On the K-theory of algebraic Cuntz-Pimsner rings

ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS THIBAUT LESCURE Abstract. W e establish a long exact sequence for the homotopy K-theory groups of the al- gebraic Cun tz-Pimsner rings introduced by Carlsen and Ortega [ CO11 ] b y adapting Pimsner’s original pro of [ Pim97 ] to C untz’s formal ism. 1. Introduction In [ Pim97 ], P imsner introduce d a cla ss of C ∗ -algebra s generalizing b oth crossed pro ducts by Z and graph C ∗ -algebra s. The co nstruction takes as input a co efficient C ∗ -algebra A a nd a r ight Hilber t mo dule H to g ether with a left ac tion of A on H by a djoin table op era tors. By seeing the elements of H as crea tion and annihilation op er ators on the F o ck space o ne defines a C ∗ -algebra called the T o eplitz(-Pims ne r ) algebr a T H . If I is a C ∗ -ideal o f A that ac ts on the left on H by compact op erator s, then quotienting b y a suitable ideal yields the r elative Cun tz-Pimsner algebra O H , I . Using Kaspar ov’s biv a riant K-theor y , Pimsner was a ble to build a n inv er se in K- theory to the inclusion map A → T H . This re s ult together with a computation of the induced maps K ∗ ( I ) → K ∗ ( A ) gives a long exa c t sequence o f K-theory g roups genera lizing both that of Pimsner- V oiculescu and that for graph C ∗ algebras . Carlsen and Ortega [ CO 1 1 ] introduced an a nalogue of P imsner’s constructio ns to the purely algebraic setting in which A is replaced b y a (non-unital, non-commutativ e) ring R and H is replaced by an R -sys tem X = ( X , X ′ , g ) which consists of a pair of R -bimo dules and an R - bimo dule map X ′ ⊗ R X → R . In [ CT06 ], Corti ˜ nas and Thom intro duced an algebraic analogue o f Kaspar ov’s biv a riant K-theo ry carrying the classical approach of Cunt z [ CMR07 , C97 ] to a purely discrete setting. Their theo ry is related to W eib el’s homotopy K-theor y [ W ei89 ] in the same way as op era tor K- theory is related to K asparov’s KK-theory . In the sa me pap er, they established a n algebraic analogue of the Pimsner–V o iculescu exa c t sequence for the homotopy K-theo ry gr oups of a crossed pro duct of the form R ⋊ Z . In [ ABC09 ] a long exac t sequence o f homoto p y K-theo r y groups w a s esta blished for Leavitt path alge br as of a row-finite quiv er o ver an a rbitrary ring. These t wo classes of rings are par ticular cases of the so -called algebr aic Cun tz-Pimsner ring s. W e r ewrite Pimsner’s or iginal pro of using a suitable quasi-homo morphism as an inv erse in K - theory to the inclusion R → T X . Instead of working directly with homotopy K -theor y o r its biv ariant version, we work with a given homotopy inv aria n t, split-exa ct a nd M -stable functor E : R ings → Ab (see [ CMR07 ] or [ CT06 ]). W e need to ensure that the under lying module structures are s ufficien tly non-degenera te. In particular the R -system will be assumed to satisfy condition (FS) of [ CO11 ] to ensure that the T o eplitz ring is universal. Moreover, for all R -system X and any M - stable functor E , if K R ( X ) denotes the ring of finite-ra nk o per ators then there should be a n induced map E ( K R ( X )) → E ( R ). Corner em bedding s should be sent to isomorphisms b y E . F or these rea sons, we ass ume that R has loca l units, that the R -system X sa tisfies conditio n (FS) and comes equipp ed with a functional homomorphism to a ca nonical R -s ystem R ( I ) (see Definition 2.12). Under these assumptions we are able to so lve thes e technical problems using the for malism int ro duced by Bur g staller [ Bur2 5 ]. W e then reca ll the construction of the T o eplitz a nd r e lative Cun tz-Pimsner ring s and pro c eed with the main computation. O ur first result is the following. Theorem A. L et R b e a ring with lo c al units. L et X b e an R -c orr esp ondenc e. L et T X b e the c orr esp onding T o eplitz ring. Every homotopy invaria nt, split-exact and M -stable functor E sends the inclusion R → T X to an isomorphism E ( R ) ∼ = E ( T X ) . 1 2 THIBA UT LESCURE Let ( E n ) n ∈ Z : Rings → Ab be a seq uence of homotopy in v aria nt and M -stable functors which satisfies excision. This means that any extension of rings I R R/I induces a lo ng exact sequence o f the co r resp onding ab elian g roups . . . E n ( I ) E n ( R ) E n ( R/I ) E n − 1 ( I ) . . . Using Theorem A and the fact that the Cun tz- Pimsner ring is a quotient o f the T o eplitz ring we obtain Theorem B. L et R b e a ring with lo c al units. L et X b e an R -c orr esp ondenc e. L et I ⊳ R b e a two-side d ide al t hat has lo c al un its and acts on X on the left by c omp act op er ators. Ther e is a long exact se quenc e ( n ∈ Z ) . . . E n ( I ) E n ( R ) E n ( O X , I ) E n − 1 ( I ) . . . E n ( i ) − E n ( X ) E n ( j ) Her e j : R → O X , I and i : I → R ar e t he natur al inclusions. Theorems A and B are b o th true if we r eplace the s equence o f functors ( E n ) n ∈ Z by either W eibel’s homotopy K-theory [ W ei89 ] or by per io dic cyclic ho mology [ CQ9 7 ]. 