence which gives an alternativ e w a y of describing the extension gro ups Ext c ( M , N ) as the set of all exact se- quences 0 ← M ← G 0 ← · · · ← G c − 1 ← N ← 0 up to an a ppropriate equiv alence relation (cf. App endix B ). T urning back to our original problem one observ es that a ny c -extension mo dule induces suc h an exact sequence with G i := E i /E i +2 . How ev er, unlik e the case c = 1, E 0 = E is not one of the mo dules in the exact seque nce and the Yoneda equiv alence cannot b e exploited in the same w ay for c > 1 as for c = 1. W e no w set L i := E i /E i +1 to simplify the nota tion a nd pro ceed inductiv ely b y making use of the data describing the sub c hains 0 ← L i ← G i ← L i +1 ← 0 of length 1, namely their 1-co cycles η L i L i +1 ∈ Ext 1 ( L i , L i +1 ). In other w ords, w e wan t to classify all c -extension mo dules E with c + 1 prescrib ed subfactors L i together with a 1-co cycle η L i L i +1 for eac h consecutiv e pair ( L i , L i +1 ). One first observ es that not eve ry c ho ice of suc h 1-co cycles leads to a c - e xtension mo dule , so it is natural to ask ab out the necessary and sufficien t conditions for suc h a mo dule with a prescribed c - tuple ( η L 0 L 1 , . . . , η L c L c +1 ) to exist. T o describ e a necessary conditio n we recall the fact that t w o compatible short exact sequence s η L i L i +1 : 0 ← L i ← G i ← L i +1 ← 0 and η L i +1 L i +2 : 0 ← L i +1 ← G i +1 ← L i +2 ← 0 can b e spliced together to a 2-extension 0 ← L i ← G i ← G i +1 ← L i +2 ← 0, whic h we will denote by η L i L i +1 ◦ η L i +1 L i +2 . In Section 5 we pro ve an existence theorem stating that a 2-extension mo dule exists, if an d onl y if η L 0 L 1 ◦ η L 1 L 2 is trivial as a 2- ex tension. Then it is easy to isolate the v anishing of η L i L i +1 ◦ η L i +1 L i +2 for all consecutiv e pairs as a necessary condition for a c -extension mo dule to exist (cf. Section 6 ). Example 8.4 sho ws that this condition is not sufficien t for c > 2. Although the Yoneda equiv alence cannot b e exploited directly for c > 1 as for c = 1 as explained ab o ve , it still prov es crucial: As a natura l transformation it also provides a wa y to rein terpret the morphisms in t he lo ng exact Ext-sequences using the so-called Yoneda pr o duct . This is the con ten t of [ ML63 , Theorem I I I.9.1] in MacLane ’s bo ok, whic h we will recall in Section 3 . It will pro vide the final step fo r the pro of of the existence theorem in Section 5 . In Section 7 we will express the in trinsic approac h used so far in a w ell-kno wn setup whic h mak es explicit computations p ossible. Using this explicit language we were able to pro v e a mo delling theorem for the case c = 2 whic h concluded Section 5 : In case the set of all 2-extension mo dules is no n-empt y , it is an a ffine space mo delled o v er the ab elian group Ext 1 ( η M L , η L N ) := Ext 1 ( M , N ) / (Ext 0 ( M , L ) ◦ η L N + η M L ◦ Ext 0 ( L, N )), whic h w e call the first extension gr oup of 1 -c o cycles . The explict language of Section 7 allows us to iden tify the set o f all c -extension mo dules for prescrib ed 1- cocycles ( η L 0 L 1 , . . . , η L c L c +1 ) as the solution space of a system of equations ov er the ring D . The cen tral observ ation of the whole Section is that it is p ossible to isolate a certain subsys tem, whic h can b e solv ed independen tly . The solution of this subsystem has an indep enden t meaning as it corresp onds to computing a lift of the first co cycle η L 0 L 1 . The system is affine for c = 2 a nd this observ atio n shows tha t it is also triangular. F o r c = 3 the system is quadratic, but due to the observ a tion it can b e reduced to solving t w o HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 3 affine systems. This puts us in the p osition to easily construct the coun ter-example 8.4 men tioned ab o ve . F or c > 3 the system is still quadratic and we w ere not able to reduce it further. Section 2 is provid ed to fix the notation rather t ha n exp osing the standard mat erial whic h can b e found in [ HS97 , ML63 , W ei ], for example, and will b e summarized in the App endix . W e are not aw are o f direct con tributions to the existence problem of higher extension mo dules in t his general frame w ork. Nev ertheless, t he w ork [ vdPR05 ] of v an de r P ut and Reversa t on the Galois theory of q -difference equations addresses the pro ble m from a more algebraic geometric p oin t of view. The connection to their approac h will b e sub ject of future joint work with v an de r Put . In computational group theory multiple extensions ha v e b een crucial in Ples ken ’s soluble quotient a lgorithm [ Ple87 ]. W e wan t to emphasize that w e we re guided by examples computed using homalg [ BR ], whic h has b een extende d b y the second author in her Diploma Thesis [ Bre08 ] to prov ide pro cedures for the computation of Yoneda pro ducts and the Yoneda equiv alence b e- t w een c -extensions and c -co cycles. Based on the results of this pap er homalg provides pro cedures to compute c -extension mo dules fo r c = 2 , 3 (ov er computable comm utative rings) if they exist. Examples of suc h computations are give n in Section 8 . The details of these computations and the thesis of the second author will app ear on the homepage of homalg [ BR08 ]. In recen t years it b ecame clear (see for example [ Ob e90 , Fli90 , Mou95 , Zer00 ], to name a f e w) how lo cal 2 linear con trol theory can b e rephrased in the language of mo dules ov er v ar ious sp ecific rings, allo wing an extensiv e use of the homolog ical mac hinery in [ Qua99 , PQ99 , CQR05 , QR08 ] for example. The results in this pap er should b e useful to analyse but also to construct con trol systems with sp ecific prop erties . Finally , the reader is encouraged to follow the line of argumen ts o n t he simple Exam- ple 8.1 . It is imp ortan t to note that w e a pply morphisms of left mo dules from the right. This leads to the use of t he row con v ention for matrices. 