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On the Cyclic Homology of Ringed Spaces and Schemes

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On the Cyclic Homology of Ringed Spaces and Schemes Doc Math J DMV On the Cyclic Homology of Ringed Spaces and Schemes Bernhard Keller Received March Revised September Communicated by Peter Schneider Abstract We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles As a byproduct of the pro of we obtain a new construction of the Chern character of a p erfect complex on a ringed space Mathematics Sub ject Classication Primary E Secondary E F Keywords Cyclic homology Chern character Ringed space Scheme Perfect complex Derived category Introduction The main theorem Let k b e a eld and X a scheme over k which admits an ample line bundle eg a quasipro jective variety Let vecX denote the category of algebraic vector bundles on X We view vecX as an exact category in the sense of Quillen By denition a short sequence of vector bundles is admissible exact i it is exact in the category of sheaves on X Moreover the category vecX is k linear ie it is additive and its morphism sets are k vector spaces such that the comp osition is bilinear In we have dened for each k linear exact category A a der A The sup erscript der indicates that the denition cyclic homology theory HC is mo deled on that of the derived category of A In lo c cit it was denoted by HC A As announced in lo c cit in this article we will show that the cyclic homology of the scheme X coincides with the cyclic homology of the k linear exact category vecX There is a canonical isomorphism cf Corollary der HC X HC vecX The denition of the cyclic homology of a scheme is an imp ortant technical p oint which will b e discussed b elow in Note that by denition Par there is an analogous isomorphism in K theory Documenta Mathematica Bernhard Keller Motivation Our motivation for proving the isomorphism is twofold Firstly it allows the computation of HC X for some nontrivial examples Indeed supp ose that k is algebraically closed and that X is a smo oth pro jective algebraic va riety Supp ose moreover that X admits a tilting bund le ie a vector bundle without higher selfextensions whose direct summands generate the b ounded derived category of the category of coherent sheaves on X Examples of varieties satisfying these hy p otheses are pro jective spaces Grassmannians and smo oth quadrics In we deduce from that for such a variety the Chern character induces an isomorphism K X HC k HC X Z Here the left hand side is explicitly known since the group K X is free and admits a basis consisting of the classes of the pairwise nonisomorphic indecomp osable di rect summands of the tilting bundle Cyclic homology of pro jective spaces was rst computed by Beckmann using a dierent metho d Our second motivation for proving the isomorphism is that it provides fur der ther justication for the denition of HC Indeed there is a comp eting and previous denition of cyclic homology for k linear exact categories due to R Mc McC Carthy Let us denote by HC A the graded k mo dule which he asso ciates McC with A McCarthy proved in lo c cit a number of go o d prop erties for HC The most fundamental of these is the existence of an agreement isomorphism McC HC A HC pro jA where A is a k algebra and pro jA the category of nitely generated pro jective A mo dules endowed with the split exact sequences In particular if we take A to b e commutative we obtain the isomorphism McC vecX HC X HC for all ane schemes X Sp ecA to identify the left hand side we use Weibels isomorphism b etween the cyclic homology of an ane scheme and the cyclic der homology of its co ordinate algebra Whereas for HC this ismorphism extends McC to more general schemes this cannot b e the case for HC Indeed for n n the group H X O o ccurs as a direct factor of HC X However the group X n McC HC vanishes for n by its very denition n Generalization Chern character Our pro of of the isomorphism actually yields a more general statement Let X b e a quasicompact separated scheme over k Denote by p er X the pair formed by the category of p erfect sheaves on X and its full sub category of acyclic p erfect sheaves The pair p er X is a lo calization pair in the sense of and its cyclic homology HC p er X has b een dened in lo c cit We will show that there is a canonical isomorphism HC X HC p er X If X admits an ample line bundle we have an isomorphism der HC vecX HC p er X so that the isomorphism results as a sp ecial case Documenta Mathematica Cyclic homology The rst step in the pro of of will b e to construct a map HC p er X HC X This construction will b e carried out in for an arbitrary top ological space X endowed with a sheaf of p ossibly noncommutative k algebras As a byproduct