On semi-infinite cohomology of finite dimensional graded algebras

ON SEMI-INFINITE COHOMOLOGY OF FINITE DIMENSIONAL GRADED ALGEBRAS ROMAN BEZRUKA VNIKO V, LEONID POSITSELSKI Abstract. W e describe a general setting for the definition of semi-infinite cohomology of finite dimensional algebras, and p rovide an int erpretation of suc h cohomology in terms of derived categories. W e apply this interpretat ion to compute semi-infinite coho mology of s ome modules ov er the small group at a ro ot of unit y , generalizing an earlier result of S. Arkhipov (conjectu red by B. F eigin). 1. Introduction Semi-infinite cohomolog y of asso ciative algebras was studied b y S. Ark hip ov in [Ar1], [Ar2], [Ar3]; see also [S] (these works are partly based on an earlier pap er by A. V o ronov [V] where the co rresp onding co nstructions were int ro duced in the context of Lie algebr as). Recall tha t the definition of semi-infinite co homology (see e.g. [Ar1], Definition 3.3.6) works in the following s et-up. W e are g iven an as so ciative graded algebr a A , t wo s ubalgebras N , B ⊂ A suc h that A = N ⊗ B as a vector space, satisfying some additional assumptions. In this situation the space of semi-infinite Ext’s, E xt ∞ / 2+ • ( X, Y ) is defined for X , Y in the appropria te derived categor ies. The definition makes use o f ex plicit complexes (a version of the ba r resolution). The aim of this no te is to show that, at least under cer tain simplifying ass umptions, E xt ∞ / 2+ • ( X, Y ) is a particular c ase of a general categor ical constr uction. T o describ e the situation in more detail, recall that sta rting from an alg ebra A = N ⊗ B as ab ov e, one can define ano ther algebr a A # , which also contains subalgebra s ident ified with N , B , so that A # = B ⊗ N . The semi-infinite Ext’s, E xt ∞ / 2+ • ( X, Y ) are then defined fo r X ∈ D ( A # − mod ), Y ∈ D ( A − mod ), where D ( A # − mod ), D ( A − mod ) are derived categories of mo dules with cer tain re s trictions on the grading. Our categ orical interpretation relies o n the following co nstruction. Giv en small categorie s A , A ′ , B with functor s Φ : B → A , Φ ′ : B → A ′ one can define for X ∈ A , Y ∈ A ′ the set of ”morphisms from X to Y through B ” ; w e denote this set by H om A B A ′ ( X, Y ). W e then show that if A = D b ( A # − mod ), A ′ = D b ( A − mod ), and B is the full triangula ted subcateg ory in A generated by N -injective A # -mo dules, then, B is identified with a full s ubca tegory in A ′ generated by N -pro jective A - mo dules, and, under certain assumptions, one has (1) E xt ∞ / 2+ i ( X, Y ) = H om A B A ′ ( X, Y [ i ]) . Notice that description (1) of E xt ∞ / 2+ i ( X, Y ) is ”internal” in the derived ca t- egory , i.e. refers o nly to the derived categ ories and their full subca tegories rather than to a particular category of complexes. 1 2 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI An example of the situatio n considered in this pap er is pr ovided by a small quantum gro up at a ro o t of unity [L], or by the res tricted env eloping a lgebra of a simple Lie algebra in p ositive c har acteristic. Computation of s emi-infinite coho- mology in the former case is due to S. Arkhip ov [Ar1 ] (the answer sug gested as a conjecture by B. F e ig in). An attempt to find a natur al interpretation of this a nswer was the s tarting p oint for the present work. In section 6 w e sketch a gener a lization of Arkhip ov’s Theorem based o n our description o f semi- infinite co homology and the results o f [ABG], [BL]. Similarly , the main r esult of [B1 ] yields a description of semi-infinite cohomo logy of tilting mo dules over the “big” q uantum gro up r e- stricted to the small qua n tum gro up as co ho mology with supp ort of coherent IC sheav es on the nilpotent cone [B2]. It should b e no ted that some definitions o f semi-infinite coho mology fo und in the literature apply in a more g e ne r al (or different) situation than the o ne cons idered in the present pa p er. An imp or tant example is pro vided by affine Lie alg ebras; in fact, semi-infinite cohomo logy has first b een defined in this con text, rela ted to the ph ysic a l notio n of BRST reduction. W e hop e that our approach can b e extended to such more gener al setting. Some of the ingredients needed for the g e neralization are provided by [P ]. The pa pe r is orga nized as follows. Section 2 is devoted to basic genera l facts ab out “ Ho m through a category ”. Section 3 contains the definition of the a lgebra A # and its pr op eties. In section 4 we re call the definition of semi-infinite coho- mology in the present context. In section 5 w e prov e the main result linking that definition to the general categor ical construction of section 2. In section 6 we discuss the example of a small quantum gro up. Ac kno wledgem en ts. W e a re gra teful to S. Arkhip ov for helpful discussio ns. This work ow es its existence to W. So ergel — whe n r efereeing the (presently unpub- lished) preprint [B3] submitted to the Journa l of Algebra he suggested to extend the results to a greater gener ality; this is accomplishe d in the present paper. W e thank W olfgang for the stimulating sugge s tion. R.B. was pa rtially supp orted by an NSF grant. He w or ked on this pap er while visiting Pr inceton IAS, the stay was funded through grants o f B ell Compa nies, Oswald V eblen F und, Ja mes D W olfen- son F und and The Am bro se Monell F oundation. L.P . acknowledges the financial suppo rt from CRDF, INT AS, and P . Deligne’s 2 004 Balzan prize. 2. Morphisms through a ca tegor y 2.1. Generalities. Let A , A ′ , B be s mall categories , and Φ : B → A , Φ ′ : B → A ′ be functors. Fix X ∈ Ob ( A ), Y ∈ O b ( A ′ ). W e define the set of ”morphisms from X to Y thro ugh B ” as π 0 of the category of diagrams (2) X − → Φ( Z ); Φ ′ ( Z ) − → Y , Z ∈ B . This set will b e denoted b y H om A B A ′ ( X, Y ). Thus elements of H om A B A ′ ( X, Y ) a re diagrams of the form (2), with tw o diagrams identified if there exists a morphism betw een them. If the categ o ries a nd the functors ar e additive (resp ectively , R - linear for a com- m utative r ing R ), then H om A B A ′ ( X, Y ) is an ab elian group (res pe ctively , a n R - mo dule); to add tw o diagra ms of the for m (2) one sets Z = Z 1 ⊕ Z 2 with the o bvious arrows. SEMI-INFINITE COHOMOLOGY 3 W e ha ve the c o mp o sition map H om A ( X ′ , X ) × H om A B A ′ ( X, Y ) × H om A ′ ( Y , Y ′ ) → H om A B A ′ ( X ′ , Y ′ ); in par ticula r, in the additive setting H om A B A ′ ( X, Y ) is