2. c -Extensions and the Yoned a Composite F o r t w o D -mo dules M , N and a natura l n um b er c , a c -extension of N with M is an exact sequence start ing at N and ending in M and running t hro ugh c in termediate mo dules. 0 M o o G 0 o o . . . o o G c − 1 o o N o o 0 . o o 2 As opp osed to what we call glob al control theory , where bo undary conditio ns must b e considered. 4 MOHAMED BAR AKA T & BAR B AR A BREMER Motiv a t ed by the homomorphism theorem one can illustrate this exact sequence by indi- cating maps b et w een the submo dule lattices of the differen t G i ’s: (Staircase) M o o G 0 • o o L 1  o o · · · o o o o G c − 1  L c − 1 • o o o o N o o W e abreviate the sequenc e by writing 0 ← M ← ( G i ) ← N ← 0. A 1-extension is nothing but a short exact sequence . A c -extension G : 0 ← M ← ( G i ) ← L ← 0 and a c ′ -extension G ′ : 0 ← L ← ( G ′ j ) ← N ← 0 ma y b e splic e d t ogether b y taking the comp osite map G c − 1 L o o G ′ 0 o o i i to giv e a ( c + c ′ )-extension of M with N : 0 ← M ← ( G i ) ← ( G ′ j ) ← N ← 0 This is called the Yoneda c omp o s i te of G and G ′ and denoted G ◦ G ′ . The Yoneda e quivalenc e Y on : Ext → YExt establishes the one-to- one correspo nde nce b et w een elemen ts of the c - th extension group and c -extensions ( cf. App endices B , C ). Elemen ts of Ext c ( M , N ) are called c -extension c o cycles and w e call them c -c o cycles for short. Now one can use the Yoneda equiv a len ce to carry the Yoned a comp osite ov er to co cycles . This is called the Yone d a pr o duct o f c o cycles : η ◦ η ′ := Y on − 1 (Y on( η ) ◦ Y o n( η ′ )) , η ∈ Ext c ( M , L ) and η ′ ∈ Ext c ′ ( L, N ) . A shorter wa y to compute the Yoneda pro duct can b e found in App endix D . W e further define L i := cok er ( G i ← G i +1 ) for i = 0 , . . . , c − 1, with G c := N =: L c (see the ( Staircase ) diagram ab o v e). Then, the c -extension G : 0 M o o G 0 o o · · · o o G c − 1 o o N o o 0 o o is the Yoneda comp osite of the c short exact sequences 0 ← L i ← G i ← L i +1 ← 0 for i = 0 , . . . , c − 1. W e denote the corr esp onding 1-co cycles by η L i L i +1 . 3. The Yoneda Product and the Connecting Homomo rphis m Let η M N b e the corr e sp onding co cycle to the short exact sequence 0 ← M π G ← − G ι G ← − N ← 0 and L a no ther mo dule. Then t he sequences · · · Ext c − 1 ( L, M ) δ c − 1 / / Ext c ( L, N ) ι G ∗ / / Ext c ( L, G ) π G ∗ / / Ext c ( L, M ) δ c / / Ext c +1 ( L, N ) · · · HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 5 and · · · Ext c − 1 ( N , L ) ∆ c − 1 / / Ext c ( M , L ) π G ∗ / / Ext c ( G, L ) ι G ∗ / / Ext c ( N , L ) ∆ c / / Ext c +1 ( M , L ) · · · are exact. They start at the left with 0 → Hom( L, N ) = Ext 0 ( L, N ) and with 0 → Hom( M , L ), resp ectiv ely , and con tinue to the righ t for all c ≥ 0. The connecting homomo- morphisms turn out to c oincide with the Yoned a pro ducts with η M N : δ c : Ext c ( L, M ) → Ext c +1 ( L, N ) , η L M 7→ η L M ◦ η M N , ∆ c : Ext c ( N , L ) → Ext c +1 ( M , L ) , η N L 7→ η M N ◦ η N L . This is the conten t of [ ML63 , Theorem I I I.9.1]. The idea is simply the univ ersalit y of b oth the connecting homomorphism and the Yoneda pro duct. F urthermore ι G ∗ ( − ) = − ◦ ι G π G ∗ ( − ) = − ◦ π G and ι G ∗ ( − ) = ι G ◦ − π G ∗ ( − ) = π G ◦ − F o r a k -co cycle η M N ∈ Ext k ( M , N ) one can a ls o define the so called iter ate d c onne cting homomorphism s using the Yoneda pr o duct: δ c k : Ext c ( L, M ) → Ext c + k ( L, N ) , η L M 7→ η L M ◦ η M N , ∆ c k : Ext c ( N , L ) → Ext c + k ( M , L ) , η N L 7→ η M N ◦ η N L . 4. c -Extension Modules W e call a mo dule E together with a chain of c submo dules E ≥ E 1 ≥ · · · ≥ E c ≥ 0 a c -extension m o dule 3 . F o r a c - ex tension mo dule E =: E 0 ≥ E 1 ≥ · · · ≥ E c ≥ E c +1 := 0, c ≥ 1 , w e denote by G ( E ) the c -extension G ( E ) : 0 M o o G 0 o o · · · o o G c − 1 o o N o o 0 , o o where M := E 0 /E 1 , G i := G i ( E ) := E i /E i +2 and N := E c , together with the natural maps E i /E i +2 → E i − 1 /E i +1 . W e sa y that a c -extension G ∈ YExt c D ( M , N ) is induced by a c - e xtension mo dule, if G = G ( E ) for some c -extension mo dule E : (ExtMo d) M o o G 0 • • o o o o • · · · o o  o o • E 1                           E · · · o o G c − 1 • • o o o o N •  E c o o • 3 This is a s pecial case of a filtered mo dule ov er a ring with the trivial filtration D = D 0 = D i . 6 MOHAMED BAR AKA T & BAR B AR A BREMER F o r c comp osable 1-extensions 0 ← L i ← G i ← L i +1 ← 0, with corresp onding 1-co cycles η L i L i +1 , we define the set o f admissible c -extension mo dules ExtMo d( η L 0 L 1 , . . . , η L c − 1 L c ) := { E c - ex tension mo dule | η L i L i +1 corresp onds to G i ( E ) f o r all i } / ≈ , where E and E ′ are equiv alen t ( E ≈ E ′ ), if there exists an isomorphism α : E → E ′ , suc h that α induces isomorphisms from E i on to E ′ i for all i . F or a c -extension mo dule in ExtMod ( η L 0 L 1 , . . . , η L c − 1 L c ) we o ccasionally use the terminology ( η L 0 L 1 , . . . , η L c − 1 L c ) -extension mo dule to emphasize the dep ende ncy on the 1- ex tension co cycles . Definition 4.1 (Rigid tuple of 1- cocycles) . W e call a c -t uple ( η L 0 L 1 , . . . , η L c − 1 L c ) o f Yoned a - comp osable 1- cocycles rigid , if | ExtMo d( η L 0 L 1 , . . . , η L c − 1 L c ) | = 1. In this case we also call the unique ( η L 0 L 1 , . . . , η L c − 1 L c )-extension mo dule rigid . In t he Section 8 w e will provide examples for b oth rigid and non-rigid extension mo dules. The natura l problem that arises is t o describ e the set ExtMo d( η L 0 L 1 , . . . , η L c − 1 L c ) and in particular to find necessary and sufficien t conditions fo r it to b e non-empty . 