Completions of Grothendieck groups

COMPLETIONS OF GR OTHENDIECK GR OUPS PRAMOD N. A CHAR AND CA THARINA STR OPPEL Abstract. F or a certain class of ab elian categories, we sho w ho w to make sense of the “Euler c haracteristic” of an i nfinite pro jectiv e resolution (or, more generally , certain chain complexes that are only b ounded abov e), b y passing to a suitab le completion of the Gr othend ieck group. W e also sho w that right-exact functors (or their left-deri v ed functors) induce con tin uous homomorphis m s of these complete d Grothendiec k groups, and we discuss examples and applica- tions coming from cat egorification. 1. Introduction Let A b e a no etherian and ar tinian ab elian categor y with eno ug h pro jectives, and let D b ( A ) b e its bo unded derived catego ry . The inclusion A → D b ( A ) g ives rise to a natural isomorphism of Grothendieck groups (1.1) K ( A ) ∼ → K ( D b ( A )) . When A has finite coho mological dimension, K ( A ) ca ptures a grea t deal of infor- mation ab out “derived” phenomena. F or instance, for an y X ∈ A , we ha ve (1.2) [ X ] = X ( − 1) i [ P i ] , where P • → X is a pro jective res olution. If B is a nother such ca tegory , then for any right-exact functor F : A → B , the derived functor L F induces a group homomorphism (1.3) [ L F ] : K ( A ) → K ( B ) . On the other hand, if A ha s infinite pro jective dimension, we s hould replace D b ( A ) by D − ( A ), the b ounded ab ov e derived ca teg ory , but then its Grothendieck group cannot see anything at all, as it is zero by an “Eilenberg swindle”- t yp e argument; see [14]. How ev er, when A and B a re mixed categ ories with a T ate t wist, a version of the statements (1.1), (1.2), a nd (1.3) can be r ecov ered. W e explain in this note ho w to replace D − ( A ) b y a certain sub categor y D ▽ ( A ) that is still large enough for derived functors, but small enough that a “topolo gical” version of (1.2) holds. F or these top ological Grothendieck g roups, cer ta in infinite sums like (1.2) conv erge, and der ived functors give rise to c ontinuous homomor- phisms. More precisely , in this setting, the Grothendieck gr oup K ( A ) is naturally a mo dule ov er the ring R = Z [ q , q − 1 ]. It a dmits a completion ˆ K ( A ) that is a mo dule ov er ˆ R = Z [[ q ]][ q − 1 ]. The main results o f the pap er a re summa r ized b elow. (F urther definitions and notation are given in Sectio n 2.) The first author receiv ed support from NSF Gran t No. DMS-1001594. 1 2 PRAMOD N. ACHAR AND CA THARINA STR OPPEL Theorem 1.1. L et A b e a n o etherian and artinian mixe d ab elia n c ate gory with enough pr oje ctives and a T ate twist. (1) The top olo gic al Gr othendie ck gr oup K ( D ▽ ( A )) is a c omplete top olo gic al R - mo dule. Mor e over, the natur al map K ( A ) → K ( D ▽ ( A )) is inje ctive and induc es an isomorphism ˆ K ( A ) ∼ → K ( D ▽ ( A )) . (2) Every obje ct X ∈ D ▽ ( A ) admits a pr oje ctive r esolution P • with asymptot- ic al ly de cr e asing weights. In ˆ K ( A ) , we have c onver gent s eries [ X ] = X i ∈ Z ( − 1) i [ H i ( X )] = X i ∈ Z ( − 1) i [ P i ] . (3) Both Irr( A 0 ) and Pro j( A ) 0 sp an dense fr e e ˆ R -submo dules of ˆ K ( A ) . If t hose sets ar e fin ite, they e ach give an ˆ R -b asis for ˆ K ( A ) . (4) L et B b e another finite-length mixe d c ate gory with a T ate twist, and let F : A → B b e a right-exact functor t hat c ommutes with the r espr e ctive T ate twist. If F has finite weigh t amplitude, then L F induc es a c ontinuous homomorph ism of ˆ R -mo dules [ L F ] : ˆ K ( A ) → ˆ K ( B ) . The idea of completing Grothendieck groups arises fro m the concept o f c ate gori- fic ation , see e.g. [10, 13]. There, Z [ q , q − 1 ]-mo dules get re a lized as Gr othendiec k groups K ( A ) o f appropria tely chosen gra ded catego r ies A . The action o f q and q − 1 arises from s hifting the gra ding (up and down). Often the Z [ q , q − 1 ]-mo dules in question come a long with standard and canonical bas es whic h then corresp ond to distinguished bases of K ( A ). So far, repr esent ation theor ists fo cused on the cases where the entries of the transforma tion matrices b et ween the different bases were element s of Z [ q, q − 1 ]. Thes e num be r s a re then usually in terpreted as Jor dan– H¨ older multiplicities or gr aded decomp osition n um b ers. Ho wev er, the theory of Lusztig’s ca nonical bases or Ka s hiw ara’s crysta l bases g iv es plent