an E nd ( X ) − E nd ( Y ) bimo dule. 2.2. Pro/Ind representable case. If the left adjoin t functor Φ L to Φ is defined on X , then we have H om A B A ′ ( X, Y ) = H om A ′ (Φ ′ (Φ L ( X )) , Y ) , bec ause in this ca s e the ab ov e catego r y cont ra cts to the sub categor y of diagrams of the form X can − → Φ(Φ L ( X )); Φ ′ (Φ L ( X )) → Y , where c an stands for the adjunction morphism. If the right adjoin t functor Φ ′ R is defined on Y , then H om A B A ( X, Y ) = H om A ( X, Φ(Φ ′ R ( Y ))) for similar reasons. More genera lly , w e hav e Prop ositi o n 1. Fix X ∈ A and Y ∈ A ′ . Assume that the functor F X : B → Sets , Z 7→ H om A ( X, Φ( Z )) c an b e r epr esente d as a filter e d inductive limit of r epr e- sentable funct ors Z 7→ H om B ( ι ( S ) , Z ) , wher e S ∈ I and ι : I → B is a fun ctor b et we en smal l c ate gories. The n we have H om A B A ′ ( X, Y ) = lim − → S ∈I H om A ′ (Φ ′ ι ( S ) , Y ) . Alternativel y, assume that the fu n ctor F Y : B op → Sets , Z 7→ H om (Φ ′ ( Z ) , Y ) c an b e r epr esente d as a filter e d inductive limit of r epr esentable funct ors Z 7→ H om B ( Z, ι ( S )) , wher e S ∈ I . Then H om A B A ′ ( X, Y ) = lim − → S ∈I H om A ( X, Φ ι ( S )) . R emark 1 . W e will only use the Prop o s ition in the case when the ca tegory I is the ordered set of po sitive (or negative) in tegers . R emark 2 . The assumptions of the Prop os ition can be rephra sed by saying, in the first cas e, tha t the functor F X is r epresented by the pr o-ob ject lim ← − ι , and in the second case, that the functor F Y is repres e n ted b y the ind-ob ject lim − → ι . R emark 3 . The r e sults of the P rop ositio n ca n b e further g eneralized as fo llows. Fix X ∈ A and Y ∈ A ′ ; let ι : B ′ → B b e a functor b etw een small cate- gories. Assume that either for an y morphism X → Φ( Z ) the category of pairs of mo rphisms X → Φ ι ( S ), ι ( S ) → Z making the triangle X → Φ ι ( S ) → Φ( Z ) commutativ e is non-empty and connected, o r for a ny morphis m Φ ′ ( Z ) → Y the category of pairs of mo rphisms Z → ι ( S ), Φ ′ ι ( S ) → Y ma king the triang le Φ ′ ( Z ) → Φ ′ ι ( S ) → Y commutativ e is non-empt y and connected. Then the natural map H om A B ′ A ′ ( X, Y ) → H om A B A ′ ( X, Y ) is an is o morphism. 4 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI Example 1 . Let M b e a Noetherian sc heme, and A = A ′ = D b ( C oh M ) b e the bo unded derived category of coherent sheav es on M ; let Φ = Φ ′ : B ֒ → A be the full embedding of the subcatego ry of co mplexes whose cohomology is suppor ted on a closed s ubset i : N ֒ → M . Then the right a djo int functor i ∗ ◦ i ! is w ell- defined as a functor to a ”la rger” derived categor y of quasi-coher ent sheav es, while the left adjoint functor i ∗ ◦ i ∗ is a well-defined functor to the Grothendieck-Serre dual ca tegory , the derived catego ry of pro-co he r ent shea ves (in tro duced in Deligne’s app endix to [H]). Let C • be a co mplex of coher ent s he aves repr esenting the o b ject X ∈ D b ( C oh M ). Let X n be the ob ject in the derived categor y represented by the c omplex C i n = C i ⊗ O M / J n N (the nonder ived tenso r pro duct) wher e J N is the idea l sheaf of N . F or F ∈ B we hav e lim − → H om ( X n , F ) g − → H om ( X, F ). Th us a pply ing Prop osition 1 to ι : Z + → B given by ι : n 7→ X n , w e get: H om A B A ( X, Y ) = lim − → H om ( X n , Y ) = H om ( i ∗ ( i ∗ ( X )) , Y ) = H om ( X , i ∗ ( i ! ( Y ))) . In particular, if X = O M is the structure sheaf, we get (3) H om A B ( O M , Y [ i ]) = H i N ( Y ) , where H • N ( Y ) stands fo r cohomolog y with supp ort on N (see e.g. [H]). 2.3. T riangulated full emb eddings. In all exa mples b elow A , A ′ , B will b e triangulated, and Φ, Φ ′ will b e full embeddings of a thick sub ca tegory . Assume that this is the case, and moreov er A = A ′ , Φ = Φ ′ . Prop ositi o n 2. W e have a long exact se quenc e H om A B A ( X, Y ) → H om A ( X, Y ) → H om A / B ( X, Y ) → H om A B A ( X, Y [1 ]) . Pr o of. The co nnecting ho momorphism H om A / B ( X, Y ) → H om A B A ( X, Y [1]) is constructed as fo llows. Let X ← X ′ → Y b e a fraction of mor phisms in A repre- senting a morphism X → Y in A / B ; the co ne K of the morphism X ′ → X b elo ngs to B . Ass ign to this fraction the diag ram X → K ; K → Y [1], wher e the morphism K → Y [1] is defined as the co mp os ition K → X ′ [1] → Y [1]. All the required verifications ar e straig htf or ward; the hardest o ne is to chec k that the sequence is exa c t a t the term H om A / B ( X, Y ). Here o ne s hows that for any tw o diagrams X → K ′ ; K ′ → Y [1] a nd X → K ′′ ; K ′′ → Y [1] co nnected b y a morphism K ′ → K ′′ making the tw o triangles commute, and for a ny t wo fractions X ← X ′ → Y and X ← X ′′ → Y to which the connecting homomorphism assig ns the resp ective diag rams, o ne can construct a morphism X ′ → X ′′ making the triangle formed by X ′ , X ′′ , X commutative, and the triangle formed by X ′ , X ′′ , Y will then co mmute up to a morphism X → Y . 3. Algebra A # and modules over it All algebras b elow will b e a sso ciative and unital alg e bras over a field k . 3.1. The s et-up. We make the fol lowing assumptions. A Z -gr aded finite dimen- sional algebra A and g r aded suba lgebras K = A 0 , B = A ≤ 0 , N = A ≥ 0 ⊂ A a re fixed and satisfy the following conditions: (1) B = A ≤ 0 , N = A ≥ 0 are gr aded b y , respectively , Z ≤ 0 , Z ≥ 0 , and K = B ∩ N is the compo nent of degre e 0 in N . (2) K = A 0 is semisimple and the ma p N ⊗ K B → A provided b y the multipli- cation map is an isomorphism. SEMI-INFINITE COHOMOLOGY 5 (3) Co nsider the K - N - bimo dule N ∨ = H om K op ( N , K ). W e require that the tensor pro duct S = N ∨ ⊗ N A is an injective r ight N -mo dule. 3.2. N -mo dul es, N ∨ -como dul e s, and N # -mo dul es. By a ”mo dule” we will mean a finite dimensional gra ded left mo dule, unless stated otherwise (though all the results of this section are als o applicable to ungr aded or infinite dimensio na l mo dules). Since N is a finitely generated pro jective right K -mo dule, the K - bimo dule N ∨ has a natural structure of a c o ring, i.e., there is a com ultiplication map N ∨ → N ∨ ⊗ K N ∨ and a counit map N ∨ → K satisfying the usual coass o ciativity and counity conditions. Consequently , there is a natural algebra structure on N # = H om K op ( N ∨ , K ) a nd an injective morphism of algebras K → N # . The category of rig ht N -mo dules is isomorphic to the ca tegory of r ight N ∨ -como dules and the category of left N # -mo dules is isomor