5. The Case of 2 - Extension Modules Since for c = 1 the notion o f 1- e xtensions and 1-extension mo dules coincide, the first in teresting case is c = 2. M o o G 0 • o o L ∼ =  o o G 1  L • o o o o N o o Here w e fo cus on the case of 2-extensions. Let G : 0 ← M ← G 0 ← G 1 ← N ← 0 with corresponding 2- c o cycle η G b e the compo s ite of 0 ← M π G 0 ← − − G 0 ι G 0 ← − − L ← 0 and 0 ← L π G 1 ← − − G 1 ι G 1 ← − − N ← 0 for L := L 1 . Let η M L and η L N denote the corresp onding 1-co cycles : η M L ◦ η L N = η G . The short exact sequence 0 ← L ← G 1 ← N ← 0 and the cov ariant functor Hom( M , − ) giv e rise to t he long exact sequence · · · Hom( M , L ) δ 0 / / Ext 1 ( M , N ) ι G 1 ∗ / / Ext 1 ( M , G 1 ) π G 1 ∗ / / Ext 1 ( M , L ) δ 1 / / Ext 2 ( M , N ) · · · The existence of a ( η M L , η L N )-extension mo dule is equiv alen t to the existence of a 1- cocycle η M G 1 ∈ Ext 1 ( M , G 1 ) with π G 1 ∗ ( η M G 1 ) = η M L : Eac h suc h eleme nt η M G 1 induces a 1-extension Y on( η M G 1 ) = 0 ← M ← E ← G 1 ← 0 suc h that the induced 1-extension 0 ← M ← E / N ← G 1 / N ← 0 corresp onds to η M L . Th us E is an extension mo dule in ExtMo d( η M L , η L N ). Due to the ex actness of the ab o v e long exact sequence the existence of η M G 1 is in turn HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 7 equiv alen t to δ 1 ( η M L ) = 0 ∈ Ext 2 ( M , N ). But since δ 1 ( η M L ) = η M L ◦ η L N (cf. Section 3 ) and η M L ◦ η L N = η G one concludes: Theorem 5.1 (Existence Theorem) . ExtMod ( η M L , η L N ) 6 = ∅ , if and only if the Yoneda pr o duct η M L ◦ η L N ∈ Ext 2 ( M , N ) vanishes. In other w ords, a 2-extension G : 0 ← M ← G 0 ← G 1 ← N ← 0 is induced b y a 2-extension mo dule, if and only if G = 0 ∈ YExt 2 ( M , N ). The ab ov e pro of pr ovides a surjection  π G 1 ∗  − 1 ( η M L ) ։ ExtMod ( η M L , η L N ) from the fib er of π G 1 ∗ o v er η M L (recall G 1 = Y on( η L N )) onto ExtMo d( η M L , η L N ). F urt her, the long exact Ext-sequence induces an action of Ext 1 ( M , N ) on  π G 1 ∗  − 1 ( η M L ) giv en b y Ext 1 ( M , N ) ×  π G 1 ∗  − 1 ( η M L ) →  π G 1 ∗  − 1 ( η M L ) : ( η , η M G 1 ) 7→ ι G 1 ∗ ( η ) + η M G 1 = η ◦ ι G 1 + η M G 1 . Because of the exactness of t he ab o v e Ext-sequenc e at Ext 1 ( M , G 1 ) this affine action is transitiv e. The k ernel of the action is the ke rnel of ι G 1 ∗ , whic h coincides due to the exactness at Ext 1 ( M , N ) with the image of δ 0 : im δ 0 = Hom( M , L ) ◦ η L N := { ϕ M L ◦ η L N | ϕ M L ∈ Hom( M , L ) } , where the first equalit y w as established in Section 3 . Hence, the ab o ve action of Ext 1 ( M , N ) turns t he fib er  π G 1 ∗  − 1 ( η M L ) in to a principal ho mo g en eous space for Ext 1 ( M , N ) / (Hom( M , L ) ◦ η L N ) . Dually , the short exact sequence 0 ← M ← G 0 ← L ← 0 and the contra v aria n t functor Hom( − , N ) giv e rise to the long exact sequence · · · Hom( L, N ) ∆ 0 / / Ext 1 ( M , N ) π G 0 ∗ / / Ext 1 ( G 0 , N ) ι G 0 ∗ / / Ext 1 ( L, N ) ∆ 1 / / Ext 2 ( M , N ) · · · with ∆ 1 ( η L N ) = η M L ◦ η L N (cf. Section 3 ). This leads to a second surjection ( ι G 0 ∗ ) − 1 ( η L N ) ։ ExtMod ( η M L , η L N ) . Ext 1 ( M , N ) acts affinely on ( ι G 0 ∗ ) − 1 ( η L N ) via Ext 1 ( M , N ) × ( ι G 0 ∗ ) − 1 ( η L N ) → ( ι G 0 ∗ ) − 1 ( η L N ) : ( η , η G 0 N ) 7→ π G 0 ∗ ( η ) + η G 0 N = π G 0 ◦ η + η G 0 N , with kernel of action b eing im ∆ 0 = η M L ◦ Hom( L, N ) . This turns the fib er ( ι G 0 ∗ ) − 1 ( η L N ) into a principal homogeneous space for Ext 1 ( M , N ) / ( η M L ◦ Hom( L, N )) . 8 MOHAMED BAR AKA T & BAR B AR A BREMER The t w o surjections motiv ate the follo wing theorem f o r whic h a “co ordinate” pro of is pro vided in Subsection 7.2 . Theorem 5.2 (Mo delling Theorem) . ExtMo d( η M L , η L N ) is a p rincip al homo gene ous sp ac e 4 for the ab elian gr oup Ext 1 ( η M L , η L N ) := Ext 1 ( M , N ) / (Hom ( M , L ) ◦ η L N + η M L ◦ Hom( L, N )) and henc e ExtMo d( η M L , η L N ) ∼ =  π G 1 ∗  − 1 ( η M L ) / ( η M L ◦ Hom( L, N )) ∼ = ( ι G 0 ∗ ) − 1 ( η L N ) / (Hom( M , L ) ◦ η L N ) . We c al l Ext 1 ( η M L , η L N ) the first extension gr oup of 1 -c o c y cles . Corollary 5.3. A p a ir ( η M L , η L N ) of 1 -c o cycles is rigid , iff a ( η M L , η L N ) -extension mo dules exists and Ext 1 ( η M L , η L N ) = 0 . The argumen t leading to Theorem 5.1 expressed in “co ordinates” will result in a linear inhomogenous system of equations (cf. Subsection 7.2 ). In Subsections 7.3 and 7.4 w e will see how Theorem 5.1 provide s an alternativ e in terpretation of this system that ev en rev eals its t riangular structure, whic h is extremely v aluable for computations. 6. The Higher c -Extension Modules ( c ≥ 3 ) A neces sary condition for the existenc e of higher c -extension mo dules f o llo ws immediately from Theorem 5.1 : Corollary 6.1. If ExtMo d( η L 0 L 1 , . . . , η L c − 1 L c ) 6 = ∅ then the Yoneda pr o ducts η L i − 1 L i ◦ η L i L i +1 ∈ Ext 2 ( L i − 1 , L i +1 ) v a n ish f o r al l i = 1 , . . . , c − 1 . In Example 8.4 w e will pro vide an example sho wing that this condition is not sufficien t. The follo wing theorem is an ob vious generalization of the a rgumen t preceedin g Theo- rem 5.1 . It prov ides necessary and sufficien t conditions for the existence of a c -extens ion mo dule with giv en ( η L 0 L 1 , . . . , η L c − 1 L c ) b y using Theorem 5.1 a s the induction step. Corollary 6.2. T h e set ExtMo d( η L 0 L 1 , . . . , η L c − 1 L c ) is non-em pty , if and only if (1) ExtMo d( η L 1 L 2 , . . . , η L c − 1 L c ) 6 = ∅ and (2) ther e exists a ( c − 1) -extension mo dule E 1 ∈ ExtMod ( η L 1 L 2 , . . . , η L c − 1 L c ) , such that the Yoneda pr o duct η L 0 L 1 ◦ η L 1 E 2 = 0 ∈ Ext 2 ( L 0 , E 2 ) , wher e the 1 -c o cycle η L 1 E 2 is i n duc e d by E 1 . 