y of examples where the entries of the transformatio n matr ix are contained o nly in the comple- tion Z [[ q ]][ q − 1 ] o f Z [ q , q − 1 ]. Theorem 1.1 provides a p ossible catego rical setup to handle such situations and could be viewed a s the abstract context for categ orifi- cations of, for instance, modules for quantum gr oups. Although this pap er focuse s on the abstr act setup, there ar e alre a dy concrete examples known, for instance in the cont ext of catego r ification of Reshetikhin–T uraev–Viro in v ariants of links and 3-manifolds; see [11]. F ollowing some set-up in Sectio n 2, the main theor em will be pro ved in Sections 3 and 4. So me examples and applications are indicated in Section 5. A cknow le dgements. The authors are grateful to Sabin Cautis for p oint ing out a flaw in an earlier version of this pape r, and to Olaf Schn¨ urer and Amnon Neeman for a n umber of very useful r emarks on a previous draft. 2. Not a tion a nd Definitions 2.1. Mixed (ab eli an) categories. All ab elian categor ies will b e assumed to b e finite-length c ate gorie s (i.e. noether ia n a nd artinian) and to be skeletally small. Let A be a n ab e lia n category . W e denote by Irr( A ) the set of isomorphism classes of simple ob jects. In this setting, the Grothendieck gro up K ( A ) is a free ab elian COMPLETIONS OF GR OTHENDIECK GROUPS 3 group o n the se t Irr( A ). Recall that K ( A ) = F ( A ) /R ( A ), where F ( A ) is the free ab elian group on isomorphism classes [ M ] of o b jects M ∈ A , and R ( A ) is the subgroup genera ted by the expressions [ A ] − [ C ] + [ B ] whenever there is a shor t exact s e quence of the for m 0 → A → C → B → 0. So the ab ov e claim follows from the existence of a Jordan– H¨ older series and the uniq ue ne s s of its m ultiset of sub q uotien ts and states that (2.1) K ( A ) = Z [Irr( A )] . Recall that A is said to b e a mixe d categor y if there is a function wt : Ir r( A ) → Z , called the weight function such that Ext 1 ( L, L ′ ) = 0 if wt([ L ]) ≤ wt([ L ′ ]) for simple ob jects L , L ′ . (2.2) In the following w e mostly write wt( L ) for wt([ L ]). A T ate twist on a mixed category A is an automorphism (1) : A → A such that wt( L (1)) = wt( L ) − 1 if L is simple. Henceforth, all ab elian categor ie s will b e mixed a nd equipp ed with a T ate twist. F or more details on mixed categories with T ate t wist we re fer to [3, 5, 1 5]. 2.2. W eigh t fil tration. Recall the following sta ndard fact ([3, Lemma 4.1.2]): Lemma 2.1 . L et A b e a mixe d ab elian c ate gory. Then any obje ct M ∈ A has a unique finite filt ra tion W • = W • ( M ) such that W i /W i − 1 is a dir e ct sum of simple obje cts, al l of weight i . This filtration is called the weight filtr ation . In case W i /W i − 1 6 = 0 w e say i o c curs as a weight in M . If only o ne weigh t o ccurs, then M is called pur e of this weigh t. In general, the maximal weight o ccurring in M is called the de gr e e of M . W e say M has weigh ts ≤ n if the degree of M is smaller or equal n . Note that morphisms are strictly co mpatible with w eight filtratio ns , in the sense that f ( W i ( M )) ⊂ W i ( N ) for any morphism f : M → N b et ween ob jects M , N in A with weigh t filtrations W • ( M ) a nd W • ( N ). It is a consequence that (2.3) Ext i ( L, L ′ ) = 0 , if wt([ L ]) − i < wt([ L ′ ]), or, equiv alently , Ext i ( L, L ′ ) 6 = 0 ⇒ wt([ L ]) ≥ wt([ L ′ ]) + i. (2.4) Example 2.2. Our standar d example of a mixed ab elian category is the categ o ry A -gmo d of finite-dimensional gr aded mo dules over a finite-dimensio na l pos itiv ely graded algebra A = ⊕ i ∈ Z ≥ 0 A i ov er the complex num ber s with semisimple A 0 . Each simple mo dule L is concentrated in a s ing le deg ree − wt( L ) and the T ate t wist (1 ) is given b y the automorphis m h 1 i which shifts the deg ree up by 1, i.e. if M = ⊕ j ∈ Z M j then M (1) = M h 1 i is the gra ded mo dule with g raded comp onents ( M h 1 i )) j = M j − 1 . Any C - linear mixed abelia n category can be realized a s the category of mo dules ov er a pro jective limit of such positively grade d algebr as; see [3, 4.1.6] for a precise statement . F or n ∈ Z , let A ≤ n (resp. A n , A ≥ n ), be the Ser r e subca tegory of A genera ted by the simple ob jects of w eight ≤ n (resp. n , ≥ n ). If m ≤ n , we also put A [ m,n ] = A ≥ m ∩ A ≤ n . F or any X ∈ A , the w eight filtration (Lemma 2.1) defines a functorial short exact sequence 0 → β ≤ n X → X → β ≥ n +1 X → 0 4 PRAMOD N. ACHAR AND CA THARINA STR OPPEL where β ≤ n X ha s weigh ts ≤ n and β ≥ n +1 X has weigh ts ≥ n + 1 . Mo reov er, the functors β ≤ n : A → A ≤ n and β ≥ n +1 : A → A ≥ n +1 are exact, so w e can apply them to a chain complex C • in A a nd get a sho rt exact sequence of chain complexes . These functors induce derived functors D − ( A ) → D − ( A ), so for a ny ob ject X ∈ D − ( A ), there is a functorial distinguished triangle β ≤ n X → X → β ≥ n +1 X → (2.5) in D − ( A ). (The sa me remarks a pply to the b ounded D b ( A ) and b ounded b elow D + ( A ) deriv ed categories as well, but we will w ork primarily with D − ( A ).) The s e functors endow D − ( A ) w ith a b aric structur e in the s e nse of [1]. The connecting homomorphism from (2.5) is in fact unique (in con trast with the co n text of weigh t structures as studied e.g. in [2], [17]). Definition 2.3. Let F : A → B b e a n additiv e functor betw een tw o mixed ab elian categorie s. The weight amplitude o f F is defined to be the infim um of the set { a ∈ Z ≥ 0 | F ( A ≤ n ) ⊂ B ≤ n + a for all n ∈ Z } ∪ { + ∞} . 2.3. Co efficient s rings and Grothendiec k groups. Mo st Grothendieck groups we consider will naturally b e modules ov er one of the follo wing r ing s (tw o of whic h were mentioned in Section 1): R 0 = Z [ q ] , R = Z [ q , q − 1 ] , ˆ R 0 = Z [[ q ]] , ˆ R = R ⊗ R 0 ˆ R 0 = Z [[ q ]][ q − 1 ] . F or instance, the T ate twist induces an automorphism q : K ( A ) → K ( A ), where [ X (1 )] = q [ X ], and so makes K ( A ) into a n R -mo dule. It also r e stricts to a fully faithful, exact functor (1) : A ≤ n → A ≤ n , but this is no long er an equiv alence. The Grothendieck gr oup K ( A ≤ n ) is naturally an R 0 -submo dule of K ( A ). It fo llows from (2.1) that K ( A ) is free as an R -mo dule (see [3, Lemma 4.3 .2 ]). In fact, for any n ∈ Z , w e hav e canonical isomorphisms (2.6) K ( A ≤ n ) ∼ = R 0 [Irr( A n )] and K ( A ) ∼ = R ⊗ R 0 K ( A ≤ n ) . The R 0 -mo dule K ( A ≤ n ) is equippe d with a natural ( q )-adic topo logy , in which the submo dules q i · K ( A ≤ n ) = K ( A ≤ n − i ) for i ≥ 0 constitute a basis of neighbo rho ods around 0. Similarly , we endo w K ( A ) with a top ology (also ca lled “ ( q )-adic”) by decla r ing the s ubmodules K ( A ≤ i ) ⊂ K ( A ) to b e a basis of neighborho o ds around 0. It follows from (2.6) that (2.7) \ m ∈ Z K ( A ≤ m ) = 0 . In other w ords, the ( q )-adic top ology on K ( A ) or K ( A ≤ n ) is Hausdorff. Let ˆ K ( A ≤ n ) a nd ˆ K ( A ) denote the completio ns of each of thes e mo dules in the ( q )-a dic topo logy . These completions are modules ov er ˆ R 0 and ˆ R , resp ectiv ely . COMPLETIONS OF GR OTHENDIECK GROUPS 5 2.4. Definition of D ▽ ( A ) . Given a mixed abelia n category with a T ate t wist, we define the following full sub catego r y o f D − ( A ): D ▽ ( A ) =  X ∈ D − ( A )    for each m ∈ Z , only finitely many o f the H i ( X ) contain a comp osition factor of weigh t > m  . It is easy to s e e that D ▽ ( A ) is closed under susp ensions (or shifts) [ i ], i ∈ Z , and cones, and hence that it is a full triangulated sub category of D − ( A ). F or n ∈ Z , we also define the follo wing full sub categories of D ▽ ( A ): D ▽ ≤ n ( A ) = { X ∈ D ▽ ( A ) | for a ll i ∈ Z , H i ( X ) has weigh ts ≤ n } , D ▽ ≥ n ( A ) = { X ∈ D ▽ ( A ) | for a ll i ∈ Z , H i ( X ) has weigh ts ≥ n } . They ar e triangulated categories . If m ≤ n , we also put D ▽ [ m,n ] ( A ) = D ▽ ≥ m ( A ) ∩ D ▽ ≤ n ( A ). It follows from the definition of D ▽ ( A ) that any o b ject in D ▽ ≥ n ( A ) has only finitely many nonze ro cohomology ob jects, so (2.8) D ▽ ≥ n ( A ) ⊂ D b ( A ) . The T ate twist induces a n auto e quiv alence (1) : D ▽ ( A ) → D ▽ ( A ) and a fully faith- ful functor (1) : D ▽ ≤ n ( A ) → D ▽ ≤ n ( A ), s o K ( D ▽ ( A )) and K ( D ▽ ≤ n ( A )) are mo dules ov er R and R 0 , resp ectively . T he catego ries D ▽ ≥ m ( A ) a re no t preser v ed b y the T ate t wist, but nevertheless we will construct in the next section an R 0 -mo dule structure on their Grothendieck gr oups. 2.5. T op ological Grothendi ec k groups. Recall that the Grothendiec k group of a small triangulated categor y C is defined as K ( C ) = F ( C ) /R ( C ), where F ( C ) is the free ab elian gr oup on isomorphism classes [ M ] of ob jects M ∈ C , and R ( C ) is the ideal genera ted b y the ex pressions [ A ] − [ C ] + [ B ] whenever there is a