phic to the catego ry of le ft N ∨ -como dules. In particular, N ∨ is an N # - N -bimo dule. Recall that the cotens or pro duct P N ∨ Q o f a r ight N ∨ -como dule P and a left N ∨ -como dule Q is defined as the kernel of the pair of maps P ⊗ K Q ⇒ P ⊗ K N ∨ ⊗ K Q one of which is induced by the coac tio n map P → P ⊗ K N ∨ and the other by the coaction map Q → N ∨ ⊗ K Q . There ar e natura l is omorphisms P N ∨ N ∨ ∼ = P and N ∨ N ∨ Q ∼ = Q . Prop ositi o n 3. a) i) F or any right N -mo dule P and any left N -mo dule Q ther e is a natur al map of k -ve ctor sp ac es P ⊗ N Q → P N ∨ ( N ∨ ⊗ N Q ) , which is an isomorphi sm, at le ast, when P is inje ctive or Q is pr oje ctive. ii) F or any right N -mo dule P and any left N # -mo dule Q ther e is a natu r al map of k -ve ctor sp ac es P ⊗ N ( N N ∨ Q ) → P N ∨ Q , which is an isomorphism, at le ast, when P is pr oje ctive or Q is inje ctive. b) The funct ors P 7→ N ∨ ⊗ N P and M 7→ N N ∨ M a r e mutual ly inverse e qu ivalenc es b etwe en the c ate gories of pr oje ctive left N -mo dules and inje ctive lef t N # -mo dules. c) The funct ors P 7→ N ∨ ⊗ N P and M 7→ N N ∨ M ar e mutual ly inverse tensor e qu ivalenc es b et we en t he tensor c ate gory of N -bimo dules that ar e pr oje ctive left N - mo dules with the op er ation of t ensor pr o duct over N and the tensor c ate gory of N # - N -bimo dules that ar e inje ctive left N # -mo dules with the op er ation of c otensor pr o duct over N ∨ . Pr o of. Both as sertions of (a) state exis tence o f a sso ciativity (iso)morphisms con- necting the tensor and cotensor pr o ducts. In particular, in (i) w e ha ve to con- struct a natural ma p ( P N ∨ N ∨ ) ⊗ N Q → P N ∨ ( N ∨ ⊗ N Q ). More gener - ally , let us consider an arbitrar y N # - N -bimo dule R and construct a natura l map ( P N ∨ R ) ⊗ N Q → P N ∨ ( R ⊗ N Q ). This map can b e defined in tw o equiv a - lent wa ys. The first approa ch is to tak e the tensor pro duct of the exact sequence of right N -mo dules 0 → P N ∨ R → P ⊗ K R → P ⊗ K N ∨ ⊗ K R with the left N -mo dule Q . Since the resulting sequence is a complex, there exists a unique map ( P N ∨ R ) ⊗ N Q → P N ∨ ( R ⊗ N Q ) making a co mm utative triangle with the natural maps of ( P N ∨ R ) ⊗ N Q and P N ∨ ( R ⊗ N Q ) into P ⊗ K R ⊗ N Q . It is clear that this map is an iso morphism whenever Q is a flat N -mo dule. Analogously , for a ny P , Q , R there is a natur al iso morphism ( P N ∨ R ) ⊗ K Q ∼ = P N ∨ ( R ⊗ K Q ), since K is semisimple. The se c ond w ay is to tak e the cotenso r pro duct of the ex act sequence of left N ∨ -como dules R ⊗ K N ⊗ K Q → R ⊗ K Q → R ⊗ N Q → 0 with the 6 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI right N ∨ -como dule P . Aga in, since the resulting sequence is a co mplex, there exists a unique ma p ( P N ∨ R ) ⊗ N Q → P N ∨ ( R ⊗ N Q ) making a commut ative triang le with the na tural maps fro m P N ∨ R ⊗ K Q to ( P N ∨ R ) ⊗ N Q and P N ∨ ( R ⊗ N Q ). Clearly , this map is an isomorphism whenever P is a coflat N ∨ -como dule (i.e., the cotensor pro duct with P pr eserves exactness). Now any injective rig ht N -mo dule is a coflat right N ∨ -como dule, since it is a direct summand of a direct sum o f copies of N ∨ . The tw o asso cia tivity maps that we have constructed coincide, since the relev ant sq uare diagra m commutes. The pro o f of (ii) is analo gous. T o prov e (b), no tice the isomor phis ms N N ∨ ( N ∨ ⊗ N P ) ∼ = N ⊗ N P ∼ = P and N ∨ ⊗ N ( N N ∨ M ) ∼ = N ∨ N ∨ M ∼ = M for a pr o jective left N -mo dule P and an injective left N # -mo dule M . S ince a pro jectiv e left N -mo dule is a direc t summand o f an N -mo dule of the form N ⊗ K V and an injectiv e left N # -mo dule is a direct summand of an N # -mo dule of the form Hom K ( N # , V ) ∼ = N ∨ ⊗ K V for a K - mo dule V , the functors in question tra nsform pro jective N -mo dules to injective N # -mo dules and vice versa. T o deduce (c), notice the isomorphism ( N ∨ ⊗ N P ) N ∨ ( N ∨ ⊗ N Q ) ∼ = N ∨ ⊗ N P ⊗ N Q for a N -bimo dule P and a pro jective left N -mo dule Q . It is straightforward to c heck that this isomorphism preserves the ass o ciativity constraints. 3.3. Definition of A # . It follows fro m the condition (2) tha t A is a pr o jective left N -mo dule. B y Prop osition 3(c), the tenso r product S = N ∨ ⊗ N A is a ring o b ject in the tensor catego r y o f N ∨ -bicomo dules with resp ect to the cotensor pro duct ov er N ∨ . By the condition (3) and the rig ht analogue of Prop osition 3(c), the cotensor pro duct A # = S N ∨ N # is a ring ob ject in the tensor categ ory of N # -bimo dules with resp ect to the tenso r pro duct ov er N # . The embedding N → A induces injectiv e maps N ∨ → S and N # → A # ; these are unit mor phisms of the ring ob jects in the co rresp onding tensor ca tegories. So A # has a natur al asso cia tive algebra struc tur e and N # is identified with a subalgebra in A # . Notice that A # is a pro jective right N # -mo dule by the definition. Prop ositi o n 4. Ther e is a n atu r al isomorphism b etwe en the N # - A -bimo dule S = N ∨ ⊗ N A and the A # - N -bimo dule S # = A # ⊗ N # N ∨ , making S an A # - A -bimo dule. Mor e over, ther e ar e isomorphisms: A # ∼ = E nd A op ( S ) , A op ∼ = E nd A # ( S # ) . Pr o of. By the definition, w e hav e S # = ( S N ∨ N # ) ⊗ N # N ∨ ∼ = S N ∨ N ∨ ∼ = S , since S is an injectiv e right N -mo dule. Let us sho w that the righ t A -module and the left A # -mo dule structures on S ∼ = S # commute. The isomor phis m S ⊗ N A ∼ = S ⊗ N ( N N ∨ S ) ∼ = S N ∨ S transforms the rig ht action map S ⊗ N A → S into the map S N ∨ S → S defining the structure of ring ob ject in the tensor ca tegory of N ∨ - bicomo dules on S . Analog ously , the isomo rphisms A # ⊗ N # S # ∼ = ( S N ∨ N # ) ⊗ N # S ∼ = S N ∨ S and S # ∼ = S transform the left action ma p A # ⊗ N # S # → S # int o the same map S N ∨ S → S . Finally , there is an isomorphism A # ⊗ N # S ⊗ N A ∼ = ( S N ∨ N # ) ⊗ N # S ⊗ N ( N N ∨ S ) ∼ = S N ∨ S N ∨ S , so the r ight a nd left actions commute since S is an asso c ia tive ring ob ject in the tensor categ ory of N ∨ -bicomo dules. Now we hav e H om A op ( N ∨ ⊗ N A, N ∨ ⊗ N A ) ∼ = H om N op ( N ∨ , N ∨ ⊗ N A ) ∼ = ( N ∨ ⊗ N A ) N ∨ N # = A # and H om A # ( A # ⊗ N # N ∨ , A # ⊗ N # N ∨ ) ∼ = H om N # ( N ∨ , A # ⊗ N # N ∨ ) ∼ = N N ∨ ( A # ⊗ N # N ∨ ) ∼ = A . SEMI-INFINITE COHOMOLOGY 7 3.4. N -pro jectiv e (inje ctive) mo dules. By A − mod w e denote the category of (graded finite dimensional) left A -mo dules. Consider the full s ub ca tegories A − mod N − pr