4 Here w e do not mak e any statement ab out the ex istence of a natura l group structure for ExtMo d( η M L , η L N ), i.e. the ex is tence of a naturally disting uis hed element in the “affine” space ExtMo d( η M L , η L N ) and its uniquene s s. W e leave this for future work. HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 9 A t the end of Subsec tion 7.5 w e will use Theorem 5.1 to pro vide a simple example show ing that there may v ery w ell exist a ( c − 1) - ex tension mo dule E 1 ∈ ExtMo d( η L 1 L 2 , . . . , η L c − 1 L c ) with η L 0 L 1 ◦ η L 1 E 2 6 = 0 ∈ Ext 2 ( L 0 , E 2 ), i.e. that do es not lead to a c -extension mo dule, whereas a differen t choice of E 1 do es. This narro ws the rang e of applicability of Corollary 6.2 considerably for c > 3. This limitation will b e explained in Subsection 7.6 . The sp ecial case c = 3 will b e discussed in Subsection 7.5 . 7. The “Coordi na te ” Des cription By a “co ordinate description” of a mo dule M w e simply mean a finite fr e e pr esentation giv en by a matrix d 1 ∈ D p × q , where d 1 is view ed as a morphism of free mo dules d 1 : D 1 × p → D 1 × q . M is then the cok ernel of d 1 . d 1 is therefore called pr ese n t ation m a t rix o r matrix of r elations . This yields the b eginning of a free resolution 0 ← M ← F 0 d 1 ← − F 1 , where F 0 := D 1 × q and F 1 := D 1 × p . In most of the argumen ts used b elo w the mo dules need not b e finitely generated and one can replace the w ord matrix b y morphism or matrix of mo r phisms. In particular, the pro of of Theorem 5.2 give n in Subsection 7.2 a pplie s without the r estriction of b eing finitely presen ted. 7.1. Ext 1 in “co ordinates”. In this subsection w e recall w ell-kno wn facts ab out Ext 1 . The Yoned a correspo ndence b et w een c -co cycles and c -extension is f o r c = 1 summed up in the diag ram (cf. App e ndix C ) 0 M o o F 0 o o η 0   K 1 d 1 o o η   0 o o 0 M o o E π o o N ι o o 0 , o o whic h sho ws how to compute η b y lifting the iden tit y M id − → M twic e. Conv ersely , M is in the a bov e diagram the pushout of N η ← − K 1 d 1 − → F 0 , i.e. the cok ernel of K 1 “ d 1 η ” − − − − − − − → F 0 ⊕ N 0 @ η 0 − ι 1 A − − − − − → E → 0 . F o llo wing the notational con ve ntion in [ BR ]: If we write M for the cok ernel of t he relation matrix d 1 then we write M for d 1 . The mo dule E in the a b ov e sequence is then the cok ernel of the matr ix (recall, we use the ro w con v ention) E :=  M η 0 N  . The upp er ro w is the mor phism  d 1 η  , whereas the second ro w is a presen tation matrix for t he mo dule F 0 ⊕ N . 10 MOHAMED BAR AKA T & BAR B AR A BREMER No w we will ma ke an attempt to deriv e the defining prop erties of a 1 -cocycle (cf. Ap- p endix A ) (Ext 1 ) η ∈ Ext 1 ( M , N ) := { η : F 1 → N | 0 = d 2 η } { d 1 ϕ | ϕ : F 0 → N } in an elemen tary w a y: Recall, an extension of N b y M is described b y a mo dule E of whic h N is a submo dule and M = E / N is the facto r mo dule mo dulo N . No w w e w ant to explicitly construct suc h an E as t he cok ernel of a matrix E . W e wan t the standard basis row v ectors of the form  0 · · · 0 0 · · · 1 · · · 0  to b e the represen tativ es of generato r s o f N in the cok ernel of E , so the lo w er part of the matrix can no w b e set to  0 N  , where N is a relation matrix for N . No w computing mo dulo these v ectors  0 1  states that E / N is presen ted b y the left hand side of the matrix E . Putting a presen tation matrix M with cok ernel M in the upp er left corner (ab o v e the zeros) th us leads to a factor mo dule isomorphic to M . No w E has the form ( M η 0 N ) a nd the conditions on η remain to b e determined: N is describ ed b y al l t he relatio ns a mong the v ectors  0 1  and to isolate them one needs the most gener al row op eration matrix, whic h applied t o the upp er ro ws leads t o a zero matrix on the left hand side. This is precise ly the first syzygies matrix d 2 : F 2 → F 1 satisfying d 2 M = d 2 d 1 = 0 :  d 2 0  E =  0 d 2 η 0 N  . Th us d 2 η must not in tro duce new relations to N , whic h means d 2 η = 0 mo dulo the relations N . In other w ords d 2 η = 0 a s a morphism F 2 → N . This give s bac k the numerator of ( Ext 1 ). T o explain the denominator w e pro ceed as follo ws: W e now consider the most general invertible ro w a nd column op erations on E pr eserving the ab o ve situation, i.e. preserving the submo dule N with factor mo dule M . These op erations lead to a congruen t extension (cf. ( Cong ) in App endix B ). In particular w e w ant to prese rve the upp er triangular structur e of the matrix E , i.e. zeros in the lo w er left corner. Without lo ss o f generality (neglecting p ossible “co ordinate changes” of M and N ) one can ev en consider only those op erations that leav e the submatrices M and N in E fixed. This lea v es us out with the follo wing tw o p ossibilities :  M η 0 N  x + χ · and  M η 0 N  . + · ϕ y The row op eration only replaces η b y η + χ N , whic h do es not c hange η considered as a morphism F 1 → N . The column op eration r eplaces η b y η + M ϕ = η + d 1 ϕ , giving bac k the denominato r in ( Ext 1 ). Since w e are workin g in co ordinates w e actually ha v e to destinguish b et w een co cycles and the matrices represen ting them. In the next subsection a strict distinction