distinguished triangle of the form A → C → B → A [1]. Supp o se now that C is a sub category of the derived catego ry of A that is stable under β ≤ m and β ≥ m for all m . Let I ( C ) = { f ∈ K ( C ) | [ β ≥ m ] f = 0 in K ( C ) for all m ∈ Z } . W e define the top olo gic al Gr othendie ck gr oup of C to be K ( C ) = K ( C ) /I ( C ) . The reason fo r the terminolo gy will b ecome clear in Rema rk 3.4. W e will even- tually prove a g eneral result (cf. Theorem 1.1(4)) ab out der iv ed functors and the top ological Grothendieck group. F or now, let us note simply that any functor of triangulated categor ies that commutes with all β ≥ m induces a homomorphis m of top ological Grothendieck gro ups. R emark 2 .4 . If C = D b ( A ), the topolog ical Gr othendieck group coincides with the ordinary Gro thendieck gr oup. Indeed, it ca n b e shown that K ( A ) ∼ = K ( C ), a nd then, in view o f (2.1) and (2.6), it follows fr o m (2.7) that I ( C ) = 0 . Therefore, K ( D b ( A )) ∼ = K ( D b ( A )). 3. The Grothendieck Group of D ▽ ( A ) The main goal of this section is to prov e part (1) of Theorem 1.1. 6 PRAMOD N. ACHAR AND CA THARINA STR OPPEL 3.1. Sequences of R 0 -mo dules . The c a tegories A ≥ m and D ▽ ≥ m ( A ) ar e not pre- served by the T a te twist (1), s o we use a different functor to make K ( A ≥ m ) and K ( D ▽ ≥ m ( A )) into R 0 -mo dules: we put q · [ X ] = [ β ≥ m ( X (1 ))] for X ∈ A ≥ m or X ∈ D ▽ ≥ m ( A ). This definition makes sense b ecause β ≥ m ◦ (1 ) is an exa ct functor that pres e r ves A ≥ m and D ▽ ≥ m ( A ). The s a me definition also ma k es se ns e for A [ m,n ] and D ▽ [ m,n ] ( A ). Lemma 3.1 . F or any n ∈ Z , ther e is c ommutative diagr am of R 0 -mo dules (3.1) 0 / / K ( A ≤ n ) / /   K ( A ) [ β ≥ n +1 ] / /   K ( A ≥ n +1 ) / /   0 0 / / K ( D ▽ ≤ n ( A )) / / K ( D ▽ ( A )) [ β ≥ n +1 ] / / K ( D ▽ ≥ n +1 ( A )) / / 0 in which the r ows ar e short exact se quenc es. Mor e over, the first t wo vertic al maps ar e inje ctive, and the last one is an isomorphism. Pr o of. W e begin by treating the second r ow in this diagram. Consider the sur- jective map γ : F ( D ▽ ( A )) → F ( D ▽ ≤ n ( A )) ⊕ F ( D ▽ ≥ n +1 ( A )) defined as [ X ] 7→ ([ β ≤ n X ] , [ β ≥ n +1 X ]). Giv en a distinguished tr iangle A → X → B → in D ▽ ( A ), we hav e γ ([ X ] − [ A ] − [ B ]) = ([ β ≤ n X ] − [ β ≤ n A ] − [ β ≤ n B ] , [ β ≥ n +1 X ] − [ β ≥ n +1 A ] − [ β ≥ n +1 B ]) . Since the functors β ≤ n and β ≥ n +1 are functors of triangulated categor ies, this calculation shows that γ ( R ( D ▽ ( A ))) ⊂ R ( D ▽ ≤ n ( A )) ⊕ R ( D ▽ ≥ n +1 ( A )). Since the restriction of β ≤ n , re sp. β ≥ n +1 , to D ▽ ≤ n ( A ), resp. D ▽ ≥ n +1 ( A ), is the identit y functor, we actually have γ ( R ( D ▽ ( A ))) = R ( D ▽ ≤ n ( A )) ⊕ R ( D ▽ ≥ n +1 ( A )). W e conclude that γ induces an isomorphism of ab elian groups K ( D ▽ ( A )) = K ( D ▽ ≤ n ( A )) ⊕ K ( D ▽ ≥ n +1 ( A )) . Because β ≥ m preserves each of D ▽ ≤ n ( A ) and D ▽ ≥ n +1 ( A ), w e hav e I ( D ▽ ( A )) = I ( D ▽ ≤ n ( A )) ⊕ I ( D ▽ ≥ n +1 ( A )). On the o ther hand, the fa c t that β ≥ n +1 is the identit y functor on D ▽ ≥ n +1 ( A ) implies that I ( D ▽ ≥ n +1 ( A )) = 0 . W e de duce that (3.2) K ( D ▽ ( A )) = K ( D ▽ ≤ n ( A )) ⊕ K ( D ▽ ≥ n +1 ( A )) . In this direct sum, the inclus io n K ( D ▽ ≤ n ( A )) → K ( D ▽ ( A )) is in fact induced by the inclusion functor ι : D ▽ ≤ n ( A ) → D ▽ ( A ), since β ≤ n ◦ ι = id a nd β ≥ n +1 ◦ ι = 0. Since ι commutes with the T ate t wist, the inclusion ma p K ( D ▽ ≤ n ( A )) → K ( D ▽ ( A )) is a homomo rphism of R 0 -mo dules. On the other hand, the setup is such that [ β ≥ n +1 ] : K ( D ▽ ( A )) → K ( D ▽ ≥ n +1 ( A )) commutes with the action of q ∈ R 0 on bo th groups. Thus, (3.2) giv es r ise to the desired short ex a ct seq uence (the second line of (3.1)) of R 0 -mo dules. Since β ≤ n and β ≥ n +1 are t - exact (for the t -structure induced fro m the standard t -structure), the same arg ument can be r epea ted with the ab elian categories A ≤ n , A , and A ≥ n +1 (but skipping the passage to the top ologica l Grothendieck group), yielding the ˆ R 0 -mo dule structure a nd e x actness of the first row in the diag ram (3 .1). Since a ll maps in that diagram ar e induced b y inclusio n functors or by β ≥ n +1 , it is easy to see that the diagram commutes. COMPLETIONS OF GR OTHENDIECK GROUPS 7 Recall from (2.8) that D ▽ ≥ n +1 ( A ) ⊂ D b ( A ). The fact that K ( A ≥ n +1 ) → K ( D ▽ ≥ n +1 ( A )) is an isomorphism follows from the fact that A ≥ n +1 is the heart of a b ounde d t -str ucture on D ▽ ≥ n +1 ( A ). Next, consider an element f ∈ K ( A ). W e may write f = a 1 [ X 1 ] + · · · + a k [ X k ] for suitable simple o b jects X i ∈ A . Now, choose n such that n < wt( X i ) fo r all i . It is clear fro m Lemma 2.1 that [ β ≥ n +1 ] f 6 = 0. In view of the preceding par agraph, it follows fro m the comm utativity of (3.1) that the imag e of f in K ( D ▽ ( A )) is nonzero. Thus, the middle v ertical arrow in (3.1) is injective. The injectivity of the first vertical arr ow is then clear as well.  