2. Preliminaries 2.1 Burgstaller’s functional mo dules and M -stable functors W e fix a (not necessarily unital or commutativ e) ring R . Definition 2.1 . A right functional R -mo dule is a triple ( X , X ′ , g ) where X is a rig ht R -mo dule, X ′ is a left R -mo dule and g : X ′ ⊗ Z X → R is a n R -bimo dule map (often called the scalar pro duct of the functional mo dule). F o r all f ∈ Hom R ( X, X ) we say that f is adjointable with adjoint f ∗ ∈ Hom R ( X ′ , X ′ ) if ∀ x ∈ X , ∀ φ ∈ X ′ , g ( φ ⊗ f ( x )) = g ( f ∗ ( φ ) ⊗ x ) W e ca ll g non-deg enerate if ( ∀ φ ∈ X ′ , g ( φ ⊗ x ) = 0) = ⇒ x = 0 and ( ∀ x ∈ X, g ( φ ⊗ x ) = 0 ) = ⇒ φ = 0 W e will write g ( φ ⊗ x ) = φ ( x ) for x ∈ X and φ in X ′ . One ea sily defines the direct sum of tw o right functional R -mo dules X = ( X , X ′ , g ) and Y = ( Y , Y ′ , h ) as X ⊕ Y = ( X ⊕ Y , X ′ ⊕ Y ′ , g + h ). The following prop osition is easy to verify . Prop ositio n 2.2. F or e ach right fun ctional mo dule X = ( X , X ′ , g ) over R , t he set of adjointable endomorphi sms of X forms a ring. If g is non-de gener ate t hen the adjoint is u nique and the adjoints satisfy ( f 1 f 2 ) ∗ = f ∗ 2 f ∗ 1 . W e denote by L R ( X ) the ring of adjointable right R -mo dule endomorphisms o n the functional mo dule X , with p oint wise sum and comp osition as o per ations. W e will always b e working with functional mo dules with non- degenerate sca lar pro duct. Hence for t wo functional R -mo dules X and Y we will often write f : X → Y fo r an R -mo dule map f : X → Y admitting an a djoint. Definition 2.3. Le t X = ( X , X ′ , g ) b e a right functional mo dule ov er R . W e define the ab elian group of compact op erator s of X to b e K R ( X ) = X ⊗ R X ′ . It is a ring with the pro duct defined by ( x 1 ⊗ φ 1 )( x 2 ⊗ φ 2 ) = x 1 ⊗ ( φ 1 ( x 2 ) · φ 2 ). K R ( X ) acts on the left on X by ( x ⊗ φ ) · y = x · φ ( y ) but also on X ′ on the right by ψ · ( x ⊗ φ ) = ψ ( x ) · φ . This g ives a ring homomor phism from K R ( X ) into the adjointable op e rators of X . Definition 2.4 . [ CO11 ] A right functional R -mo dule X = ( X, X ′ , g ) is said to satisfy condition (FS) if f or all x 1 , . . . x n ∈ X and φ 1 , . . . φ n ∈ X ′ there exist Θ 1 , Θ 2 ∈ K R ( X ) such that Θ 1 · x i = x i and φ i · Θ 2 = φ i for all 1 ≤ i ≤ n . Recall that a r ing R is said to hav e lo cal units if for an y r 1 , . . . , r n ∈ R there exis ts an idemp otent e ∈ R such that f or all 1 ≤ i ≤ n we ha ve er i = r i e = r i . Recall that a right R -mo dule M is called non-degenera te if M · R = M . If R has lo cal units then non- degeneracy is eq uiv alent to a sking that the right action map M ⊗ R R → M is bijective. ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS 3 Prop ositio n 2.5. [ CO11 ] L et X b e a right functional mo dule over R satisfying c ondition (FS). (i) The u nderlying sc alar pr o duct of X is non-de gener ate. (ii) The underlying R - mo dule structur es of X ar e non-de gener ate. (iii) K R ( X ) is emb e dde d in L R ( X ) as a two-side d ide al by the ring homomorph ism j : x ⊗ φ 7→ θ x,φ define d by θ x,φ ( y ) = x · φ ( y ) . Pr o of. Let x ∈ X be such that for all φ ∈ X ′ , φ ( x ) = 0. There exists a Θ ∈ K R ( X ) such that Θ · x = x . W e can write Θ = P i x i ⊗ φ i th us x = P i x i · φ i ( x ) = 0. Similarly if φ ∈ X ′ is such that ∀ x ∈ X, φ ( x ) = 0 then φ = 0. The formulas f ◦ θ x,φ = θ f ( x ) ,φ and θ x,φ ◦ f = θ x,f ∗ ( φ ) ensure that the image of K R ( X ) is a t wo-sided idea l of L R ( X ). Let k ∈ K R ( X ) b e such that j ( k ) = 0, i.e. k · x = 0 for all x ∈ X . W rite k = P i x i ⊗ φ i . Let e ∈ K R ( X ) b e such that φ i · e = φ i for all 1 ≤ i ≤ n . W e have k e = P i x i ⊗ ( φ i · e ) = k . W rite now e = P j y j ⊗ ψ j . Finally , k = k e = P j ( k · y j ) ⊗ ψ j = 0.  In the rest of this article we will alw ays be w o rking with functional mo dules s atisfying conditio n (FS). Hence w e will iden tify the elemen t x ⊗ φ ∈ X ⊗ R X ′ with θ x,φ ∈ L R ( X ). Example 2.6. F o r any s et I we will denote b y R ( I ) the R -bimo dule of finitely supported sequences ( r i ) i ∈ I of elements of R . W e will also denote b y R ( I ) the functional mo dule ( R ( I ) , R ( I ) , h , i R ) equipp e d with the scalar pro duct given b y h ( α i ) , ( β i ) i = P i α i β i . W e will ma ke use of the identi- fication K R ( R ( I ) ) ∼ = M I ( R ) with the r ing of matr ices with v alues in R having only a finite num b er of non-zer o v alues. When R has loc al units this functional mo dule