o j ⊂ A − mod , A # − mod N # − inj ⊂ A # − mod consisting of modules whos e restrictio n to N is pro jectiv e (r esp ectively , restriction to N # is injectiv e). W e abbrev iate D ( A ) = D b ( A − mod ), D ( A # ) = D b ( A # − mod ), and let D ∞ / 2 ( A ) ⊂ D ( A ), D ∞ / 2 ( A # ) ⊂ D ( A # ) b e the full triangulated sub c ategories generated b y A − mod N − pr o j , A # − mod N # − inj resp ectively . Prop ositi o n 5 . We have c anonic al e quivalenc es: A − mod N − pr o j ∼ = A # − mod N # − inj , D ∞ / 2 ( A ) ∼ = D ∞ / 2 ( A # ) . Pr o of. Let us show that the a djoint functors P 7→ S ⊗ A P and M 7→ H om A # ( S, M ) betw een the catego ries A − mod and A # − mod induce an equiv a lence b e t ween their full s ub ca tegories A − mod N − pr o j and A # − mod N # − inj . It suffices to chec k that the adjunction mor phisms P → H om A # ( S, S ⊗ A P ) and S ⊗ A H om A # ( S, M ) → M are isomor phisms when an A -module P is pro j ective over N and a n A # -mo dule M is injective over N # . Ther e ar e natural is omorphisms S ⊗ A P ∼ = N ∨ ⊗ N P a nd H om A # ( S, M ) ∼ = H om N # ( N ∨ , M ) ∼ = N N ∨ M , s o it r emains to apply Prop o si- tion 3(b). T o obtain the equiv alence o f c a tegories D ∞ / 2 ( A ) ∼ = D ∞ / 2 ( A # ), it suffices to chec k that D ∞ / 2 ( A ) is equiv alent to the bounded derived catego ry of the exact category A − mod N − pr o j and D ∞ / 2 ( A # ) is eq uiv alent to the b o unded derived cat- egory of the exact categ o ry A # − mod N # − inj . Let us pr ov e the former; the pr o of of the latter is analogous. It suffices to chec k that for an y b ounded complex of N -pro jectiv e A -mo dules P and an y bo unded complex of A -mo dules X together with a q uasi-isomo rfism X → P there exists a bo unded complex of N -pro jectiv e A -mo dules Q tog ether with a quasi- isomorphism Q → X . Let Q ′ be a b o unded ab ov e complex of pro jective A -mo dules mapping quasi-isomor phically into X ; then the canonical truncatio n Q ′ ≥− n for lar g e enough n pr ovides the desired co mplex Q . 3.5. The case of an in vertible ent wi ning map. Cons ider the multiplication map φ : B ⊗ K N → A ∼ = N ⊗ K B . It yields a ma p ψ : N ∨ ⊗ K B → H om K op ( N , B ) ∼ = B ⊗ K N ∨ . Assume that the ma p ψ is an iso morphism and consider the inv erse map ψ − 1 : B ⊗ K N ∨ → N ∨ ⊗ K B . By the analog ous ”low ering of indices” we obtain from it a map N # ⊗ K B → H om K op ( N ∨ , B ) = B ⊗ K N # that will be deno ted b y φ # . Then the algebra A # can b e a lso defined a s the unique ass o ciative a lgebra with fixed em b eddings o f N # and B into A # such that i) the embeddings N # → A # and B → A # form a commut ative s quare with the embeddings K → N # and K → B ; ii) the mult iplicatio n map induces an isomorphism B ⊗ K N # → A # ; iii) the map induced b y the multiplication map N # ⊗ K B → A # ∼ = B ⊗ K N # coincides with φ # . Indeed, the existence of an algebra A with subalgebras N and B in ter ms of which the map φ is defined can b e eas ily seen to b e equiv a lent to the ma p ψ sa tisfying the equations of a r ight ent wining structure for the coring N ∨ and the algebra B (see [BW] or [P] for the definition). When ψ is inv ertible, it is a right ent w ining structure if and o nly if ψ − 1 is a left e n twining s tructure, a nd the latter is equiv alent to the existence of an algebra A # satisfying (i-iii). 8 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI T o s how that the t wo definitions of A # are equiv alent, it suffices to chec k that the ring ob ject S in the tensor category of N ∨ -bicomo dules ca n b e constructed in terms of the en twining structure ψ in the w ay expla ined in [Brz] or [P]. 3.6. The case of a sel f-injectiv e N . Assume that N is self-injective. In this cas e A # is canonically Morita equiv alent to A ; the equiv a lence is defined by the A # - A - bimo dule S , so it sends A − mod N − pr o j = A − mod N − inj to A # − mod N # − pr oj = A # − mod N # − inj . Indeed, N ∨ is obviously an injective g e ne r ator of the category of right N - mo dules. Since every injective N -mo dule is pro jective, N ∨ is a pro jectiv e right N -mo dule. Since N is an injective right N - mo dule, it is a dir ect summand o f a finite direct s um o f copies of N ∨ . So N ∨ is a pro jective gener a tor of the categor y of right N - mo dules; hence S = N ∨ ⊗ N A is a pro jective g enerator of the c ategory of right A -mo dules . No w it r e mains to use Pr op osition 4. Analogously , N # is Mo r ita equiv a lent to N ; hence N # is also self-injective. If N is F rob enius, N # is isomorphic to N and A # is iso morphic to A . Indeed, K is a lso F rob enius; choose a F ro b enius linear function K → k ; then the right K - mo dule Hom k ( K, k ) is isomor phic to K . Hence a F rob enius linear function N → k lifts to a right K -mo dule ma p N → K . Now the comp osition N ⊗ K N → N → K of the multiplication map N ⊗ K N → N and the r ight K -module map N → K defines an isomorphism of right N -mo dules N → H om K op ( N , K ) = N ∨ . By Prop o s ition 4, this lea ds to the isomor phism A # ∼ = A a nd analog ously to the isomorphism N # ∼ = N ; these isomo rphisms are compatible with the embeddings N → A and N # → A # , but not with the em b eddings of K to N a nd N # , in general. 4. Definitions of E xt ∞ / 2 by explicit complexes 4.1. Conca ve and conv ex resolutions. A complex of gr aded mo dules will b e called c onvex if the gra ding ”go es down”, i.e. for a ny n ∈ Z the sum of g raded comp onents of degr ee more than n is finite dimensiona l; it will b e ca lled non-s trictly c onvex if the grading ” do es not go up” , i.e. the gra ded comp onents of high enough degree v anish. A complex o f g raded mo dules will be ca lled c onc ave (resp ectively non-strictly c onc ave ) if the grading ”go es up” (r esp ectively ”do es not go down”) in the similar sense. An A # -mo dule M will b e called we akly pr oje ctive r elative to N # if for a ny A # - mo dule J whic h is injective as an N # -mo dule one has E xt i A # ( M , J ) = 0 for a ll i 6 = 0. Analogo usly one defines A -mo dules we akly inje ctive r elative to N . Notice that an y A # -mo dule induced fro m an N # -mo dule is weakly pro jectiv e rela tive to N # . The cla ss o f A # -mo dules weakly pro jective relative to N # is c losed under extensions and kernels of surjective morphisms. Lemma 1. i) Any A -mo dule admits a left c onc ave r esolution by A -m o dules which ar e pr oje ctive as N -mo dules. Any A # -mo dule admits a left non-strictly c onvex r esolution by A # -mo dules