in the notation will b e una voidable. So from now on w e denote a mat rix represen tativ e of a co cycle η b y ˜ η . Con v ersely , the co cycle represen ted b y a matrix ˜ η is denoted b y η = [ ˜ η ]. HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 11 7.2. ExtMo d( η M L , η L N ) in “co ordinates” ( c = 2 ). In this subsection w e provide a co ordi- nate description and pro of of Theorem 5.2 . Let M , L and N b e as in Section 5 . F ollo wing the line of arg ume nt of the previous subsec- tion one can assume that a ma t r ix of relations of a 2 - ex tension mo dule in ExtMo d( η M L , η L N ) has the upp er triangular form E :=   M ˜ η M L η · L ˜ η L N · · N   . The “co ordinate” description o f ExtMo d( η M L , η L N ) b oils dow n to classifying the admissible η ’s. F or the cok ernel of E to b e in ExtMo d( η M L , η L N ) it is necessary and suffic ient for  ˜ η M L η  to describe a co cycle in Ext 1 ( M , G 1 ) = Ext 1 (cok er ( M ) , cok er  L ˜ η L N 0 N  ) i.e. d M 2  ˜ η M L η  =  X 1 X 2   L ˜ η L N 0 N  , where d M 2 is the first syzygies matrix of M = cok er( M ). This leads to the line ar inhomo ge- nous system of equations with co effic ien ts in D : (ExtMo d 2 ) d M 2 ˜ η M L = X 1 L d M 2 η = X 1 ˜ η L N + X 2 N Equiv alen tly ,  η ˜ η L N  m ust describe a co cycle in Ext 1 ( G 0 , N ) = Ext 1 (cok er  M ˜ η M L 0 L  , cok er N ) , whic h leads to the s a me system of equations. Again, without loss of generality one can assume M , L and N fixed. First w e define the set of all matrix represen tativ es of t he fixed pair of 1-co cycle s ( η M L , η L N ) B ( η M L , η L N ) :=  ( ˜ η M L , ˜ η L N ) | [ ˜ η M L ] = η M L , [ ˜ η L N ] = η L N  . Second w e consider the disjoint union ^ ExtMo d( η M L , η L N ) := a ( ˜ η M L , ˜ η L N ) ∈ B ( η M L ,η L N ) { η | t he matrix η satisfies ( ExtMo d 2 ) for the matrices ˜ η M L , ˜ η L N } 12 MOHAMED BAR AKA T & BAR B AR A BREMER together with the obvious pro jection π : ^ ExtMo d( η M L , η L N ) ։ B ( η M L , η L N ). W e wan t to iden tify elemen ts of ^ ExtMo d( η M L , η L N ) with relation matrices of the form E =   M ˜ η M L η · L ˜ η L N · · N   . This also emphasizes the dep ende ncy of the tw o sets just in tro duced on the choice of the presen tation matrices M , L , and N . F urther w e intro duce the double-unip oten t group U :=      1 κ χ 0 1 ν 0 0 1      ×      1 µ ϕ 0 1 λ 0 0 1      , together with it s action on ^ ExtMo d( η M L , η L N ) given by U × ^ ExtMo d( η M L , η L N ) → ^ ExtMo d( η M L , η L N ) : (( r , c ) , E ) 7→ r E c − 1 . Since U is the bigg es t group fixing t he diagonal and preserving the tr ia ngular structure of E , there is a 1- 1-correspondence b et w een the set ExtMo d( η M L , η L N ) a nd t he global quotien t 5 ^ ExtMo d( η M L , η L N ) /U . Before w e pro ceed w e illustrate the action by the follo wing ro w and column o perations:   M ˜ η M L η · L ˜ η L N · · N   + χ · − · ϕ ,   M ˜ η M L η · L ˜ η L N · · N   + κ · − · λ ,   M ˜ η M L η · L ˜ η L N · · N   . + ν · − · µ Since U acts o n the set of fib ers of π it acts equiv alen tly on the set B ( η M L , η L N ) and this action turns o ut to b e transitiv e: ˜ η M L → ˜ η M L + κ L − M µ and ˜ η L N → ˜ η L N + ν N − L λ . The o rbit space ^ ExtMo d( η M L , η L N ) /U is due to the tr ansitiv ity o f the induced action on B ( η M L , η L N ) natura lly bijectiv e to the global quotien t π − 1 (( ˜ η M L , ˜ η L N )) / Stab ( ˜ η M L , ˜ η L N ) ( U ) for a n arbitrary but fixed pa ir of matrices ( ˜ η M L , ˜ η L N ) ∈ B ( η M L , η L N ). Stab ( ˜ η M L , ˜ η L N ) ( U ) is thus the larg e st subgroup which fixes the secondary dia g onal and w e conclude that Stab ( ˜ η M L , ˜ η L N ) ( U ) =    (   1 κ χ 0 1 ν 0 0 1   ,   1 µ ϕ 0 1 λ 0 0 1   ) | κ L − M µ = 0 and ν N − L λ = 0    . 5 This a rgumen t generalizes to c > 2 in the ob vio us way . HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 13 There a r e no conditions on the matr ic es ϕ and χ , while the t wo sp ecifi ed conditions are in terpreted as follow s: L λ = ν N states that λ defines a morphism in Hom( L, N ) and hence 6 ˜ η M L λ ∈ η M L ◦ Hom( L, N ) . κ L = M µ states that µ is a morphism in Hom ( M , L ) and κ is a lif t o f µ in the diagram 0 M µ   o o F 0 I rank( F 0 ) o o µ   F 1 d M 1 = M o o κ   µ ◦ η L N   4 - &     0 L o o F ′ 0 I rank( F ′ 0 ) o o F ′ 1 d L 1 = L o o ˜ η L N   N th us κ ˜ η L N ∈ Hom( M , L ) ◦ η L N . Hence, the stabilizer leads to the four op erations η → η + χ N ( χ arbitrary) η → η − M ϕ ( ϕ arbit r a ry) η → η + κ ˜ η L N ( κ lift) η → η − ˜ η M L λ ( λ morphism ) defining the c ongruenc e r elation b et w een different η ’s t hat represen t the same elemen t in ExtMo d( η M L , η L N ): η ≈ η ′ iff η − η ′ = χ N − M ϕ + κ ˜ η L N − ˜ η M L λ, where ϕ , χ, κ, λ as ab o v e . Summing up, one can iden tify ExtMo d( η M L , η L N ) with the set of all η ’s satisfying the ab o v e system of equations ( ExtMo d 2 ) mo dulo the congruence relation ≈ . Pr o of of The or em 5.2 . It is now easy to see that η 7→ η + ˜ η M N defines a natural action of Ex t 1 ( M , N ) on the set of all η satisfying ( ExtMo d 2 ) with Hom( M , L ) ◦ η L N + η M L ◦ Hom( L, N ) b eing the largest subgroup tha t acts trivially: A co cycle in Ext 1 ( M , N ) is represen ted b y a morphism η M N ∈ Hom( K 1 , N ) whic h is in turn represen ted b y a matrix ˜ η M N that fullfills the equation d M 2 ˜ η M N = X 3 N fo r some matrix X 3 (compare with the numerator of ( Ext 1 )). Th us η + ˜ η M N still satisfies the ab ov e equations ( ExtMo d 2 ), with X 2 replaced b y X 2 + X 3 . Moreo ve r, a second matrix ¯ η M N represen ting the same morphism η M N ∈ Hom( K 1 , N ) differs from ˜ η M N b y a matrix of the form χ N and hence η + ¯ η M N ≈ η + ˜ η M N . F o r Ext 1 ( M , N ) to act w e need to ve rify t hat the subgroup d 1 Hom( F 0 , N ) acts trivially (compare with the denominator o f ( Ext 1 )). But since d 1 = M , the congruence relation 6 F or the computation of the Yoneda pro duct cf. Appe ndix D . 