The same reaso ning yie lds the following related statemen t. Lemma 3.2 . Sup ose m ≤ n . Ther e is a c ommutative diagr am of R 0 -mo dules (3.3) 0 / / K ( A ≤ m ) / /   K ( A ≤ n ) / /   K ( A [ m +1 ,n ] ) / /   0 0 / / K ( D ▽ ≤ m ( A )) / / K ( D ▽ ≤ n ( A )) / / K ( D ▽ [ m +1 ,n ] ( A )) / / 0 in which the r ows ar e short exact se quenc es. Mor e over, the first t wo vertic al maps ar e inje ctive, and the last is an isomorphism.  3.2. ( q ) -adi c top ology . Recall from (3.1) that K ( D ▽ ≤ m ( A )) ca n naturally b e iden- tified with a R 0 -submo dule of K ( D ▽ ( A )) (or of K ( D ▽ ≤ n ), if m ≤ n ). T hus, w e are at last able to define the ( q )-a dic top ology on these mo dules: w e take the set o f submo dules of the form K ( D ▽ ≤ m ) to b e a basis of neighbo rho ods of 0. It follows from the pro of of Lemma 3.1 that for s ∈ Z (3.4) q s · K ( D ▽ ≤ m ( A )) = K ( D ▽ ≤ m − s ( A )) , so K ( D ▽ ≤ n ( A )) and K ( D ▽ ( A )) are natura lly top olog ical R 0 - and R - modules , re- sp ectiv ely . Lemma 3.3 . In the ( q ) -adic top olo gy, t he gr oups K ( D ▽ ≤ n ( A )) and K ( D ▽ ( A )) ar e Hausdorff. Pr o of. Being Hausdorff is equiv alen t to the condition that \ m ∈ Z K ( D ▽ ≤ m ( A )) = { 0 } . If f ∈ T m ∈ Z K ( D ▽ ≤ m ( A )), it fo llo ws fro m Lemma 3.1 and (2.6) that [ β ≥ m ] f = 0 for all m , but then it is clear from the definition of K ( D ▽ ( A )) that f = 0.  R emark 3.4 . The statements of Se c tio n 3 .1 are still true if w e r eplace K ( D ▽ ( A )) b y K ( D ▽ ( A )), and the de finitio n of the ( q )-adic topo logy makes sense for K ( D ▽ ( A )) as well, but the resulting s pace is not Hausdorff. In fact, K ( D ▽ ( A )) is the universal Hausdorff quo tient of K ( D ▽ ( A )), in the sense that every contin uous homo morphism from K ( D ▽ ( A )) to a Hausdorff abelian group factors thr o ugh K ( D ▽ ( A )). This construction is w ell-known in the context of top ologica l groups, see e.g [18, Note after 3.22]. 8 PRAMOD N. ACHAR AND CA THARINA STR OPPEL Lemma 3.5. The R 0 -mo dule K ( D ▽ ≤ n ( A )) is c omplete in the ( q ) -adic top olo gy. Inde e d, the natu r al map K ( A ≤ n ) → K ( D ▽ ≤ n ( A )) induc es an isomorphism ˆ K ( A ≤ n ) → K ( D ▽ ≤ n ( A )) . Pr o of. Since q s · K ( D ▽ ≤ n ( A )) = K ( D ▽ ≤ n − s ( A )) fo r any s ∈ Z ≥ 0 , it follo ws fro m Lemma 3.2, with m = n − s , that K ( D ▽ ≤ n ( A )) /q s · K ( D ▽ ≤ n ( A )) ∼ = K ( D ▽ [ n − s +1 ,n ] ( A )) . Suppo se w e have a sequence of elements f i ∈ K ( D ▽ [ n − i +1 ,n ] ( A )), i ∈ Z ≥ 0 satisfying the co ndition that [ β ≥ n − j +1 ] f i = f j when j < i . T o show that K ( D ▽ ≤ n ( A )) is complete, we must exhibit an element g ∈ K ( D ▽ ≤ n ( A )) such that [ β ≥ n − i +1 ] g = f i for all i . By Lemma 3.2 again, we ide ntify K ( D ▽ [ n − i +1 ,n ] ( A )) with K ( A [ n − i +1 ,n ] ), viewed as a subgroup of K ( A ). Regarding a ll the f i as elements of K ( A ), w e can form the elements a i = f i − f i − 1 = f i − [ β ≥ n − i +2 ] f i ∈ K ( A n − i +1 ) . Then f i = a 1 + a 2 + · · · + a i for all i . Since K ( A n − i +1 ) is the free ab elian g roup on Irr( A n − i +1 ), we can wr ite a i = c i 1 [ L i 1 ] + · · · + c i,r i [ L i,r i ] − d i 1 [ M i 1 ] − · · · − d i,s i [ M i,s i ] for unique (up to renum ber ing) [ L ij ] , [ M ij ] ∈ Irr( A n − i +1 ), and c ij , d ij > 0. Now, let X • be the chain complex with trivia l differen tials and (3.5) X k =      0 if k ≥ 0 , L r i j =1 L ⊕ c ij ij if k = − 2 i < 0 is even , L s i j =1 M ⊕ d ij ij if k = − 2 i + 1 < 0 is o dd . By co nstruction, H k ( X • ) ∼ = X k v anis he s for k ≥ 0, and is pure of w eight n + 1 + ⌊ k / 2 ⌋ for k < 0 , so X • ∈ D ▽ ≤ n ( A ). It is easy to see that [ β ≥ n − i +2 X • ] = f i , so g = [ X • ] is the element we were lo oking for. Finally , w e also see from Lemma 3.2 that K ( A ≤ n ) /q i · K ( A ≤ n ) ∼ = K ( D ▽ ≤ n ( A )) /q i · K ( D ▽ ≤ n ( A )) for each i , so K ( A ≤ n ) and K ( D ▽ ≤ n ( A )) hav e the s ame completion.  