which ar e we akly pr oje ct ive r elative to N # . ii) Any fi nite c omplex of A -mo dules is a quasiisomorphic quotient of a b ou n de d ab ove c onc ave c omplex of N -pr oje ctive A -mo dules. Any finite c omplex of A # - mo dules is a quasiisomorphic quotient of a b oun de d ab ove non-s t rictly c onvex c om- plex of A # -mo dules we akly pr oje ctive r elative to N # . SEMI-INFINITE COHOMOLOGY 9 Pr o of. T o deduce (ii) from (i) choos e a quasiisomo rphic surjection onto a given co m- plex C • ∈ C om b ( A − m od ) from a complex o f A -pro jective mo dules P • ∈ C om − ( A − mod ) (notice that condition (2) of 3.1 implies that an A -pro jectiv e mo dule is als o N -pro jectiv e), and apply (i) to the mo dule of co cyc les Z n = P − n /d ( P − n − 1 ) for large n . T o chec k (i) it suffices to find for any M ∈ A − mod a surjection P ։ M , where P is N - pro jective, a nd if n is s uch that all gr aded comp onents M i for i < n v anish, then P i = 0 for i < n and P n g − → M n . It suffices to tak e P = I nd A B ( Res A B ( M )). It is indeed N -pro jectiv e, b eca use of the equality (4) Res A N ( I nd A B ( M )) = I nd N K ( M )) , which is a consequenc e of assumption (2). The second a ssertions of (i) and (ii) ar e prov en in the a na logous w ay , except that one us e s the induction from N # (this is even simpler, as weak rela tive pro jectivit y of the relev a nt mo dules is just obvious). 4.2. Definition of se mi-infini te Ext’s . Definition 1 . (cf. [FS], § 2.4) The assumptions (1–3) of 3.1 are enforced. Let X ∈ D ( A # ) and Y ∈ D ( A ). Let P X ւ be a non-strictly conv ex bounded ab ov e complex of A # -mo dules w eak ly pro jective rela tive to N # that is quasiisomorphic to X , and P Y տ be a concav e bo unded ab ov e complex of N - injective A -mo dules that is quasiisomor phic to Y . Then we set (5) E xt ∞ / 2+ i ( X, Y ) = H i ( H om • A # ( P X ւ , S ⊗ A P Y տ )) . Independenc e of the right-hand side of (5) on the choice of P X ւ , P Y տ follows from Theorem 1 be low. R emark 4 . Notice that H om in the r ight-hand side o f (5) is H om in the category of graded mo dules . As usual, it is often c onv enient to denote b y E xt ∞ / 2+ i ( X, Y ) the graded space which in present notations is written down as L n E xt ∞ / 2+ i ( X, Y ( n )), where ( n ) refers to the shift of gra ding by − n . R emark 5 . Definition 1 is co mpatible with [Ar1], Definition 3 .3 .6 in the sense ex- plained below. In this re ma rk w e will free ly use the notatio n of lo c. cit. F or a finite dimensional algebra A the definition of the algebra A # given in [Ar1], 3.3.2 r educes to A # = E nd A op ( S ), where S is defined b y S = H om k ( N , k ) ⊗ N A , so according to Prop osition 4 this agrees with our definition (see also 3 .5 ). Notice that in lo c. cit. it is presumed that K = k , so one has N # = N . Let L ∈ C om b ( A # − mod ) a nd M ∈ C om b ( A − mod ). Then the restricted Bar - resolution Bar • ( A # , N # , L ) is a non-str ic tly co nv ex bo unded ab ove r esolution of L by A # -mo dules weakly pro jective re la tive to N # ; and B a r • ( A, B , M ) is a concav e bo unded ab ov e re solution of M by N -pro jectiv e A -mo dules. Thu s the definition o f semi-infinite cohomolo gy E xt ∞ / 2+ i ( L, M ) = H om • A #  Bar • ( A # , N # , L ) , S ⊗ A Bar • ( A, B , M )  from lo c. cit. is a particular case of our definition whenever b oth ar e applicable. 10 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI 4.3. Alternativ e assumptions. The conditions on the res olutions P X ւ , P Y տ used in (5) are for mu lated in terms of the subalgebras N ⊂ A and N # ⊂ A # ; the sub- algebra B ⊂ A is not mentioned there (and the left-hand side o f (9) in Theor em 1 below do es not depend on it either). Howev er, existence of a ”complemental” sub- algebra B is use d in the construction of a resolution P Y տ with r equired prop erties . Moreov er , the next s tandard Lemma shows that conditions on the resolutions P X ւ , P Y տ can be alter natively rephrased in terms of the s ubalgebra B and any nonp os- itively graded subalgebr a B # ⊂ A # such that B # ⊗ K N # ∼ = A # , assuming tha t such a suba lgebra exists (e.g ., in the assumptions of 3.5 or when N is F rob enius and B ⊗ K N ∼ = A ). Lemma 2. i) A n A -mo dule is N -pr oje ctive iff it has a filt r ation with sub quotients of the form I nd A B ( M ) , M ∈ B − m od . ii) Assume that B # ⊂ A # is a sub algebr a gr ade d by nonp ositive int e gers such that K ⊂ B # and the multiplic ation map induc es an isomorphism B # ⊗ K N # → A # . Then an A # -mo dule is N # -inje ct ive iff it has a filtr ation with sub quotients of the form C oI nd A # B # ( M ) , M ∈ B # − mod . Pr o of. The ” if” dire ction follows fro m semisimplicity of K , and e quality (4) ab ove. T o show the ” only if” pa rt let M b e a pro jective N -mo dule. Let M − be its g raded comp onent of minimal degree; then the cano nical morphism (6) I n d N K M − → M is injective. If M is actually an A -mo dule, then the injection M − → M is an embedding of B - mo dules, hence yields a morphism of A -mo dules (7) I n d A B M − → M . (4) shows that Res A N sends (7) into (6); in pa rticular (7) is injective. Thus the bo ttom submo dule of the required filtration is constructed, and the pro o f is finished by induction. The pro of of (ii) is analogous. R emark 6 . Replacing the a ssumption o f e xistence o f a suba lgebra B ⊂ A (assuming only that A is a pro jectiv e left N - mo dule) with the assumption of ex istence of a nonp ositively gr aded subalgebra B # ⊂ A # such that B # ⊗ K N # ∼ = A # , one can define E xt ∞ / 2+ i ( X, Y ) in terms of injectiv e re s olutions rather than pro jective ones. Namely , for X ∈ D ( A # ) and Y ∈ D ( A ), let J X ց be a conv ex bo unded be low complex of N # -injective mo dules quasiisomor phic to X , a nd J Y ր be a non-strictly concav e bo unded b elow complex of A -mo dules weakly injectiv e relative to N . Then set E xt ∞ / 2+ i ( X, Y ) = H i ( H om • ( H om A # ( S, J X ց ) , J Y ր )) . The analogue of Theorem 1 b elow holds fo r this definition as well, hence it follows that the t wo definitions are equiv alent whene ver b o th are applicable. 