sa tisfies condition (FS). Definition 2.7. [ Bur25 ] Let X = ( X , X ′ , g ) and Y = ( Y , Y ′ , h ) be tw o r ight functional R -mo dules. A fu nctional homomorphis m is the data of t wo R -mo dule maps U : X → Y and V : X ′ → Y ′ such that ∀ φ ∈ X ′ , x ∈ X , V ( φ )( U ( x )) = φ ( x ) If the scalar pro duct of Y is non-degenera te then the maps U and V a re injectiv e. (If U ( x ) = 0 then φ ( x ) = 0 for all φ ∈ X ′ ). Example 2.8. Assume that w e ha ve three right fun ctional R -mo dules such that Y = X ⊕P . In this case we let U : X → Y and V : X ′ → Y ′ be the inclusions asso ciated to the splittings Y = X ⊕ P and Y ′ = X ′ ⊕ P ′ . In other words if ι : X → Y and π : Y → X are the obvious inclusion and pro jection, we let U = ι and V = π ∗ . W e hav e V ( φ )( U ( x )) = ( π ∗ ( φ ))( ι ( x )) = φ ( π ( ι ( x ))) = φ ( x ). Let φ 1 , . . . φ n ∈ X ′ and y ∈ Y . V ( φ i )( y ) = ( π ∗ ( φ i ))( y ) = φ i ( π ( y )). Hence ( ι, π ∗ ) is a functional homomorphism. Example 2. 9. Let M b e a finitely ge ner ated r ight R -mo dule. Then M is pro jective if and o nly if it is a direct summand o f a right R -mo dule of the form ( eR ) n for an idempo tent e ∈ R (see [ Abr83 ]). There is a right functional R -mo dule M = ( M , M ∗ , g ), where M ∗ = Hom R ( M , R ) and g is the ev aluation ma p. W e hav e (( eR ) n ) ∗ ∼ = ( Re ) n . By a slig ht abuse of notation we denote b y ( eR ) n the right functional R -mo dule (( eR ) n , ( Re ) n , h , i ). There exists a right functional R -mo dule N such that M ⊕ N ∼ = ( eR ) n . Moreover there is an ob vious functional homomorphism ( eR ) n → R n given by the inclusions. Hence there is a functional homomorphism M → R n . Let ( U, V ) : X = ( X , X ′ , g ) → Y = ( Y , Y ′ , h ) b e a functional ho momorphism b etw een t w o rig h t functional R -mo dules. The form ula of Definition 2.7 makes the map ι X , Y = U ⊗ V : K R ( X ) → K R ( Y ) a ring homomorphism. Mo reov e r one can c heck that if X satisfies condition (FS) then ι X , Y is injective. Let Rings b e the categ ory of as s o ciative rings with ring homomorphisms. Definition 2.10. Let E : Rings → Ab b e a functor . E is called M -stable if for any ring S , any set I a nd a n y distinguished element i ∈ I the diagonal embedding σ i : S → M I ( S ) at the i -th co ordinate is sent to an iso morphism. According to [ CMR07 , Prop os ition 3.16] an y such functor is in v ariant under inner isomo rphisms, and thus E ( σ i ) do es not dep end on the choice of i . Burgstaller [ Bur25 ] recently extended this iso morphism to more g eneral corner em bedding s b y using functional homomo rphisms. F or the rea der’s convenience, w e reca ll his result to gether with its pro of. 4 THIBA UT LESCURE Prop ositio n 2.11. Le t E : Ring s → Ab b e an M -stable functor. L et R b e a ring. L et X b e a right functional R - mo dule admitting a fun ct ional homomorphism to a functional mo dule of the form R ( I ) , I b eing a set. The map induc e d by the upp er left c orner emb e dding E ( ι R,R ⊕X ) : E ( R ) → E ( K R ( R ⊕ X )) is an isomorphi sm. Pr o of. W rite D X = K R ( R ⊕ X ). Ther e is a functional homo mo rphism ( U, V ) : X → R ( I ) . Let a b e a forma l sy m bo l no t in I , write simply I ∪ a for I ∪ { a } . ( U, V ) g ives another functional homomorphism (id R ⊕ U, id R ⊕ V ) : R ⊕ X ֒ → R ( I ∪ a ) which induces a ring homomor phism ρ = ι X ,R ( I ∪ a ) : D X → M I ∪{ a } ( R ). Denote by e : R ֒ → D X the upp e r left corner embedding induced by the split embedding of functional mo dules R ֒ → R ⊕ X . Consider the following diagram M 2 ( D X ) D X M I ∪ a ( R ) M I ∪ a ( D X ) M I ∪ a ∪ I ′ ∪ a ′ ( D X ) R D X φ ρ j 1 ∼ = M I ∪ a ( e ) j 2 ∼ = e i a e ∼ = i ′ a ∼ = Here e = ι R,R ⊕X denotes the c o rner embedding R → D X . i a and i ′ a denote the embeddings int o the corner of the co ordinate lab elled a in the c o rresp onding matr ix ring. I ′ ∪ a ′ is just a copy of I ∪ a and j 2 embeds M I ∪ a ( D X ) via identification with the coo rdinates o f I ′ ∪ a ′ . j 1 is the low er right corner embedding. W e hav e the following iden tifica tions M 2 ( D X ) ∼ = K R (( R ⊕ X ) 2 )) and M I ∪ a ∪ I ′ ∪ a ′ ( D X ) ∼ = K R (( R ⊕ X ) I ∪ a ∪ I ′ ∪ a ′ ). The map φ : M 2 ( D X ) → M I ∪ a ∪ I ′ ∪ a ′ ( D X ) in the diagram ab ove is asso ciated to the following functional ho momorphisms ( R ⊕ X ) 2 R ⊕ X ⊕ R ( I ∪ a ) ( R ⊕ X ) ( I ∪ a ) ⊕ ( R ⊕ X ) ( I ′ ∪ a ′ ) id R ⊕X ⊕ id R ⊕ f α ⊕ β Here f = ( U, V ) : X → R ( I ) and α : R ⊕ X ֒ → ( R ⊕ X ) ( I ∪ a ) is the em b edding at the a -th co ordinate. β : R ( I ∪ a ) ֒ → ( R ⊕ X ) ( I ′ ∪ a ′ ) is the embedding R ֒ → R ⊕ X at each co ordinate. In the ab ov e diagra m the isomorphisms are to b e understo o d at the level of the cor resp onding a belia n groups: by M -stability the maps E ( i a ), E ( i ′ a ), E ( j 1 ), E ( j 2 ) ar e bijective. A direct co mputation pr ov es that this diagra m co mm utes ( β induces the ring homo morphism M I ∪ a ( e )). It r emains to show that E ( e ) is a n isomorphism with inverse E ( i a ) − 1 E ( ρ ). W e hav e E ( i a ) − 1 E ( ρ ) E ( e ) = id E ( R ) bec ause i a = ρe . E ( e ) E ( i a ) − 1 E ( ρ ) = E ( i ′ a ) − 1 E ( M I ∪{ a } ( e )) E ( ρ ) = E ( i ′ a ) − 1 E ( j 2 ) − 1 E ( j 2 ) E ( M I ∪{ a } ( e )) E ( ρ ) = E ( i ′ a ) − 1 E ( j 2 ) − 1 E ( φ ) E ( j 1 ) The trick is now to write E ( j 2 ) = E ( j ′ 2 ) and E ( j 1 ) = E ( j ′ 1 ) where j ′ 1 : D X ֒ → M 2 ( D X ) is the upp er left cor ner embedding and j ′ 2 : M I ∪ a ( D X ) ֒ → M I ∪ a ∪ I ′ ∪ a ( D X ) is the embedding ob- tained by identifying the co or dinates I ∪ a in I ∪ a ∪ I ′ ∪ a ′ . As φj ′ 1 = j ′ 2 i ′ a we finally g et E ( i ′ a ) − 1 E ( j 2 ) − 1 E ( φ ) E ( j 1 ) = E ( i ′ a ) − 1 E ( j ′ 2 ) − 1 E ( φ ) E ( j ′ 1 ) = id E ( D X ) .  Definition 2.12. Let R and S b e t wo rings. An R - S corr esp ondence is the data of a qua druple ( X , ∆ , U , I ) where (i) X is a right functional S -mo dule sa tisfying condition (FS) (ii) ∆ : R → L S ( X ) is a ring ho momorphism (ca lled the left action) such that the induced left R -mo dule s tructure on X and right R -mo dule s tructure on X ′ are no n-degenerate: ∆( R ) · X = X a nd X ′ · ∆( R ) = X ′ (iii) I is a set and U : X → S ( I ) is a functional homomor phism An R -corres po ndence is an R - R corr esp ondence. The notio n of cor resp ondence se r ves here a s the minimal a lgebraic analogue o f the cla ssical notion of a C ∗ -corre spo ndence. ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS 5 W e will often simply write X for the corr esp ondence ( X , ∆ , U , I ). Cho ose an R - S cor resp ondence X = ( X , X ′ , g ) with functional homomorphism U 1 : X → S ( I ) and an S - T corr esp ondence Y = ( Y , Y ′ , h ) with functional homomorphism U 2 : Y → T ( J ) . One defines the tensor pro duct X ⊗ S Y as the R - T cor r esp ondence given b y X ⊗ S Y = ( X ⊗ S Y , Y ′ ⊗ S X ′ , k ) with k defined by the for m ula ( ψ ⊗ φ )( x ⊗ y ) = ψ ( φ ( x ) · y ) for a ll x ∈ X , y ∈ Y , φ ∈ X ′ , ψ ∈ Y ′ . The functional homo mo rphism underlying X ⊗ S Y is given by X ⊗ S Y S ( I ) ⊗ S Y Y ( I ) R ( I × J ) Y ′ ⊗ S X ′ Y ′ ⊗ S S ( I ) Y ′ ( I ) R ( I × J ) U 1 ⊗ S id Y ⊕ I U 2 id Y ′ ⊗ S V 1 ⊕ I V 2 Let X b e an R - S corresp ondence. Denote b y ∆ : R → L S ( X ) the left action. Assume that R is such that ∀ r ∈ R, ∆( r ) ∈ K S ( X ). Let E : Ri ngs → Ab be an M -stable functor. As E ( ι R,R ⊕X ) is an isomor phism w e se e that X induces a ma p E ( X ) : E ( R ) → E ( S ) by the following diagr am R K S ( X ) K S ( S ⊕ X ) S ∆ ι X ,S ⊕X ι S,S ⊕X F or the applications, we will need another wa y of computing E ( X ). The functional homomor- phism of X yields the existence of a ring ho momorphism ι X ,S ( I ) : K S ( X ) → M I ( S ). As E is an M - s table functor there is a map E ( K S ( X )) → E ( S ) obtained by comp osing with the inv erse of the map induced by any corner embedding ι S,S ( I ) : S → M I ( S ). Prop ositio n 2.13. E ( ι S,S ⊕X ) − 1 E ( ι X ,S ⊕X ) = E ( ι S,S ( I ) ) − 1 E ( ι X ,S ( I ) ) . Pr o of. The following diag r am K S ( X ) K S ( S ⊕ X ) S M I ∪ a ( S ) ι X ,S ⊕X h ρ ι S,S ⊕X ι a is co mm utative. ρ = ι S ⊕X ,S ( I ∪ a ) is induced b y the splitting S ( I ∪ a ) = ( S ⊕ X ) ⊕ Y . h = ι X ,S ( I ∪ a ) is the co mp os itio n of ι X ,S ( I ) : K S ( X ) → M I ( S ) with the embedding σ = ι S ( I ) ,S ( I ∪ a ) : M I ( S ) ֒ → M I ∪ a ( S ). ι a = ι S,S ( I ∪ a ) is the em bedding at the co or dinate a . Its comm utativity yields E ( ι S,S ⊕X ) − 1 E ( ι X ,S ⊕X ) = E ( i a ) − 1 E ( h ). The diagr am K S ( X ) M I ( S ) S M I ∪ a ( S ) ι X ,S ( I ) h σ ι S,S ( I ) i a is also commut ative. Similarly , w e get E ( ι S,S ( I ) ) − 1 E ( ι X ,S ( I ) ) = E ( i a ) − 1 E ( h ).  2.2 Algebraic Cun tz-Pim sner rings In this section R is a ring with lo ca l units and I is a tw o-s ided ideal of R whic h als o admits lo cal units. Let X = ( X, X ′ , g ) be an R -corr esp ondence such that I acts on X on the left by compact op erators . This means that if ∆ : R − → L R ( X ) is the left action then ∆( I ) ⊂ K R ( X ). Let T ( X ) = L n ∈ N X ⊗ n be the N -gra ded R - R bimo dule known as the F o ck space (where the tensor pro ducts are taken ov er R ). There is an R -corres po ndence T ( X ) = ( T ( X ) , T ( X ′ ) , e g ) where e g ( φ 1 ⊗ · · · ⊗ φ n , x 1 ⊗ · · · ⊗ x m ) = 0 if n 6 = m , and e g ( φ 1 ⊗ · · · ⊗ φ n , x 1 ⊗ · · · ⊗ x n ) = φ 1  φ 2 ( · · · ( φ n ( x 1 ) x 2 ) · · · ) x n  . 