14 MOHAMED BAR AKA T & BAR B AR A BREMER asserts that η ≈ η + M ϕ and the action of this subgroup is indeed trivial. The remaining t w o op erations η → η + κ ˜ η L N and η → η − ˜ η M L λ of the congruence relation state that the subgroup Hom( M , L ) ◦ η L N + η M L ◦ Hom( L, N ) coincides with the k ernel of the a ction of Ext 1 ( M , N ) on ExtMo d( η M L , η L N ). T o complete the pro of of Theorem 5.2 w e still need to see that Ext 1 ( M , N ) acts tran- sitiv ely on ExtMo d( η M L , η L N ). T o this end let ¯ η , ¯ X 1 , ¯ X 2 b e a second solution of the system ( ExtMo d 2 ). Then ( X 1 − ¯ X 1 ) L = ( d M 2 ˜ η M L − d M 2 ˜ η M L ) = 0 and hence there exists a Y suc h that X 1 − ¯ X 1 = Y d L 2 , where d L 2 is the first syzygies matrix of L = cok er ( L ). It follo ws that d M 2 ( η − ¯ η ) = ( X 1 − ¯ X 1 ) ˜ η L N + ( X 2 − ¯ X 2 ) N = Y d L 2 ˜ η L N + ( X 2 − ¯ X 2 ) N and since d L 2 ˜ η L N = 0 mo d N w e conclude d M 2 ( η − ¯ η ) = 0 mo d N . This means η − ¯ η is indeed a matrix represen ting a co cyc le.  Refining the congruence relations ≈ in t w o differen t w ays and fixing a pa ir ( ˜ η M L , ˜ η L N ) with [ ˜ η M L ] ∈ Ext 1 ( M , L ) and [ ˜ η L N ] ∈ Ext 1 ( L, N ) w e r ecov er the tw o fib ers fr om Section 5 : The set of all  ˜ η M L η  in terpreted as 1 -cocycles in Ext 1 ( M , G 1 ) with η satisfying ( ExtMo d 2 ) giv es bac k  π G 1 ∗  − 1 ( η M L ). This means that w e refine the congruence relation by dropping the op eration η → η − ˜ η M L λ (or equiv alen tly restricting the action to the subgroup of U defined b y λ = 0 and ν = 0 ), whic h w ould in general a lter [  ˜ η M L η  ] ∈ Ext 1 ( M , G 1 ). Similarly with the set of all [  η ˜ η L N  ] ∈ Ext 1 ( G 0 , N ) and ( ι G 0 ∗ ) − 1 ( η L N ), dropping the op eration η → η + κ ˜ η L N . W e illustrate t he tw o situations in the matrix E :     M ˜ η M L η · L ˜ η L N · · N     and     M ˜ η M L η ˜ η L N · L · · N     . 7.3. How to compute t he Yoneda pro duct of tw o 1 -co cycles. As explained in Ap- p endix D , the Yoneda pro duct of t w o 1-co cycle s η M L ∈ Ext 1 ( M , L ) and η L N ∈ Ext 1 ( L, N ) can b e computed b y lifting η M L to X 1 in the diag ram: F 1 η M L z z u u u u u u u u u u u η M L   F 2 d M 2 o o ♠ X 1   η M L ◦ η L N   4 - &     0 L o o F ′ 0 I rank( F ′ 0 ) o o F ′ 1 d L 1 = L o o η L N   N This means w e hav e to solve the D -linear inhomogenous equation d M 2 η M L = ✒✑ ✓✏ X 1 L HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 15 and then simply compute the pr o duct X 1 η L N . This is a matr ix represen ting the Yoned a pro duct of the t w o co cycles: η M L ◦ η L N = X 1 η L N . 7.4. The system ( ExtMo d 2 ) is triangular. Now w e wan t to study the solv ability of the system ( ExtMo d 2 ): d M 2 η M L = ✒✑ ✓✏ X 1 L d M 2 η = X 1 η L N + X 2 N b y trying to describ e its compatibility conditions. Using the insight of Subsection 7.3 , w e can rein terpret the system in the fo llo wing w a y: The upp er equation defines X 1 whic h alw a ys exists as the matrix of the lift (cf. Subsection 7.3 ), even though this equation is inhomogenous. F urthermore, f or any solution X 1 , the pro duct matrix X 1 η L N is a represen- tativ e of the 2- cocycle η M L ◦ η L N (cf. Subsection 7.3 ). The low er equation precisely states that the Yoned a pro duct η M L ◦ η L N is zero as a 2-co cycle, and hence the solv abilit y of the system only dep ends on the Yoneda pro duct and not on the c hoice of X 1 . The ab o v e discussion “ co ordinatizes” the pr o of of Theorem 5.1 , since it shows t ha t the v anishing of the Yoneda pro duct is expressed by the solv ability of the system. Hence, one has to admit, that a t first glance nothing is w on if one considers the v anishing of the Yoneda pro duct a s the compat ibility condition of the system 7 . But in the course of sho wing this w e discov ered the system to b e triangular 8 with respect to the shap e giv en ab o v e, i.e. the system is successiv ely solv able b y first solving the upp er equation and then the low er. This is of considerable computational v alue. Moreo v er, this triang ula r structure will play a decisiv e role for the case c = 3 in Subsection 7.5 .  It is imp ortant to note t hat the our no tion of “triangular system” o v er a ring is we ll defined only with resp ect to a giv en triangular shap e and w e wan t to emphasize that our statemen t only applies to the tria ngular shap e given ab o v e. 7.5. ExtMo d( η M L , η L K , η K N ) in “co ordina tes” ( c = 3 ) . In Coro llary 6.2 we gav e an induc- tiv e condition for the existence of a c - ex tension mo dule. This tra nslates for c = 3 into “co ordinates” as follow s: In the relation matr ix        M η M L η M K η M N · L η L K η L N · · K η K N · · · N        7 As if one would say , the system is solv able, if and only if the system is s olv able. 8 Recall that ov er a ring a triang ular shap e do es not neces sarily imply triang ular structure , cf. App en- dix E . 