Pr o of of The or em 1.1 (1) . The injectivit y of K ( A ) → K ( D ▽ ( A )) was established in Lemma 3.1. Since every Cauch y sequence in K ( A ) or K ( D ▽ ( A )) is cont ained in some s ubmo dule K ( A ≤ n ) o r K ( D ▽ ≤ n ( A )), L e mma 3.5 implies tha t K ( D ▽ ( A )) is complete, and that ˆ K ( A ) → K ( D ▽ ( A )) is an isomor phism.  4. Projective Resolutions and D erived Functors W e will prov e the r emaining parts of Theorem 1.1 in this section. Henceforth, A is a ssumed to ha ve eno ugh pro jectiv es. Since A is also as sumed to be a finite-length category , Fitting’s lemma and its c o nsequences hold; for instance, each pro jective is a direct sum of finitely many indecomp osable ones. The deg ree of an indecom- po sable pro jective P is the integer deg( P ) = wt(the unique simple quotient of P ) . COMPLETIONS OF GR OTHENDIECK GROUPS 9 F or n ∈ Z , let Pr o j( A ) n denote the se t of isomor phism cla sses o f indecomp osable pro jectives of degree n . Obviously , the map P 7→ P / ra d P induces a bijection Pro j( A ) n ∼ → Irr( A n ) . By considering the weigh t filtration (Lemma 2 .1, see also Example 2 .2), o ne can see that (4.1) deg( P ) = n implies P ∈ A ≤ n , with the co n ven tion deg (0) = − ∞ . More gener ally , the de g ree of a pr o jective ob ject is s imply the maxim um of the degrees of its indecomp osa ble summands, and the degree of an arbitrary ob ject is the degree of its pro jective cover. Definition 4.1. A bounded-ab ov e c o mplex P • of pro jectives is said to have asymp- totic al ly de cr e asing weights if for ea c h m ∈ Z , all but finitely many of the terms P i hav e deg r ee ≤ m . Lemma 4.2 . The fol lowing c onditions on an obje ct X ∈ D − ( A ) ar e e quivalent: (1) X ∈ D ▽ ≤ n ( A ) . (2) X is qu asi-isomorp hic to a b ounde d-ab ove c omplex of pr oje ctives P • with asymptotic al ly de cr e asing weights wher e e ach term P i is of de gr e e ≤ n . Pr o of. In view o f (4.1), it is obvious that condition (2) implies co ndition (1). F or the o ther implication, w e firs t consider the sp ecial case where X ∈ A . Let d b e the degree of X , a nd let Q • be a minimal pro jectiv e r esolution of X . The pro jective cov er of a simple o b ject L o ccurs as a direct summa nd of Q i (for i ≤ 0 ) if and only if Ext − i ( X, L ) 6 = 0. This can only happ en if wt( L ) ≤ d + i , so Q i is of degre e ≤ d + i . Using (4.1) aga in, w e see that the complex Q • satisfies condition (2). F or g eneral X , c ho ose a minimal pro jective reso lution Q • i for each coho mology ob ject H i ( X ). Then X is quasi-is omorphic to a complex P • with terms of the for m P i = M k ≥ 0 Q − k i + k . Let N be the lar g est in teger such that H N ( X ) 6 = 0. (Such an N exists bec a use X is b ounded ab ov e.) F or i ≤ N , let d i be the degree of H i ( X ), o r let d i = − ∞ if H i ( X ) = 0. Next, let a i = ma x { d i , d i +1 − 1 , d i +2 − 2 , . . . , d N − ( N − i ) } . Note that P i is of degree ≤ a i . In particular , each P i is o f de g ree ≤ n . Next, given m ∈ Z , there is a k 0 such that d i ≤ m for all i ≤ k 0 . Le t k = min { i − d i + m | k 0 ≤ i ≤ N } . W e cla im that for all i ≤ k , a i ≤ m . Indeed, if i ≤ k a nd 0 ≤ j ≤ N − i , then d i + j − j ≤ d i + j ≤ m if i + j ≤ k 0 , d i + j − j = i + m − ( i + j − d i + j + m ) ≤ i + m − k ≤ m if k 0 ≤ i + j ≤ N . Hence P • is a b ounded ab ov e complex of pro jectives with a symptotically decreasing weigh ts, so it sa tisfies condition (2), as desired.  10 PRAMOD N. ACHAR AND CA THARINA STR OPPEL Lemma 4.3. L et F : A → B b e a right-exact functor c ommuting with the r esp e ctive T ate twist. If F has weight amplitude α < ∞ , then the left-derive d funct or L F : D − ( A ) → D − ( B ) has the pr op erty that (4.2) L F ( D ▽ ≤ n ( A )) ⊂ D ▽ ≤ n + α ( B ) . Pr o of. Given X ∈ D ▽ ≤ n ( A ), choose a pro jective reso lution P • satisfying the condi- tion in Lemma 4.2(2). It is clear from (4.1) that all terms of the complex F ( P • ) hav e weigh ts ≤ n + α , and that for any m , only finitely terms hav e a composi- tion factor of w eight > m . The sa me then holds for its c o homology ob jects, so L F ( X ) ∈ D ▽ ≤ n + α ( B ).  