6 THIBA UT LESCURE for all φ k ∈ X ′ and x i ∈ X . F or φ ∈ X ′ , x ∈ X a nd p ∈ T ( X ) a pure tensor, define T x ( p ) := x ⊗ p, T φ ( p ) := ( 0 if deg( p ) = 0 φ ( p 1 ) · p 2 ⊗ · · · ⊗ p n if p = p 1 ⊗ · · · ⊗ p n , n ≥ 1 . Then T x , T φ ∈ L R ( T ( X )) with adjoin ts given on a ll pure tenso rs ψ ∈ T ( X ′ ) by T ∗ x ( ψ ) := ( 0 if deg( ψ ) = 0 , ψ 1 ⊗ · · · ⊗ ψ n − 1 · ψ n ( x ) , if ψ = ψ 1 ⊗ · · · ⊗ ψ n T ∗ φ ( ψ ) := ψ ⊗ φ, in L R ( T ( X ′ )). Definition 2.14. The T o epl itz ring T X of X is the subring gener a ted b y the T x , T φ , r · id T ( X ) ∈ L R ( T ( X )) , φ ∈ X ′ , x ∈ X , r ∈ R . Let J X , I be the tw o -sided ideal of T X generated by I · P 0 where P 0 ∈ L R ( T ( X )) is the map that sends x to 0 if deg( x ) ≥ 1 a nd is the identit y o n R . The Cun tz-Pimsner ring of X with resp ect to the ideal I is the quotient ring O X , I = T X /J X , I . T ( X ) is an N -graded R -bimo dule. The rings T X and O X , I are Z -graded. Observe that these rings hav e lo cal units if the base ring R has lo cal units. Remark 2.15 . Be ca r eful that P 0 is no t in general a n element of T X . The reas on why I · P 0 is included in T X is beca use I acts on X b y compact op erator s . Hence for all i ∈ I there ar e x i ∈ X and φ i ∈ X ′ such that ∆( i ) = X i x i ⊗ φ i and i · P 0 = i · id T ( X ) − X i T x i T φ i ∈ T X Prop ositio n 2.16. J X , I is the ide al of c omp act op er ators of the right fun ctional R - m o dule T ( X ) · I = ( T ( X ) · I , I · T ( X ′ ) , e g ) . Pr o of. W e ca n repres e n t elemen ts of L R ( T ( X )) by (infinite) matrices by giving the action of the op erator on each subspace X ⊗ n . In pa rticular T x =       0 0 0 0 . . . x 0 0 0 . . . 0 x 0 0 . . . 0 0 x 0 . . . . . .       , T φ =       0 φ 0 0 . . . 0 0 φ 0 . . . 0 0 0 φ . . . 0 0 0 0 . . . . . .       , π 0 · i =       i 0 0 0 . . . 0 0 0 0 . . . 0 0 0 0 . . . 0 0 0 0 . . . . . .       Here the symbol x (r e sp. φ ) in the above matrix is to b e understo o d a s the op erator T x (resp T φ ) restricted to the corresp onding subspace X ⊗ n . Moreover K R ( T ( X ) · I ) = T ( X ) · I ⊗ R I · T ( X ′ ) = T ( X ) · I ⊗ R T ( X ′ ) b ecause I has lo cal units. Distributing the tenso r pro duct we get K R ( T ( X · I )) = M ( n,m ) ∈ N 2 X ⊗ n ⊗ I ⊗ ( X ′ ) ⊗ m , which is an ideal of L R ( T ( X )). W e have J X , I ⊂ K R ( T ( X ) · I ) beca us e P 0 ∈ K R ( T ( X · I )) and any matrix of the form T φ or T x m ultiplied by a finite matrix is still a finite ma trix. In addition, for any i ∈ I , p = x 1 ⊗ . . . ⊗ x n ∈ X ⊗ n , ψ = φ 1 ⊗ . . . ⊗ φ m ∈ ( X ′ ) ⊗ m :       0 . . . 0 0 . . . . . . . . . 0 0 . . . 0 ( p · i ) ⊗ ψ 0 0 . . . 0 0 0 0 . . .       = T x 1 . . . T x n ( i · P 0 ) T φ 1 . . . T φ m ∈ T X .  ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS 7 Example 2.17. Let X = ( M , M ∗ , ev) with M an R - R bimo dule that is finitely gene r ated and pro jective o n the r ight. W e have O X ∼ = lim − → k ∈ N M d ∈ Z Hom − ,R ( M ⊗ k , M ⊗ k + d ) as rings. W e take the conv ention that M ⊗ n is the z ero mo dule for neg ative n . The co limit is taken over tensoring by the identit y of M and the pro duct is g iven by the comp osition. In- deed, beca use M R is finitely generated and pro jective we hav e a n isomor phism (see [ A ´ AM87 ]) Hom − ,R ( M ⊗ k , M ⊗ l ) ∼ = M ⊗ l ⊗ ( M ∗ ) ⊗ k . If we let O b e the ring o btained a s the ab ove direct sum of colimits then there is a ring ho momorphism T X → O obtained by restriction to the mo dules M ⊗ k for sufficien tly large k . The kernel of this map is clearly J X = T ( M ) ⊗ T ( M ∗ ). If R is unital then O X coincides with the stro ng cov aria nce ring of the bimo dule M (see [ Mey ]). Example 2.18. Let X = ( R , R, µ ), I = R . Let R act on R by multiplication. Then T ( X ) = R ( N ) . Each T x acts on T ( X ) by mult iplication by x in R and by adding one to the deg ree, each T φ m ultiplies by φ and subtracts one from the degree. Mor e over J X = K R ( T ( X ) ) = M ∞ ( R ) a nd T X ∼ = R h x, y i / ( xy − 1) , O X ∼ = R [ x, x − 1 ] . More genera lly , if α is a ring a utomorphism of R and each r ∈ R acts on R o n the left by m ultiplication by r and o n the rig ht by m ultiplica tion by α ( r ) then O X ∼ = R ⋊ α Z is the cro ssed pro duct ring of R b y Z with α . The T o eplitz r ing has a universal characterization in ter ms of representations analogo us to the one for C ∗ -algebra s. Definition 2.19. [ CO11 , Definition 1.2] A cov ar iant r epresentation of ( R, X ) is a quadruple ( S, T , σ , D ) where (i) D is a ring; (ii) σ : R → D is a ring homomorphism; (iii) S : X ′ → D and T : X → D are R -bimo dule homo morphisms with respe c t to the bimo dule structure induced by m ultiplication and by σ ; (iv) ∀ φ ∈ X ′ , ∀ x ∈ X , σ ( g ( φ ⊗ x )) = S ( φ ) T ( x ). If j : R → T X is the natur a l inclus ion, T X , S X : X , X ′ → T X map resp ectively x to T x and φ to T φ then ( S, T , j, T X ) is a cov ar ia nt representation. The ring ho momorphism j is injective by no n- degeneracy . The r ing T X is g enerated by T X ( X ), S X ( X ′ ) and j ( R ). The T oeplitz ring is universal in the following sense: Theorem 2 .20. Le t R b e a ring with lo c al un its and let X b e an R -c orr esp ondenc e. If ( S, T , σ, D ) is a c ovariant re pr esentation of ( R, X ) , then ther e exists a unique ring homomorphism η : T X → D such that η ◦ j = σ , η ◦ T X = T , and η ◦ S X = S . Pr o of. See [ CO11 , T heo rem 1.7 a nd Pr op osition 4.2].  