4.4. Comparison with ordinary Ext and T or. In four sp ecial cases E xt ∞ / 2+ i ( X, Y ) coincides with a combination of traditiona l derived functors . First, supp ose that R es A N ( Y ) has finite pro j ective dimension; then o ne ca n use a finite complex P Y տ in (5) ab ov e. It follows immediately , that in this ca se w e hav e E xt ∞ / 2+ i ( X, Y ) ∼ = H om D ( A # ) ( X, S L ⊗ A Y [ i ]) . SEMI-INFINITE COHOMOLOGY 11 Analogously , in the assumptions of Remar k 6 ab ov e, whenever Re s A # N # ( X ) has finite injectiv e dimension o ne has E xt ∞ / 2+ i ( X, Y ) ∼ = H om D ( A ) ( RH om A # ( S, X ) , Y [ i ]) . On the other hand, suppo se that the complex P X ւ in (5) can b e chosen to b e a finite co mplex of A # -mo dules whose ter ms hav e filtrations with sub quotients b e ing A # -mo dules induced from N # -mo dules. W e claim that in this cas e we have E xt ∞ / 2+ i ( X, Y ) ∼ = H i ( RH om A # ( X , S ) L ⊗ A Y ) . This isomor phism is a n immediate co nsequence of the next Lemma. Analogo usly , in the situation of Remark 6, whenev er J Y ր can be c hosen to b e a finite co mplex of A - mo dules who s e terms hav e filtrations with s ubq uo tient s being A - mo dules co induced from N -mo dules, one has E xt ∞ / 2+ i ( X, Y ) ∼ = H i ( X ∗ L ⊗ A # RH om A ( S ∗ , Y )) . Here we denote by V 7→ V ∗ the pass a ge to the dual vector space, V ∗ = H om k ( V , k ), and the corresp onding functor on the level o f derived categor ies. Lemma 3. L et L ∈ A # − mod , M ∈ A − mod b e such that L has a filtr ation with sub quotients b eing A # -mo dules induc e d fr om N # -mo dules, while M is N -pr oje ctive. Then we have a) i) E xt i A # ( L, S ) = 0 and T or A i ( H om A # ( L, S ) , M ) = 0 for i 6 = 0 . ii) T or A i ( S, M ) = 0 and E xt i A # ( L, S ⊗ A M ) = 0 for i 6 = 0 . b) The natura l map (8) H om A # ( L, S ) ⊗ A M − → H om A # ( L, S ⊗ A M ) is an isomorphism. Pr o of. The first equality in (i) holds because S is an injectiv e N # -mo dule. T o chec k the seco nd o ne, notice that if L = I nd A N # L 0 , then H om A # ( L, S ) ∼ = H om N # ( L 0 , N ∨ ⊗ N A ) ∼ = H om N # ( L 0 , N ∨ ) ⊗ N A is a rig ht A -mo dule induced from a right N -mo dule. The firs t equalit y in (ii) ho lds b eca use the right A - mo dule S is induced fro m a rig ht N -mo dule, and the seco nd one is verified since S ⊗ A M is N # -injective. Let us no w deduce (b) from (a). Notice that (a) implies that b o th sides o f (8) a re ex a ct in M (and also in L ), i.e. send exact sequences 0 → M ′ → M → M ′′ → 0 with M ′ , M ′′ being N -pro jective into exact s e q uences. Also (8) is evidently an isomorphism for M = A . F or any N -pro jectiv e M there exists an exact sequence A n φ − → A m → M → 0 with the image a nd kernel of φ be ing N -pro jectiv e. Th us b oth s ides of (8) turn int o exact sequence s, which shows that (8 ) is an iso morphism for any N -pro jectiv e M . 5. Main resul t Theorem 1 . L et D ∞ / 2 ⊂ D ( A # ) , D ∞ / 2 ⊂ D ( A ) b e the ful l t riangulate d sub- c ate gory of D ( A # ) gener ate d by N # -inje ct ive mo dules, which is e quivalent to t he 12 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI ful l triangulate d sub c ate gory of D ( A ) gener ate d by N -pr oje ctive mo dules. F or X ∈ D b ( A # − mod ) , Y ∈ D b ( A − mod ) we have a natur al isomorphism (9) H om D ( A # ) D ∞ / 2 D ( A ) ( X, Y [ i ]) ∼ = E xt ∞ / 2+ i ( X, Y ) . Example 2 . 1 Assume that A = N is a F r ob enius algebra and K = k . Then A # ∼ = A , and according to section 4.4 we have E xt ∞ / 2+ i ( X, Y ) = T or A − i ( X ∗ , Y ). In this c a se we can ident ify A ′ with A , so that Φ ′ = Φ is the em b edding of the category of p er fect complexes. The long exact sequence of P rop osition 2 b ecomes a standard sequence linking Ext, T or and Hom in the stable categ ory A/ B ; in particular , for mo dules ov er a finite gr o up w e re cov er the description of T ate coho mo logy as Hom’s in the stable category . R emark 7 . Notice that the definition of the left hand side in (9) a pplies also to non-gra ded a lgebras a nd mo dules. Th us the Theor em allows one to extend the definition of semi-infinite cohomo lo gy to nong raded algebras. Another definition of the semi-infinite cohomolo gy of nongr aded a lgebras was given in [P ]. Let us p oint out that these tw o definitions are not equiv alent: fo r example, when k is a finite or a count able field, the left hand side of (9) in the no ngraded ca se is no more than coun table, while the semi-infinite coho mology defined in lo c. cit. can hav e the ca rdinality of contin uum. The pro of of Theorem 1 is based on the following Lemma 4. i) Every N -pr oje ctive A -mo dule admits a non-strictly c onvex left r eso- lution c onsisting of A -pr oje ct ive mo dules. ii) A finite c omplex of N -pr oje ctive A -mo dules is quasiisomorphic to a non- strictly c onvex b ounde d ab ove c omplex of A -pr oje ct ive mo dules. Pr o of. (ii) follo ws fro m (i) as in the pro of o f Lemma 1 . (Recall that, according to a well-known ar gument due to Hilbert, if a bo unded ab ove co mplex o f pro jectiv es represents an ob ject of the derived categor y which has finite pro jectiv e dimension, then for large negative n the mo dule of co cycles is pro jective.) T o prove (i) it is enough for any N - pro jective mo dule M to find a sur jection Q ։ M , where Q is A -pro jective, and Q n = 0 fo r i > n provided M i = 0 fo r i > n . (Notice that the kernel of such a surjection is N -pro jectiv e, b ecause Q is N -pro jectiv e by co ndition (2).) W e can take Q to b e I nd A N ( Res A N ( M )), and the condition on grading is clearly satisfied. Prop ositi o n 6. a) L et P տ b e a c onc ave b ounde d ab ove c omplex of A -mo dules r epr esenting an obje ct Y ∈ D − ( A − mod ) . L et P n տ b e the ( − n ) -th stupid trunc ation of P տ (thus P n տ is a sub c omplex of P տ ). L et Z b e a finite c omplex of N -pr oje ctive A -mo dules. Then we have (10) H om D − ( A − mod ) ( Z, Y ) g − → lim − → H om D ( A ) ( Z, P n տ ) . In fact, for n lar ge enough we have H om D − ( A − mod ) ( Z, Y ) g − → H om D ( A ) ( Z, P n տ ) . 