W e finish now this section b y proving the r emaining parts of Theorem 1.1. Pr o of of The or em 1.1 (2) . Giv en m ∈ Z , let k b e such that for all i ≤ k , the cohomolog y H i ( X ) has weights ≤ m . Then [ X ] = [ τ ≤ k X ] + [ τ ≥ k +1 X ] wher e τ ≤ k , τ ≥ k denote the usual truncatio n functors in triangulated ca tegories. Since X is b ounded ab ov e, τ ≥ k +1 X has only finitely many nonzer o c ohomology ob jects, and it is clear that [ τ ≥ k +1 X ] = P ∞ i = k +1 ( − 1) i [ H i ( X )]. Moreov er, by construction, τ ≤ k X ∈ D ▽ ≤ m ( A ), so [ X ] − ∞ X i = k +1 ( − 1) i [ H i ( X )] ∈ K ( D ▽ ≤ m ( A )) . Thu s, the series P ( − 1) i [ H i ( X )] conv erges to [ X ]. The argument for P ( − 1) i [ P i ] is similar.  Pr o of of The or em 1.1 (3) . A description of the c o mpletion of a fre e module ca n be found in [16, § 2.4]. I t follows from that des cription that the basis of a free R 0 -mo dule spa ns a dense free ˆ R 0 -submo dule o f its completion, a nd that the t wo coincide if the basis is finite. Th us, it follo ws fr om (2.6) that Irr( A 0 ) spa ns a de ns e free ˆ R 0 -submo dule of ˆ K ( A ≤ 0 ). The cas e of Pro j( A ) 0 is somewhat differe nt, s ince this set do es not give an R 0 - basis for K ( A ≤ 0 ) in general. How ever, recall that if P ∈ Pro j( A ) 0 and if L ∈ Ir r( A 0 ) is its unique irreducible quotient, then in K ( A ≤ 0 ), we ha ve [ P ] = [ L ] + (terms in q · K ( A ≤ 0 )) . It is easy to deduce from this that the elemen ts of Pro j ( A ) 0 are linearly indep endent in K ( A ≤ 0 ): an y r elation would give r ise to a rela tion a mong elements of Irr( A 0 ). Thu s, the R 0 -submo dule K pf ( A ≤ 0 ) ⊂ K ( A ≤ 0 ) genera ted by Pro j( A ) 0 is free. It follows that the corre s ponding R -submo dule K pf ( A ) ⊂ K ( A ) is free as w ell. Since completion is left-exact, we have a natural inclus ion ˆ K pf ( A ) ⊂ ˆ K ( A ). The argu- men t of the previous paragr aph sho ws that P ro j( A ) 0 spans a free dense submo dule of ˆ K pf ( A ), so it remains only to show that this submo dule is a lso dense in ˆ K ( A ). But this fo llo ws from the fact that the class of every ob ject in D ▽ ( A ) ca n be written as a conv ergent series of pro jectives.  Pr o of of The or em 1.1 (4) . W e see from Lemma 4 .3 that L F ( D ▽ ( A )) ⊂ D ▽ ( B ), s o we certainly hav e an induced map [ L F ] : K ( D ▽ ( A )) → K ( D ▽ ( B )). Moreov er, if f ∈ K ( D ▽ ( A )) is such that [ β ≥ m ] f = 0, it follows from Lemmas 3.1 and 4.3 that [ β ≥ m + α ][ L F ] f = 0. In particular , if f ∈ I ( A ), then [ L F ] f ∈ I ( B ), so we a ctually COMPLETIONS OF GR OTHENDIECK GROUPS 11 hav e a n induced ma p [ L F ] : K ( D ▽ ( A )) → K ( D ▽ ( B )). The a ssertion that it is contin uous is then just a restatement of (4.2).  5. Examples and Applica tions 5.1. Graded mo dules o v er a graded l o cal ring. Let k b e a field and H = L i ∈ Z ≥ 0 H i a finite- dimens io nal p ositively graded connected (i.e., H 0 = k ) k -a lgebra. Then H is gr aded lo cal with maximal ideal m = L i ∈ Z > 0 H i and has, up to isomor - phism and grading shift, a unique ir reducible (finite-dimensio nal) graded H -mo dule, namely the trivial mo dule L = k = H 0 . Let H -gmod b e the categor y of finite dimen- sional Z -gra ded H -mo dules with gr ading shift functor h j i defined as ( M h j i ) i = M i − j for M = L i ∈ Z M i ∈ H -gmo d. Prop osition 5. 1. • H - gmo d with wt( L h i i ) = − i and (1) = h 1 i is a n o ether- ian and artinian mixe d ab elian c ate gory with T ate twist. • L et p ( q ) = P i ≥ 0 (dim H i ) q i b e t he Poinc ar ´ e p olynomial of H . It has non- trivial c onstant term, so it c an b e inverte d in the ring ˆ R . In fact, [ L ] = p ( q ) − 1 [ H ] in ˆ K ( A ) , and e ach of [ L ] and [ H ] give s an ˆ R -b asis for ˆ K ( A ) (which is a fr e e ˆ R -mo dule of r ank 1 ). Pr o of. The first statement is just E x ample 2.2. By Theor em 1 .1(3), each of [ L ] and [ H ] gives an ˆ R -basis, since H is lo cal, hence has up to isomor phism and grading shift a unique simple mo dule. The formula [ L ] = p ( q ) − 1 [ H ] follows then just by a basis transformation.  