The relative algebraic Cunt z-Pimsner ring O X , I also has a universal prop erty [ CO11 , Theor em 3.18] but w e will no t use it in this article. 3. Proofs of main resul ts Let X = ( X , X ′ , g ) b e a corresp ondence ov er a r ing with loca l units R , let I b e a t wo-sided ideal of R that has lo ca l units and acts on X on the left b y c o mpact op erator s . Let E : Rings → Ab be a ho motopy in v ariant, split-exact, M -stable functor (w e refer to [ CMR07 ] a nd [ CT06 ] for the definitions of these no tions and class ical computatio ns us ing quasihomomor phisms). There is a natural short exact sequence of rings 0 − → K R ( T ( X · I )) − → T X − → O X , I − → 0 W rite T ( X · I ) = I ⊕ T 1 ( X · I ) where T 1 ( X · I ) = L n ≥ 1 ( X · I ) ⊗ n . By Prop ositio n 2.11 and M - s tabilit y of E the upp er le ft cor ner embedding e : I → K R ( T ( X · I )) induces an iso mo rphism E ( K R ( T ( X · I ))) ∼ = E ( I ) 8 THIBA UT LESCURE The ma in difficulty is th us to co mpute the v a lue o f E o n T X . W e hav e a map j : R → T X by definition of the T o eplitz ring. W e wan t to build a map that is the inv erse to j once w e a pply E . W e define a quas i-homomorphism π = T X L R ( T ( X ) ) ⊲ K R ( T ( X ) ) . π 0 π 1 by letting π 0 be the natural inclusio n and by defining π 1 to b e such that for all pure tensor s p ∈ T ( X ), π 1 ( T x )( p ) = ( 0 if deg( p ) = 0 , x ⊗ p otherwise , π 1 ( T φ )( p ) = ( 0 if deg( p ) ≤ 1 , φ ( p 1 ) ⊗ p 2 ⊗ · · · ⊗ p n otherwise , π 1 ( a )( p ) = ( 0 if deg ( p ) = 0 , a · p otherwise . One can check that b oth π 0 and π 1 preserve the pairing and th us define a ho momorphism o n T X according to Theorem 2.2 2. Lemma 3.1. F or al l τ ∈ T X , π 0 ( τ ) − π 1 ( τ ) ∈ K R ( T ( X )) Pr o of. Every element o f the T o eplitz r ing can be written as a finite sum of elements of the form τ = T p 1 . . . T p k T φ 1 . . . T φ l for k , l ≥ 0, p 1 , . . . p k ∈ X and φ 1 , . . . , φ l ∈ X ′ . π 0 ( τ ) − π 1 ( τ ) v anishe s on homogeneous tensor s of degree ≥ l + 1 and of degree ≤ l − 1. Let q = q 1 ⊗ . . . ⊗ q l be a pure tensor of degree l . W e hav e ( π 0 ( τ ) − π 1 ( τ ))( q ) = ( p 1 ⊗ . . . ⊗ p k · ( φ 1 ( . . . φ l ( q 1 ) . . . ) q l ) Hence π 0 ( τ ) − π 1 ( τ ) b elongs to K R ( T ( X )) = L n,m ∈ N X ⊗ n ⊗ R ( X ′ ) ⊗ m .  Theorem 3.2. L et X b e an R -c orr esp ondenc e over a ring with lo c al u nits R . Every homotopy invariant, split-exact, M -stable functor E : Rings → Ab sends the inclusion j : R → T X to an isomorphi sm E ( T X ) ∼ = E ( R ) . Pr o of. First we compute E ( π ◦ j ). Let ι = ι R,T ( X ) : R → K R ( T ( X )) b e the inclusion in the upper left co r ner ( ι ( r )( p ) = 0 if deg ( p ) ≥ 1 a nd ι ( r )( p ) = r · p other wise). The map E ( ι ) is a n isomorphism by Prop osition 2.1 1. W e co mpute E ( π ◦ j ) = E ( j, π 1 ◦ j ) = E ( ι + π 1 ◦ j, π 1 ◦ j ) = E ( ι ) as π 1 ◦ j and ι are orthogo nal quasi-homomo rphisms (see [ CMR07 , Pr op osition 3.3]). Now w e compute E ( j ) ◦ E ( π ). W rite T X = T for s implicity . The map j : R → T makes ( T , T , µ ) an R - T corresp ondence. Consider the following diagra m : T L R ( T ( X ) ) ⊲ K R ( T ( X ) ) L T ( T ( X ) ⊗ R T ) ⊲ K T ( T ( X ) ⊗ R T ) π 0 π 1 [ −⊗ id T ] [ −⊗ id T ] Let us first verify that [ − ⊗ id T ] maps the c o mpact op erator s to the compac t o p er ators. K T ( T ( X ) ⊗ R T ) = T ( X ) ⊗ R T ⊗ T T ⊗ R T ( X ′ ) ∼ = T ( X ) ⊗ R T ⊗ R T ( X ′ ) and K R ( T ( X ) ) = T ( X ) ⊗ T ( X ′ ) ∼ = T ( X ) ⊗ R ⊗ T ( X ′ ). Let p ∈ X ⊗ n , q ∈ X ′⊗ m and r ∈ R . The compact op- erator p ⊗ r ⊗ q acting on T ( X ) ⊗ R T is equal to p ⊗ j ( r ) ⊗ q ∈ T ( X ) ⊗ R T ⊗ R T ( X ′ ). Hence [ − ⊗ id T ] restric ted to K = K R ( T ( X ) ) is the map induced by j on ea ch co ordinate. W e w r ite [ − ⊗ id T ] : L R ( T ( X ) ) → L T ( T ( X ) ⊗ R T ) as ˜ j . E ( ˜ j | K ) equa ls E ( j ) up to stabilization isomor- phisms. E ( j ) = E ( ι T ,T ( X ) ⊗ R T ) − 1 E ( ˜ j | K ) E ( ι ) . ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS 9 Here ι T ,T ( X ) ⊗ R T : T → K T ( T ( X ) ⊗ R T ) is the upp er left corner embedding. W e no w define λ 0 , λ 1 : T → L T ( T ( X ) ⊗ R T ) t wo r ing homomorphisms. W e let λ 1 ( τ ) be zer o on p ⊗ τ with deg( p ) ≥ 1 and the op erator o f left multiplication b y τ on R ⊗ R T ∼ = T (this is omorphism holds bec ause we assume the left action to b e non-degenerate). Let λ 0 ( T x ) s end p ⊗ τ to zero if deg( p ) ≥ 1 and to x ⊗ p ⊗ τ otherwise. W e let λ 0 ( T φ ) send p ⊗ τ to zer o if deg( p ) ≥ 2 or if deg( p ) = 0 and to φ ( p ) · τ if deg ( p ) = 1. Finally w e let λ 0 ( r ) be zero on tensor s of degree ≥ 1 and send τ ∈ T to r ⊗ τ . One easily chec ks that λ 0 preserves the pairing of X . W e can write elements of L T ( T ( X ) ⊗ R T ) as infinite matrices by giving their action on each subspace of the form X ⊗ n ⊗ T . W e hav e λ 0 ( T x ) =     0 0 0 . . . T x 0 0 . . . 0 0 0 . . . . . .     , λ 0 ( T φ ) =     0 T φ 0 . . . 0 0 0 . . . 0 0 0 . . . . . .     , λ 0 ( r ) =     r 0 0 . . . 0 0 0 . . . 0 0 0 . . . . . .     The la s t step o f the pro of is to build a po lynomial homo to p y H : π 0 ⊗ id ∼ λ 1 + π 1 ⊗ id . W e define, using a “rotationa l” homotopy similar to that o f [ CT06 ] H ( T x ) = (1 − t 2 ) λ 0 ( T x ) + (2 t − t 3 ) λ 1 ( T x ) + ( π 1 ⊗ id)( T x ) , H ( T φ ) = (1 − t 2 ) λ 0 ( T φ ) + t λ 1 ( T φ ) + ( π 1 ⊗ id)( T φ ) , H ( r ) = r · id . In other words H ( T x ) =       (2 t − t 3 ) T x 0 0 0 . . . (1 − t 2 ) T x 0 0 0 . . . 0 T x 0 0 . . . 0 0 T x 0 . . . . . .       H ( T φ ) =       tT φ (1 − t 2 ) T φ 0 0 . . . 0 0 T φ 0 . . . 0 0 0 T φ . . . 0 0 0 0 . . . . . .       A direct c omputation s hows that H indeed preser ves the pair ing and hence defines a ring homomorphism H : T → L T ( T ( X ) ⊗ R T )[ t ] such that H (0) = π 0 ⊗ id and H (1) = λ 1 + π 1 ⊗ id. Hence, using the homotopy in v aria nce of E E ( ˜ j ◦ π ) = E ( π 0 ⊗ id , π 1 ⊗ id) = E ( λ 1 + π 1 ⊗ id , π 1 ⊗ id) = E ( λ 1 ) as λ 1 and π 1 ⊗ id ar e orthogona l homomor phisms. But λ 1 = ι T ,T ( X ) ⊗ R T is the upp er left corner embedding o f T in K T ( T ( X ) ⊗ R T ). Th us E ( λ 1 ) is an isomo r phism by Pr op osition 2.11 a nd M - s tabilit y of E . Since E ( ˜ j ◦ π ) = E ( ˜ j | K ) ◦ E ( π ), this finishes the pro of be c ause we no w hav e tw o iso morphisms which are inv ers e s of each other : E ( j ) E ( ι ) − 1 E ( π ) = id T , E ( π ) E ( j ) E ( ι ) − 1 = id R .  Lemma 3 .3. L et E b e a split-ex act and M -stable functor. L et ι R,T ( X ) b e the upp er left c orner emb e dding R → K R ( T ( X ) ) . We write i : I → R , and i ′ : K R ( T ( X ) · I ) → T X for t he inclusion maps. We have E ( π ) ◦ E ( i ′ ◦ ι I , T ( X ) ·I ) = E ( ι R,T ( X ) ) ◦ ( E ( i ) − E ( X )) wher e E ( X ) is define d to b e the map induc e d by the structur e of I - R c orr esp ondenc e on R . 10 THIBA UT LESCURE Pr o of. Recall that E ( X ) = E ( ι R,R ⊕X ) − 1 ◦ E ( ι X ,R ⊕ X ) ◦ E (∆) where ∆ : I → K R ( X ) is the left action, ι X ,R ⊕ X : K R ( X ) → K R ( R ⊕ X ) is the lower rig ht corner embedding and ι R,R ⊕X : R → K R ( R ⊕ X ) is the upp e r left corner embedding. There is a nother natural inclusion ι X ,T ( X ) : K R ( X ) → K R ( T ( X ) ). E ( π ) ◦ E ( i ′ ◦ ι I , T ( X ) ·I ) = E ( i ′ ◦ ι I , T ( X ) ·I , π 1 ◦ i ′ ◦ ι I , T ( X ) ·I ) F or all a ∈ I , write ∆( a ) = P k x k ⊗ φ k with x k ∈ X and φ k ∈ X ′ . W e have i ′ ( ι I , T ( X ) ·I ( a )) = a · p 0 = a · id T ( X ) − X k T x k T φ k By using the definition of π 1 we get π 1 ( i ′ ( ι I , T ( X ) ·I ( a ))) = ι X ,T ( X ) (∆( a )). The ring homomorphisms π 0 ◦ i ′ ◦ ι I , T ( X ) ·I and π 1 ◦ i ′ ◦ ι I , T ( X ) ·I : I → L R ( T ( X ) ) map into K R ( T ( X ) ). [ CMR07 , Prop osition 3.3] gives E ( π ) ◦ E ( i ′ ◦ ι I , T ( X ) ·I ) = E ( ι R,T ( X ) ◦ i ) − E ( ι X ,T ( X ) ) ◦ E (∆) . It r e mains to s how that E ( ι R,T ( X ) ) ◦ E ( ι R,R ⊕X ) − 1 ◦ E ( ι X ,R ⊕ X ) = E ( ι X ,T ( X ) ). This is a direct consequence of the commutativit y of the following diagram K R ( X ) K R ( R ⊕ X ) R K R ( T ( X ) ) ι X ,R ⊕X ι X ,T ( X ) ι R ⊕X ,T ( X ) ι R,R ⊕X ι R,T ( X )  Theorem 3.4 . L et ( E n ) n ∈ Z : Rings → Ab b e a se quenc e of homotopy invariant and M - stable functors satisfying excision. L et R b e a ring with lo c al units. Le t X b e an R -c orr esp ondenc e and I ⊳ R a two-side d ide al that has lo c al units and acts on X on the left by c omp act op er ators. Ther e is a long exact s e quenc e . . . E n ( I ) E n ( R ) E n ( O X , I ) E n − 1 ( I ) . . . E n ( i ) − E n ( X ) E n ( j ) Her e j : R → O X , I and i : I → R ar e t he natur al inclusions. W e now show how so me lo ng exact sequences of KH and HP gro ups of so me s pecific cla sses of r ings ca n b e obtained directly when realizing these sp ecific rings as algebr aic Cuntz-Pimsner rings. W e fix ( E n ) n ∈ Z : Rings → Ab a sequence of homotop y in v ariant, s table functors satisfying excision. As a direct co rollary of Theorem B we get (see [ CO11 , Exa mple 5.5] for details) Theorem 3.5. (Pimsner-V oiculescu) L et R b e a ring with lo c al units, let α : R → R b e a ring automorphism. L et R ⋊ α Z b e the cr osse d pr o duct ring of R by α . Ther e is a long exact se quenc e : . . . E n ( R ) E n ( R ) E n ( R ⋊ α Z ) E n − 1 ( R ) . . . 