1 W e thank A. Beilinson who suggested to us this example. SEMI-INFINITE COHOMOLOGY 13 Pr o of. Let Q ւ be a non-strictly conv ex b ounded a b ov e complex o f A -pr o jectiv e mo dules qua siisomorphic to Z (which exists b y Lemma 4(ii)). Then the le ft-ha nd side o f (1 0) equa ls H om H ot ( Q ւ , P տ ), where H ot stands for the homotopy category of c omplexes o f A -mo dules . The conditions o n g radings o f o ur complexes ensure that there ar e only finitely many degrees for whic h the corresp onding graded com- po nents b oth in Q ւ and P տ are nonzero; thus a ny morphism b etw een the grade d vector spaces Q ւ , P տ factors through the finite dimensional sum o f the co rre- sp onding gr aded co mpo nent s. In particular , H om • ( Q ւ , P n տ ) g − → H om • ( Q ւ , P տ ) for large n , and hence H om D ( A ) ( Z, P n տ ) = H om H ot ( Q ւ , P n տ ) g − → H om H ot ( Q ւ , P տ ) for large n . Corollary 1. L et P տ b e a c onc ave b ounde d ab ove c omplex of N -pr oje ctive A - mo dules, and X b e the c orr esp onding obje ct of D − ( A − mod ) . Then the functor on D ∞ / 2 given by Z 7→ H om D − ( A − mod ) ( Z, Y ) is r epr esente d by the ind-obje ct lim − → P n տ . Pr o of of the Theor em. W e keep the notation of Definition 1. It follows from the Prop ositio n that H om D ( A # ) D ∞ / 2 D ( A ) ( X, Y [ i ]) = lim − → n H om D ( A # ) ( X, S ⊗ A ( P Y տ ) n ) . The right-hand side of (9) (defined in (5)) equals H i ( H om • A # ( P X ւ , S ⊗ A P Y տ )). The conditions on gradings of P X ւ , P Y տ show that for lar ge n w e hav e H om • A # ( P X ւ , S ⊗ A ( P Y տ ) n ) g − → H om • A # ( P X ւ , S ⊗ A P Y տ ) . Since E xt i A # ( L, S ⊗ A M ) = 0 for i > 0 if L is weakly pro jective relative to N # and M is N -pr o jective, we have H om D ( A # ) ( X, S ⊗ A ( P Y տ ) n ) = H om • A # ( P X ւ , S ⊗ A ( P Y տ ) n ) . The Theorem is prov ed. R emark 8 . There is a version of Theore m 1 applica ble in the situatio n when the condition tha t K is the comp onent of de g ree 0 of N in (1) o f 3.1 is repla ced with the condition that K is the co mpo nent of degr ee 0 of B . One just has to change the conditions on the complexes P X ւ , P Y տ in Definition 1, requiring that P X ւ be convex and P Y տ be non-strictly concav e, and ma ke the related changes in the pr o of. 6. Semi-infinite cohomol ogy of the small quantum group This s ection concerns with the example provided by a small qua ntum gr oup. So let g be a simple Lie algebr a ov er C wit h a fixed tr iangular decompo sition g = n ⊕ t ⊕ n − . Le t q ∈ C b e a ro ot of unit y of order l , a nd let A = u q = u q ( g ) be the corresp onding sma ll quantum g r oup [L]. W e assume that l is large eno ugh (larger than Coxeter num b er) a nd is prime to t wice the maximal m ultiplicity of a n edge in the Dynkin diagra m of g . Let A ≥ 0 = u + q ⊂ u q and A ≤ 0 = u − q ⊂ u q be resp ectively the upp er a nd the low er triangula r subalg ebras. The algebra u q carries a canonic a l gr a ding by the weigh t lattice. W e fix an arbitra ry element in the dual cow eight lattice, which is a dominant coweigh t, thus we obta in a Z -grading o n u q . Then the a bove conditions (1–3) are satisfied. 14 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI F or an augmented k algebra R we write H • ( R ) for E xt R ( k , k ); we abbreviate H • = H • ( u q ). The coho mology alge br a H • , and the semi-infinite cohomolog y E xt ∞ / 2+ • ( k , k ) were computed r esp ectively in [GK] a nd [Ar1]. Let us reca ll the results of these computations. Be low b y “Hom” we will mean gr aded Hom as in Remark 4 ab ov e. Let N ⊂ g be the cone of nilp otent elements, a nd n ⊂ N b e a maximal nilp otent subalgebra . Then the Theor em of Ginzburg and Kumar asserts the existence of canonical isomorphisms (11) H • ∼ = O ( N ) , (12) H • ( u + q ) = O ( n ) , such that the restriction map O ( N ) → O ( n ) coincides with the map arising from functoriality of cohomo logy with resp ect to ma ps of augmented alge bras. Also, a Theorem of Arkhip ov (conjectur ed by F eigin) asser ts that (13) E xt ∞ / 2+ • ( I , I ) ∼ = H d n − ( N , O ) , where d is the dimensio n of n − , and H n − denotes cohomolog y with supp ort in n − ; one also has H i n − ( N , O ) = 0 for i 6 = d . The aim of this se ction is to show how (a gene r alization of ) this isomor phism follows from Theorem 1. 6.1. D ∞ / 2 and cohomol ogical supp ort. Let D • denote the category defined by O b ( D • ) = O b ( D ), H om D • ( X, Y ) = H om • ( X, Y ) = L i H om ( X, Y [ i ]). Then D • is an H H • -linear categor y , i.e. w e hav e a canonica l homomor phism H H • → E nd ( I d D • ), wher e H H • denotes the Ho chschield coho mology of u q . Since u q is a Hopf algebr a, we hav e a cano nical homo morphism H • → H H • , thus D • is an H • linear catego ry . F or an o b ject X ∈ D • its c ohomolo gic al supp ort supp ( X ) ⊂ S pec ( H • ) is the set-theoretic supp ort of the H • mo dule E nd • ( X ). Prop ositi o n 7. F or an obje ct X ∈ D we have: X ∈ D ∞ / 2 ⇐ ⇒ su pp ( X ) ⊆ n . Pr o of. It is well known that su pp ( X ) ⊃ supp ( Y ) ∪ supp ( Z ) provided that there exists a dis tinguished tria ngle Y → X → Z → Y [1], th us the set of ob jects with cohomolog ical suppor t contained in n forms a full triang ulated sub catego ry . In vie w of Lemma 2, to chec k the implication ⇒ it sufficies to chec k that supp ( X ) ⊂ n if X = C oI nd u q u + q ( M ) for some M . F or such X we have E xt • u q ( X, X ) = E xt • u + q ( X, M ). Moreov er , it is not har d to chec k that this isomorphis m is co mpatible with the H • action, w he r e the action on the right ha nd side is obtained as the compo sition H • → H • ( u + q ) → E xt • u + q ( M , M ) a nd the canonica l right a ction of E xt • u + q ( M , M ). Thu s in this ca se E xt • u q ( X, X ) is set- theo retically suppo rted on n . Assume no w tha t X ∈ D is such tha t supp ( X ) ⊆ n . T o c heck that X ∈ D ∞ / 2 it suffices to show that E xt • u − q ( M , X ) is finite dimensional fo r a ny M ∈ D b ( u + q − m od ). It is a standar d fact that E xt • ( M 1 , M 2 ) is a finitely gener ated H • ( u − q ) module for any M 1 , M 2 ∈ D b ( u − q − mod ), th us it s uffices to see that the H • ( u − q )-mo dule E xt • u − q ( M , X ) is s upp o r ted at { 0 } ⊂ n − = S pe c ( H • ( u − q )). T his is clear, since viewed as a H • ( X ) it is supp or ted on n . SEMI-INFINITE COHOMOLOGY 15 6.2. A description of the deriv ed u q -mo dul es category via coheren t shea v es. Let ˜ N = T ∗ ( B ) = { ( b , x ) | b ∈ B , x ∈ ra d ( b ) } , where B = G/B is the flag v ariety of G identified with the set of Borel subalgebra s in g , and rad stands for the nil-radical. Let π : ˜ N → N b e the Spring er map, π : ( b , x ) 7→ x . The r esult of [BL] (based on [ABG]) yields a triangulated functor Ψ : D b ( C oh G m ( ˜ N )) → D b ( u q − M od ), where G m acts on ˜ N by t : ( b , x ) 7→ ( b , t 2 x ), and u q − M od stands for the category of finite dimensional modules. Notice that in con tra st with the definition of u q − mod , the modules in u q − M od do not carry a grading. 