16 MOHAMED BAR AKA T & BAR B AR A BREMER the submatrix  η L K η L N  m ust b e a 1-co cycle in Ext 1 ( L, E 2 ) = Ext 1 (cok er( L ) , cok er  K η K N 0 N  ) ( ∗ ) and its Yoned a pro duct η M L ◦  η L K η L N  with η M L m ust v anish ( ∗∗ ). This is summed up in the quadratic system of equations (cf. Subsections 7.1 a nd 7.3 ): ( ∗ ) d L 2  η L K η L N  =  Y 1 Y 2   K η K N 0 N  ( ∗∗ ) d M 2 η M L = X 1 L d M 2  η M K η M N  = X 1  η L K η L N  +  X 2 X 3   K η K N 0 N  . Again, as in Subsection 7.4 , the middle equation is a lw ays solv able and indep enden t from the r e st. Therefore, the rest of the quadratic system is in fa c t inhomo genous l i ne ar : d L 2 η L K = Y 1 K d L 2 η L N = Y 1 η K N + Y 2 N d M 2 η M L = ✒✑ ✓✏ X 1 L d M 2 η M K = X 1 η L K + X 2 K d M 2 η M N = X 1 η L N + X 2 η K N + X 3 N . As men tioned a t the end of Section 6 w e will prov ide a w a y to build simple examples sho wing that the r est of the system is not succes siv ely solv able in the sense that the upp er t w o equations ( ∗ ), whic h define all p ossible η L N ↔ E 1 ∈ ExtMod ( η L K , η K N ) cannot b e solv ed indep end ently from the low er tw o: W e c ho ose K = (1) ∈ D 1 × 1 (i.e. K = coker( K ) = 0) whic h implies η L K = 0 and η K N = 0 as 1-co cycles. Then t he t w o succ essiv e Yoneda pro ducts η M L ◦ η L K and η L K ◦ η K N trivially v anish and therefore η M K and η L N exist. Since η M K = 0 as a 1-co cycle a n yw a y , the situation is reducible to the case c = 2:       M η M L 0 η =? · L 0 η L N · · 1 0 · · · N       ❀     M η M L η =? · L η L N · · N     . Theorem 5.1 sho ws t ha t for η t o exist the admissible c hoices of η L N dep end on the Yoned a pro duct with η M L . Cho osing η L N = 0 prov es ExtMo d( η M L , 0 , 0) 6 = ∅ . How eve r, in Example 8.3 HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 17 w e giv e tw o 1- c o cycles η M L and η L N with non-v anishing Yone d a pro duct. So starting with η M L this choice of η L N do es not lead to a solution η . 7.6. The cases c ≥ 4 in “co ordinates”. As for c = 3, the cases c ≥ 4 lead to quadratic systems . W e only demonstrate this for c = 4. Corollary 6.2 applied to the relations matrix           M η M L η M K η M H η M N · L η L K η L H η L N · · K η K H η K N · · · H η H N · · · · N           leads to t he following quadratic system of equations: d K 2 η K H = Z 1 H d K 2 η K N = Z 1 η H N + Z 2 N d L 2 η L K = Y 1 K d L 2 η L H = Y 1 η K H + Y 2 H d L 2 η L N = Y 1 η K N + Y 2 η H N + Y 3 N d M 2 η M L = ✒✑ ✓✏ X 1 L d M 2 η M K = X 1 η L K + X 2 K d M 2 η M H = X 1 η L H + X 2 η K H + X 3 H d M 2 η M N = X 1 η L N + X 2 η K N + X 3 η H N + X 4 N . The middle equation d M 2 η M L = X 1 L is as alw ay s solv able and indep enden t from the rest. An analogous argumen t to the one giv en at the end of Subsection 7.5 sho ws that in general the remaining blocks of equations (three f or c = 4) cannot be treated indep ende nt from eac h other. This still leav es us with a quadratic system. 8. Examples The followin g examples hav e b een computed using homalg [ BR ], whic h w as extended b y the second author to include the Yoneda equiv alence and the Yoned a pro duct. The detailed computations and more examples can b e found on the homepage of homalg [ BR08 ]. See also [ QR08 ] for explicit computations with 1-extension mo dules. 18 MOHAMED BAR AKA T & BAR B AR A BREMER 8.1. The most simple example ( c = 2 ). W e illustrate the mo delling theorem 5 .2 using this simple example. Let D = Z . Since D is a principal ideal ring, Ext 2 Z = 0 and the condition of t he existence theorem 5.1 is alwa ys f ullfille d. W e set M = L = N = Z / 2 Z and consider the a sso ciated relation matrix of a corresp onding 2-extension mo dule E :=   2 η M L η =? 0 2 η L N 0 0 2   . Since Hom( Z / 2 Z , Z / 2 Z ) ∼ = Ext 1 Z ( Z / 2 Z , Z / 2 Z ) ∼ = Z / 2 Z w e conclude fo r the first extension group Ext 1 ( η M L , η L N ) of 1- cocycles η M L , η L N ∈ { (0) , (1) } that Ext 1 ( η M L , η L N ) =  Z / 2 Z , if η M L = (0) a nd η L N = ( 0 ) 0 , els e. This means | ExtMo d( η M L , η L N ) | =  2 , if η M L = (0) a nd η L N = ( 0 ) 1 , els e, so only the pair ( η M L , η L N ) = ( (0) , (0)) is non-rig id. 8.2. The condition of Theorem 5.1 is sufficien t ( c = 2 ). Let D = Q [ x, y , z ]. Let M , L and N b e relation matrices for the mo dules M , L and N : M :=   x y z   , L = N :=  x 5 z  . Let 0 6 = η M L ∈ Ext 1 ( M , L ) and 0 6 = η L N ∈ Ext 1 ( L, N ) b e 1-co cycles represen ted by t he matrices ˜ η M L and ˜ η L N : ˜ η M L :=   0 x 4 0   , ˜ η L N :=  0 x  . A represen ting matrix for the Yoneda pro duct of the 1- cocycles is ^ η M L ◦ η L N =   0 x 0   , whic h is trivial in Ext 2 ( M , N ), so w e kno w t hat ExtMo d( η M L , η L N ) 6 = ∅ . Indeed, a particular solution η in ( ExtMo d 2 ) is η =   0 1 0   . and since Ext 1 ( η M L , η L N ) = 0 (ev en though Ext 1 ( M , N ) 6 = 0) we get | ExtM o d( η M L , η L N ) | = 1 and the pa ir ( η M L , η L N ) is rigid. HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 19 8.3. The condition of Theorem 5.1 is necessary ( c = 2 ) . Consider the ring D = Q [ x, y ]. Let M := ( x y ), L :=  x y  , N := ( x ), further ˜ η M L := ( 1 0 0 − 1 ) and ˜ η L N := (1). In this situation the Yoneda pro duct η M L ◦ η L N ∈ Ext 2 ( M , N ) do es not v anish and indeed the system ( ExtMo d 2 ) has no solution. 8.4. The necessary condition of Corollary 6.1 is not sufficien t for c ≥ 3 . Let D = Q [ x, y ], M := ( x y ), L := ( x y ), K := ( x 2 y ) and N := ( y ). If w e c ho ose η M L =  0 1  , η L K = ( xy 0 ) and η K N = ( x ), b oth Yoneda pro ducts η M L ◦ η L K and η L K ◦ η K N v anish. Ho w ev er, there is no sim ultaneous solution for η M K , η L N and η M N , so no 3-extension mo dule exists. Appendix A. Ext ’s as Sa tellite s Let 0 ← M d 0 ← − P 0 d 1 ← − P 1 d 2 ← − · · · d c ← − P c d c +1 ← − − P c +1 b e t he b eginning of a pro jectiv e resolution of M . A c -co cycle is b y definition a morphism η : P c → N with d ∗ c +1 ( η ) = d c +1 η = 0. I.e. η factors o ve r. ( c -Syz) P c / im( d c +1 ) = P c / k er ( d c ) d c ∼ = k er( d c − 1 ) =: K c . K c is called the c - th syzygies mo dule of M , whic h is due to Schanuel ’s Lemma uniquely defined up to pr oje ctive e quivalenc e . This establishes the w ell-known equiv alence b et w een the c - th deriv ed functor R c Hom( − , N )( M ) := def (Hom R ( P c − 1 , N ) d ∗ c − → Hom R ( P c , N ) d ∗ c +1 − − → Hom R ( P c +1 , N )) = { η : P c → N | 0 = d ∗ c +1 ( η ) := d c +1 η } { d ∗ c ϕ := d c ϕ | ϕ : P c − 1 → N } and the c - th righ t satel l i te 9 S c Hom( − , N )( M ) := cok er( Ho m R ( P c , N ) d ∗ c − → Hom R ( K c , N )) = { η : K c → N } / { d ∗ c ϕ = d c ϕ | ϕ : P c − 1 → N } (cf. [ HS97 , Section I II.2 , Prop. IV.5.8, Exercises IV.7.3 and IX.3.1] and [ CE99 , I I I.