we therefore obtain a new construction of the Chern character of a p erfect complex P Indeed the complex P yields a functor b etween lo calization pairs P p er pt p er X k and hence a map HC p er pt HC p er X HC X The image of the class chk HC p er pt HC k under this map is the value of the Chern character at the class of P An analogous construction works for the other variants of cyclic homology in particular for negative cyclic homology The rst construction of a Chern character for p erfect complexes is due to BresslerNestTsygan who needed it in their pro of of SchapiraSchneiders conjecture They even construct a generalized Chern character dened on all higher K groups Several other constructions of a classical Chern character are due to B Tsygan unpublished Cyclic homology of schemes Let k b e a commutative ring and X a scheme over k The cyclic homology of X was rst dened by Lo day He sheaed the classical bicomplex to obtain a complex of sheaves CC O He then dened the X cyclic homology of X to b e the hypercohomology of the total complex of CC O X Similarly for the dierent variants of cyclic homology There arise three problems The complex CC O is unbounded to the left So there are at least two non X equivalent p ossibilities to dene its hypercohomology should one take Cartan Eilenberg hypercohomology cf or derived functor cohomology in the sense of Spaltenstein Is the cyclic homology of an ane scheme isomorphic to the cyclic homology of its co ordinate ring If a morphism of schemes induces an isomorphism in Ho chschild homology do es it always induce an isomorphism in cyclic homology Problem is related to the fact that in a category of sheaves pro ducts are not exact in general We refer to for a discussion of this issue In the case of a no etherian scheme of nite dimension Beckmann and Weibel Geller gave a p ositive answer to using CartanEilenberg hypercohomology By proving the existence of an SBIsequence linking cyclic homology and Ho chschild homology they also settled for this class of schemes whose Ho chschild homology vanishes in all suciently negative degrees Again using CartanEilenberg hypercoho mology Weibel gave a p ositive answer to in the general case in There he also showed that cyclic homology is a homology theory on the category of quasicompact quasiseparated schemes Problem remained op en Documenta Mathematica Bernhard Keller We will show in A that CartanEilenberg hypercohomology agrees with Spal tensteins derived functor hypercohomology on all complexes with quasicoherent ho mology if X is quasicompact and separated Since CC O has quasicoherent ho X mology this shows that problem do es not matter for such schemes As a byproduct of A we deduce in B a partially new pro of of Bo ekstedtNeemans theorem which states that for a quasicompact separated scheme X the unbounded derived category of quasicoherent sheaves on X is equivalent to the full sub category of the unbounded derived category of all O mo dules whose ob jects are the com X plexes with quasicoherent homology A dierent pro of of this was given by Alonso JeremasLipman in Prop In order to get rid of problem we will slightly mo dify Lo days denition Using sheaves of mixed complexes as introduced by Weibel we will show that the image of the Ho chschild complex C O under the derived global section functor is X canonically a mixed complex M X The mixed cyclic homology of X will then b e dened as the cyclic homology of M X For the mixed cyclic homology groups the answer to is p ositive thanks to the corresp onding theorem in Ho chschild homology due to WeibelGeller the answer to is p ositive thanks to the denition The mixed cyclic homology groups coincide with Lo days groups if the derived global section functor commutes with innite sums This is the case for quasicompact separated schemes as we show in Organization of the article In section we recall the mixed complex of an algebra and dene the mixed complex M X A of a ringed space X A In section we recall the denition of the mixed complex asso ciated with a lo calization pair and give a sheaable description of the Chern character of a p erfect complex over an algebra In section we construct a morphism from the mixed complex asso ciated with the category of p erfect complexes on X A to the mixed complex M X A We use it to construct the Chern character of a p erfect complex on X A In section we state and prove the main theorem and apply it to the computation of the cyclic homology of smo oth pro jective varieties admitting a tilting bundle In app endix A we prove that CartanEilenberg hypercohomology coincides with derived functor cohomology for unbounded complexes with quasicoherent homology on quasicompact separated schemes In
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