1 − E n ( α ) E n ( j ) Let Q = ( Q 0 , Q 1 , r , s ) b e a quiver. Recall that a vertex v ∈ Q 0 is regular if 0 < | s − 1 ( v ) | < ∞ , and le t ρ ( Q ) ⊆ Q 0 denote the se t of reg ular vertices. It is known [ CO 11 , Example 5.8] that the Leavitt path algebra L k ( Q ) is the algebraic Cun tz-Pimsne r ring asso c iated to the r ing R , the ideal I , a nd the R -corr esp o ndence X = ( X , X ′ , h· , ·i ) defined by R = M v ∈ Q 0 k · 1 v , I = M v ∈ ρ ( Q ) k · 1 v , X = M e ∈ Q 1 k · 1 e , X ′ = M e ∈ Q 1 k · 1 e ∗ . The ev aluation map is given, for e, f ∈ Q 1 , by h 1 e ∗ , 1 f i = δ e,f 1 r ( e ) and the bimodule structures are determined o n ge nerators by 1 e · 1 v = δ r ( e ) ,v 1 e , 1 v · 1 e = δ s ( e ) ,v 1 e , 1 e ∗ · 1 v = δ s ( e ) ,v 1 e ∗ , 1 v · 1 e ∗ = δ r ( e ) ,v 1 e ∗ . ON THE K-THEOR Y OF ALGEBRAIC CUNTZ-PIMSNER RINGS 11 The functional ho momorphism of the co rresp ondence is given by the maps U : X → R ( Q 1 ) and V : X ′ → R ( Q 1 ) defined by U (1 e ) = ( δ e,f 1 r ( e ) ) f ∈ Q 1 and V (1 e ∗ ) = ( δ e,f 1 r ( e ) ) f ∈ Q 1 for all e ∈ Q 1 . Let N ′ Q = ( n x,y ) x,y ∈ Q 0 be the adjacency matrix of Q , where n x,y is the num b er of arrows from x to y . Let N Q be the matrix obtained from N ′ Q by remo ving th e columns indexed by elemen ts of Q 0 \ ρ ( Q ). Theorem 3.6. L et k b e a ring with lo c al u n its. Ther e is a long exact se quenc e ( n ∈ Z ) . . . E n ( k ) ( ρ ( Q )) E n ( k ) ( Q 0 ) E n ( L k ( Q )) . . . E n ( i ) − N Q E n ( j ) Pr o of. W e just hav e to prove t hat the map E n ( X ) : E n ( I ) = E n ( k ) ( ρ ( Q )) → E n ( R ) = E n ( k ) ( Q 0 ) equals N Q . K R ( X ) is the set of k -linear c o mbin ations of elements of the form 1 e ⊗ 1 f ∗ for e, f ∈ Q 1 . The functional ho momorphism of X describ ed above induces the ring homomor phism ρ : K R ( X ) → M Q 1 ( R ) which sends every element of the for m 1 e ⊗ 1 f ∗ to the matrix ( δ e,e ′ 1 r ( e ) ⊗ δ f ,f ′ 1 r ( f ) ) e ′ ,f ′ ∈ Q 1 = δ e,f 1 r ( e ) . By Pro po sition 2.13, E n ( X ) e quals the following map E n ( I ) E n ( K R ( X )) E n ( M Q 1 ( R )) E n ( R ) E n (∆) E n ( ρ ) ∼ Here ∆ : I → K R ( X ) is the left a ction. F o r all v ∈ ρ ( Q ), ρ (∆(1 v )) = ρ  X e ∈ s − 1 ( v ) 1 e ⊗ 1 e ∗  = X e ∈ s − 1 ( v ) 1 r ( e ) .  Nekrashevych algebr as were int ro duced or iginally fo r C ∗ -algebra s as Cuntz-Pimsner algebr as naturally asso ciated t o self-similar groups [ Nek04 ]. Analo gues in the discr e te setting were recen tly found [ SS23 ]. L e t X b e a finite set with | X | ≥ 2 , and let X ∗ be the free monoid gene r ated by X . A self-similar gr o up ov e r X is a group G together with a fa ithful action on X ∗ by length-preserving per mut ations such that, for ev er y g ∈ G and ev ery x ∈ X , there exists an elemen t g | x ∈ G satisfying g ( xw ) = g ( x ) g | x ( w ) for all w ∈ X ∗ . Let k b e a field, let R = k G b e the convolution algebra of G with co efficient in k , let R = M g ∈ G k · g = k ( G ) , X = R ( X ) = k ( G × X ) , X ′ = R ( X ) W e define the k -bilinear scalar pro duct D X x ∈ X λ x x · g x , X x ∈ X µ x x · h x E = X x ∈ X λ x µ x g − 1 x h x Here X a nd X ′ are resp ectively rig ht and left free R -mo dules, but as a le ft R -mo dule w e define X by the re lation g · x = g ( x ) · g | x for all g ∈ G, x ∈ X (we have g | x ∈ G ) using the self-similar it y condition. The Nekrashevych algebr a [ SS23 ] asso ciated to the se lf-s imilar gr oup G (and the set X ) with co efficients in k is easily s een to b e isomo r phic to the alg ebraic Cuntz-Pimsner ring asso ciated to the ring R and the cor r esp ondence X = ( X , X ′ , h , i ) N k ( G, X ) = O X . As X is finite, the left a ction obviously maps int o the c o mpact op erators . The ring of compact op erators is just M d ( k G ) where d = | X | . Theorem 3.7. F or any self-similar gr oup ( G, X ) ther e is a long exact se qu en c e ( n ∈ Z ) . . . E n ( k G ) E n ( k G ) E n ( N k ( G, X )) . . . 1 − E n ( X ) E n ( j ) Ackno wledgements The a uthor would like to thank Professo r Ralf Me yer for insightful discus s ions and v aluable comments on earlier versions of this pap er. 12 THIBA UT LESCURE References [Abr83] G. D. Abr ams, Morita e quivalenc e for rings with lo c al units , Comm. Algeb ra 1 1 (1983), 801–837. [A ´ AM87] P . N. ´ Anh and L. 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LMNO, 6 Bo ulev ard Mar ´ echal J uin, 14000 Ca en , FR ANCE Email addr ess : thibaut.lescure@ etu.unicaen.fr