2 The functor satisfies the following prope r ties: (14) Ψ( F (1)) ∼ = Ψ( F )[1] where F (1) is the t wist o f F by the tautolog ic al character of G m ; (15) Ψ : M n ∈ Z Hom( F , G [ n ]( n )) g − → Hom(Ψ( F ) , Ψ( G )); (16) h I m (Ψ ) i = D b ( u q − M od 0 ) , where h I m (Ψ) i denotes the full triangulated sub categor y generated by ob jects o f the form Ψ( F ), a nd u q − M od 0 is the blo ck (dir e ct summand) of the category u q − M od whic h co nt ains the trivial repre s entation; (17) Ψ( O ˜ N ) = k . The following slight gener alization o f this r esult is prov ed by a straig htforward mo dification of the arg ument of [BL]. Prop ositi o n 8 . L et C b e a subt orus in the maximal torus T , and let u q − mod C b e the c ate gory of u q -mo dules c arrying a c omp atible gr ading by weights of C . Ther e exists a funct or Ψ C : D b ( C oh C × G m ( ˜ N )) → D b ( u q − mod C ) satisfying pr op erties (14) – (17 ) ab ove. 6.3. Semi-inifini te cohomology as cohomology with supp ort. F rom now on we fix C to b e a c opy of the multiplicativ e g roup corr esp onding to the cow eight used to define the grading on u q (see the beginning of this section), thu s we hav e u q − mod C = u q − mod . Theorem 2. F or F ∈ D b ( C oh C × G m ) we have a c anonic al isomorphism E xt ∞ / 2+ i u q ( k , Ψ( F )) ∼ = R Γ n ( π ∗ ( F )) . Pr o of. W e hav e R Γ • n ( π ∗ ( F )) ∼ = lim − → E xt • ( O ˜ N /π ∗ ( I ) , F ) , 2 In fact, D b ( C oh G m ( ˜ N )) can b e identified with the derived categ ory of a blo ck in the category of graded modules o ver u q compatible with a certain grading on u q , defined in [AJS]. How ev er, unlike the natural grading by weigh ts and its modi fications, this gr ading is neither explicit, nor elemen tary; it is simi lar to a grading on the catego ry O of g modules with highest weigh t arising from Ho dge we ights on the H om space betw een Hodge D -m odules, or fr om F rob enius weigh ts. 16 R OMAN BEZRUKA VNIKOV, LE ONID POSITSELSKI where I r uns ov er C × G m inv ariant ideals in O N with supp ort on n . W e hav e a canonical arrow Ψ( O ˜ N ) → Ψ( O ˜ N /π ∗ ( I )), and in view of Pr op osition 7 we have Ψ( O ˜ N /π ∗ ( I )) ⊂ D ∞ / 2 . Thus b y Theorem 1 we hav e a natural map R Γ n ( π ∗ ( F )) − → E xt ∞ / 2+ i u q ( k , Ψ( F )) . In view of Prop o sition 1, to c heck that this map is an iso morphism it suffices to show that the pro -ob ject \ Ψ( O ) in D ∞ / 2 defined by \ Ψ( O ) = lim ← − Ψ( O ˜ N /π ∗ ( I )) represents the same functor on D ∞ / 2 as the ob ject k = Ψ( O ) ∈ D . Let X ∈ D ∞ / 2 , and let f 1 , . . . , f n be a reg ular sequence in O ( N ) whose c ommon set of zer o es eq ua ls n . W e ca n and will assume that f i is a n eigen-function for the action of C × G m . There exists N such that f N i maps to 0 ∈ E nd • ( X ). Then any morphism k → X factors thro ugh k N = Ψ( O / ( f N i )). This shows that the ma p lim − → H om ( k N , X ) → H om ( k , X ) is surjective. Similarly , for large N the map H om ( k N , X ) → H om ( k 2 N , X ) kills the kernel of the map H om ( k N , X ) → H om ( k , X ). Thus the map lim − → H om ( k N , X ) → H om ( k , X ) is injective. Corollary 2. L et T b e a t ilting mo dule over Lusztig’s “ big” quantum gr oup U q . The semi-infinite c ohomolo gy E xt ∞ / 2+ i u q ( k , T ) either vanishes or is c anonic al ly iso- morphic t o R Γ n ( F ) , wher e F ∈ D b ( C oh G ( N )) is a c ert ain (explici t) irr e ducible obje ct in the he art of t he p erverse t -str u ctur e c orr esp onding t o the midd le p erversity [B2] . Pr o of. By the result of [B1] w e hav e T = Ψ( ˜ F ) for some ˜ F ∈ D b ( C oh G × G m ( ˜ N )), such that π ∗ ( ˜ F ) e ither v anishes or is a n (explicit) irreducible perverse equiv aria nt coherent sheaf as ab ov e. The s tatement now follows from Theorem 2. Example 3 . If T = k is the triv ia l mo dule, then it is clear from the construction of [B1] that we ca n set ˜ F = O ˜ N . Th us F ∼ = O N , so Corollar y y ields the main r esult of [Ar1]. References [AJS] H.H . Andersen, J.C. Jantzen , W. Soergel Rep r esentations of quantum gr oups at a p th r o ot of unity and of semisimple gr oups in char acteristic p : indep endenc e of p , Ast´ erisque 220 (1994), 321 pp. [Ar1] A rkhip o v, S. , Semiinfinite co homolo gy of quantum gr oups, Comm. Math. Ph ys. 1 88 (199 7), no. 2, 379–405. [Ar2] A rkhip o v, S., Semi-infinit e c ohomolo gy of asso ciative algebr as and b ar duality, Int ernat. Math. Res. Notices 1 7 (1997), 833–863. [Ar3] A rkhip o v, S., Semi- infinite co homolo gy of quantum gr oups. II, in: T opics in quantu m groups and finite-t yp e inv ariants, 3–42, Amer. Math. Soc. T r ansl. Ser. 2, 185, Amer. Math. Soc., Providen ce, RI, 1998. [ABG] Ar khipov, S., Bezruk a vniko v, R., Gi nzburg V., Quantum Gr oups, the lo op Gr assmannian, and the Springer r esolution, J. Amer. Math. So c. 17 (2004), 595-678. [B1] Bezruk a vniko v, R., Cohom olo gy of tilting mo dules over quantum gr oups, and t -structur es on derive d cate gories of c oher ent she aves, Inv. Math. 166 (2006), no 2, 327–357. [B2] Bezruk a vniko v, R. , Perverse c oher e nt she aves (after Deligne), electronic preprint , arXiv:math.AG/00 05152 . [B3] Bezruk a vniko v, R., On semi-infinite c ohomolo gy of finite dimensiona l algebr as, electronic preprint, arX iv:math.R T/0005148. [FS] Bezruk a vniko v, R. , Fink elb erg, M. , Sc hech tman, V., F actorizable she aves and quantum gr oups, Lecture Notes Math. 1691, Springer V erl ag, 1998. SEMI-INFINITE COHOMOLOGY 17 [BL] Bezruk avnik ov, R., Lac ho wsk a, A., The smal l quantum gr oup and the Springer reso lution, Pro ceedings of Haif a Conference on Quan tum Gr oups (J. Donin memorial volume), Con- temporary Math. 433 (2007), 89–101. [Brz] Br zezinski, T., The structur e of c orings: induction functors, Maschke-ty p e the or em, and F r ob enius and Galois-typ e pr op erties , Algebr. Represent . Theory 5 (2002), #4, p. 389–410. [BW] Brzezinski, T. , Wis bauer, R., Corings and c omo dules, London Mathematical Society Lec- ture Note Series, 309, Cambridge Universit y Press, Cambridge, 2003. [GK] Ginzburg, V., Kumar, S. , Cohomolo gy of quantum gr oups at r o ots of unity, Duke Math. J. 69 (1993), no. 1, 179–198. [H] Hartshorne, R . , R esidues and Duality, Lecture Notes M ath. 20, Springer V erl ag, 1966. [L] Lusztig, G., Finite-dimensional Hopf algebr as arising fr om quantize d universal enveloping algebr a, J. Amer. M ath. So c. 3 (1990), no. 1, 257–296. [P] P ositselski, L., Homolo gic al algebr a of semimo dules and semic ontr amo dules, electronic preprint, arX iv:0708.3398 [ math.CT]. [S] Sev osty ano v, A., Semi-infinite c ohomolo gy and H e cke algebr as, Adv. Math. 159 (2001), no. 1, 83–141. [V] V orono v, A., Se mi-infinite homolo gi ca l algebr a, Inv en t. Math. 113 (1993), no. 1, 103–146.