(6a),(6a’)]). In w ords, the first rig h t satellite of Hom R ( − , N ) applied to M is the ab elian group of all morphisms K c → N , mo dulo those whic h factor ov er P c − 1 (i.e. whic h extend to P c − 1 ). Appendix B. The Yoneda Composite of Extensions Tw o 1-extensions o r just e x t ensi ons G and G ′ of N with M are called c ongruent G ≡ G ′ if there exists a c hain map (id M , β 0 , id N ) : G → G ′ , i.e. a comm utativ e diag r a m (Cong) G : 0 M o o G 0 o o β 0   N o o 0 o o G ′ : 0 M o o G ′ 0 o o N o o 0 . o o 9 The rig h t satellites of a left ex a ct functor coincide with the r igh t derived functors (cf. [ CE99 , Theo- rem V.6.1] a nd [ HS97 , Prop. IV.5 .8 ]). 20 MOHAMED BAR AKA T & BAR B AR A BREMER In this case, the Fiv e Lemma shows that the middle homomorphism β 0 is an isomorphism, hence congruence o f extensions is a reflexiv e, symmetric, and transitive relation. F o r a short exact sequence G : 0 M o o G 0 o o N ι o o 0 o o and a morphism ϕ : N → N ′ the fo llowing diagram with the pushout G ′ 0 of G 0 ι ← − N ϕ − → N ′ (cf. [ HS97 , Exercise I I.9.2]) is comm utativ e: (Pushout) G : 0 M o o G 0 o o   N ι o o ϕ   0 o o G ′ : 0 M o o G ′ 0 o o N ′ o o 0 . o o Recall, the pushout is the cok ernel of N “ ι ϕ ” − − − − − − → G 0 ⊕ N ′ → G ′ 0 → 0 . The resulting exact sequence G ′ is called the Yoneda c omp osi te of G and ϕ and denoted b y Gϕ , it is unique up to congruence. Lik ewise, the Yoneda c omp osite G ′′ = ϑG with a morphism ϑ : M ′ → M is the pullbac k G ′ 0 of M ′ ϑ − → M π ← − G 0 : (Pullbac k) G : 0 M o o G 0 π o o N o o 0 o o G ′′ : 0 M ′ o o ϑ O O G ′ 0 o o O O N o o 0 . o o Recall, the pullback is the k ernel of 0 → G ′ 0 → G 0 ⊕ M ′ 0 @ π − ϑ 1 A − − − − − − → M . W rite a c -extension G as the comp osite of c short exact sequences (cf. Section 2 ): G = G 0 ◦ G 1 ◦ . . . ◦ G c − 1 . The Yoned a c omp osite of this c -extension with morphisms is defined as ( Gϕ ) Gϕ := G 0 ◦ G 1 ◦ . . . ◦ ( G c − 1 ϕ ) and ( ϑG ) ϑG := ( ϑG 0 ) ◦ G 1 ◦ . . . ◦ G c − 1 . F o r c > 1, congruence is a wider relation. W rite any c - ex tension G as the comp osite of c short exact sequences: G = G 0 ◦ G 1 ◦ . . . ◦ G c − 1 . Then congruence of c -extensions is defined as follo ws (cf. [ ML63 , Section I I I.5]): • G 0 ◦ G 1 ◦ . . . ◦ G c − 1 ≡ G ′ 0 ◦ G ′ 1 ◦ . . . ◦ G ′ c − 1 if G i ≡ G ′ i for a ll i ; HIGHER EXTENSION MODULES AN D THE YONEDA PRODUCT 21 • G 0 ◦ . . . ◦ ( G i − 1 β ) ◦ G i ◦ . . . ◦ G c − 1 ≡ G 0 ◦ . . . ◦ G i − 1 ◦ ( β G i ) ◦ . . . ◦ G c − 1 for a matc hing homomorphism β . This congruence relation defines a mo dule fo r ev ery c ≥ 1: YExt c ( M , N ) := { c -extensions of N with M } / ≡ . F o r c = 0 one sets YExt 0 ( M , N ) := Hom( M , N ). The natural equiv alence of functors Y on c : Ext c ( − , − ) / / YExt c ( − , − ) is explicitly described in App endix C . Appendix C. From c -Exte nsions to c -Cocycles and back T o a c -extension or shortly an extension of N b y M corresp onds a co cycle in Ext c ( M , N ). T o describe this we need the follo wing t wo ingredien ts, whic h b oth app eared in App e ndix A . First, the represen tation of the c -co cycle η b y a map K c → N (recall, K c is defined up to pro jectiv e equiv alence). Second, the c - ex tension G c M : 0 ← M ← P 0 ← · · · ← P c − 1 ← K c ← 0 , dep endin g only on M up to pro jectiv e equiv alence. The natura l equiv alence Y on c satisfies the defining equation G c M η = Y o n c ( η ) , where G c M η is the Yoneda comp osite defined ab o ve (see ( Gϕ )). The inv erse of Y on c is giv en b y lifting the iden tity map id M to η : K c → N : (Lift c ) 0 M o o P 0 o o η 0   · · · o o P c − 1 o o η c − 1   K c d c o o η   0 o o 0 M o o G 0 o o · · · o o G c − 1 o o N o o 0 . o o See [ HS97 , Theorem I II.2.4 ] for c = 1 and [ HS97 , Theorem IV.9.1] and [ ML63 , Theorem I II.6.4] for the general case. Appendix D . c - C ocycles and the Yoned a Pr oduct Let η M L ∈ Ext c ( M , L ) and η L N ∈ Ext c ′ ( L, N ) b e co cycles for c, c ′ ≥ 0 , 0 ← M ← P and 0 ← L ← Q pro jectiv e resolutions of M and L . Lift η M L to η as in the diagram P c η M L           η 0   P c +1 o o η 1   . . . o o P c + c ′ − 1 o o η c ′ − 1   P c + c ′ o o η   η M L ◦ η L N   4 - &     0 L o o Q 0 o o Q 1 o o . . . o o Q c ′ − 1 o o Q c ′ o o η N L   L 22 MOHAMED BAR AKA T & BAR B AR A BREMER then the comp osite of morphisms η η N L is an elemen t of Ext c + c ′ ( M , N ). This defines a pro duct Ext c ( M , L ) × Ext c ′ ( L, N ) → Ext c + c ′ ( M , N ) ( η M L , η L N ) 7→ η M L ◦ η L N := η η N L whic h is called the Yoned a pr o duct of co cyc les. Appendix E. Triangular systems over rings It is a w ell know n fact that a system in triangular shap e ov er a field can b e solv ed success iv ely . W e call systems with this feature “triangula r systems”. 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MR MR17811 75 (2001e:9 3002) 3 Lehrstuhl B f ¨ ur Ma thema tik, R WTH-Aa chen University, 52062 G e rmany E-mail addr ess : mohamed. barakat@rwth-aachen.de , barbara. bremer@rwth-aachen.de