A natura l example arising in this context is the cohomology ring H = H ∗ ( X ) of a smo oth pro jective complex algebraic v a riety X . If w e choose for instance X = CP 1 then H = H ∗ ( X ) = C [ x ] / ( x 2 ) with Poincare p olynomia l p ( q ) = 1 + q 2 , and we obtain the equation [ L ] = 1 1+ q 2 [ H ] = (1 − q 2 + q 4 − q 6 + . . . )[ H ] in ˆ K ( A ). More genera lly , if H = H ∗ ( X ), where X = Gr( i, n ) is the Grassmannia n v ariety of complex i -pla nes in C n , or any partial flag v ariety X = GL( n, C ) /P for some parab olic subgroup P , then the complex cohomolog y rings H ∗ ( X ) are explicitly known (see for instance [8], [9]). W e hav e the equality [ H ] =  n d 1 ,...,d r  [ L ] in the Grothendieck gr o up of g raded H -mo dules, where  n d 1 , . . . , d r  = [ n ]! [ d 1 ]![ d 2 ]! · · · [ d r ]![( n − d 1 − · · · − d r )]! denotes the quant um binomial c o efficient defined by taking the quantum n umbers [ n ] = q 2 n − 1 q 2 − 1 = 1 + q 2 + · · · + q 2( n − 1) for n ∈ Z > 0 and their facto rials [ n ]! = [1][2][3] · · · [ n ] with [0]! = 1. In terpreting this q ua n tum binomial coefficient as a formal pow er series in q , we o btain the equation [ L ] = 1  n d 1 ,...,d r  [ H ] in ˆ K ( A ). By Theorem 1 .1, L and [ H ] eac h form a n ˆ R -basis of ˆ K ( A ), and the transformatio n matrix is given b y quan tum binomial coefficients and their in verses. This transforma tion matrix also o ccurs in the r epresentation theor y of the sma lle st quantum gro up U q ( sl 2 ), as we will see in the next section. 12 PRAMOD N. ACHAR AND CA THARINA STR OPPEL 5.2. Categorification of fini te-dimensio nal i rreducible m o dules for quan- tum s l 2 . Let C ( q ) b e the field o f ra tional functions in an indeterminate q . Let U q = U q ( sl 2 ) b e the asso ciative algebr a over C ( q ) generated b y E , F, K , K − 1 sub- ject to the relations: K K − 1 = K − 1 K = 1 , K E = q 2 E K , K F = q − 2 F K, E F − F E = K − K − 1 q − q − 1 . Let ¯ V n be the unique (up to isomorphism) irreducible mo dule for sl 2 of dimension n + 1. Denote by V n its quantum analogue (of type I), that is the ir reducible U q ( sl 2 )- mo dule with basis { v 0 , v 1 , . . . , v n } such that (5.1) K ± 1 v i = q ± (2 i − n ) v i E v i = [ i + 1 ] q − i − 1 v i +1 F v i = [ n − i + 1] q 1 − i v i − 1 . Note that it is defined ov er R . The c hosen basis is the canonical basis in Lusztig’s theory o f canonical bas e s ([12], [6]) and pairs via a bilinear form with the dual c anonic al b asis given by v i = q − i ( n − i ) 1 ( n i,n − i ) v i . Hence, pass ing to the completion ˆ V n of V n we hav e an isomorphism of ˆ R -modules ˆ V n 7→ n M i =0 ˆ K ( A i ) (5.2) v i 7→ [ H ∗ (Gr( i, n )) h i ( n − i ) i ] v i 7→ [ L i ] where A i denotes the mixed ab elian category H ∗ (Gr( i, n ))-gmo d with unique simple ob ject L i of weigh t ze ro. The action of the quantum group can then be realized via corres p ondences : if we let Gr( i, i + 1 , n ) b e the v ar iet y of partial flags ( i -plane) ⊂ (( i + 1)-pla ne) ⊂ C n , then H ∗ (Gr( i, i + 1 , n )) is natura lly a ( H ∗ (Gr( i, n )) , H ∗ (Gr( i + 1 , n ))-bimodule or a ( H ∗ (Gr( i + 1 , n )) , H ∗ (Gr( i, n ))-bimo dule. T ens oring (with appropria te grading shifts) with these bimo dules defines exact endo functor s on ⊕ n i =0 ( A i ) which induce the action o f E and F on ⊕ n i =0 ˆ K ( A i ) given b y the formula (5.1) via the isomorphism (5.2). F or details se e [7, Section 6 ] and [4 ] in the non-graded v ersio n. 5.3. Quotien t categories. Let k b e an algebra ically closed field, and let A = L i ∈ Z ≥ 0 A i be a finite-dimensional p ositively graded k -algebra , semisimple in de- gree zer o. Let A = ⊕ r i =1 Ae i be the decomp osition in to indec o mpos able pro jective mo dules with simple q uotien ts L i , 1 ≤ i ≤ r . Let A = A -gmo d b e the mixed category of finite-dimensional g raded right A -mo dules with T ate twist (1) = h 1 i . Assume w e are given a Serre subcateg ory S I of A stable under T ate t wist. That is, S I is a full sub catego ry co nsisting of all mo dules which have comp osition factor s only of the form L i h j i , where j ∈ Z and i ∈ I for so me fixed subset I o f { 1 , . . . , r } . Let Q : A → A / S I be the quotient functor to the Serre quo tien t A / S I . Under the identification o f A / S I with gra ded modules ov er E nd A ( P I ), where P I = L i / ∈ I Ae i we hav e Q = Hom A ( L i ∈ I Ae i , − ). In particular, Q is exact and has left a djoin t Q ′ : M 7→ M ⊗ End A ( P I ) P I . Now the following is just a direct applica tion of our main result: COMPLETIONS OF GR OTHENDIECK GROUPS 13 Prop osition 5.2. The functors Q : A → A / S I and Q ′ : A / S I → A ar e exact and right exact r esp e ctively, c ommute with T ate twist and have finite weight amplitude. 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