Bass $NK$ groups and $cdh$-fibrant Hochschild homology

BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. G. COR TI ˜ NAS, C. HAESEMEYER, MARK E. W ALKER, AND C. WEIBEL Abstract. The K -theory of a p olyno mial r ing R [ t ] con tains the K -theory of R as a summand. F or R commutativ e and con taining Q , w e describ e K ∗ ( R [ t ]) /K ∗ ( R ) in terms of Hochsc hild homology and the cohomology of K¨ ahler differen tials for the cdh topol ogy . W e use this to address Bass’ question, whether K n ( R ) = K n ( R [ t ]) im plies K n ( R ) = K n ( R [ t 1 , t 2 ]). The answ er to this question is affir mativ e when R is essen tially of finite type ov er the complex num bers, but negative in general. In 1 972, H. Bass p osed the following question (see [2], question (VI) n ): Do e s K n ( R ) = K n ( R [ t ]) imply that K n ( R ) = K n ( R [ t 1 , t 2 ])? One can rephrase the question in terms of Ba ss’ gr oups N K n , intro duced in [1]: Do e s N K n ( R ) = 0 imply that N 2 K n ( R ) = 0? More generally , for any functor F from rings to a n ab elian catego ry , Bass defines N F ( R ) as the kernel of the map F ( R [ t ]) → F ( R ) induced by ev aluation at t = 0, and N 2 F = N ( N F ). Bass ’ q uestion was ins pired by T r av erso’s theorem [24], fro m which it follows that N Pic( R ) = 0 implies N 2 Pic( R ) = 0 . In this pap er, we give a new in terpretatio n of the groups N K n ( R ) in terms of Ho chschild homology a nd the cohomolog y of K ¨ ahler differe n tials for the cdh top ology , for commut ative Q -a lgebras. This a llows us to give a co unterexample to Bass’ question in the companion pape r [6] (see Theo rem 0 .2 below). T o state our main structura l theorem, r ecall from [2 8] that ea ch N K n ( R ) has the structure of a mo dule over the r ing of big Witt vectors W ( R ). It is conv enie nt to use the countably infinite-dimensional Q -vector spaces t Q [ t ] and Ω 1 Q [ t ] . If M is any R - mo dule, then M ⊗ t Q [ t ] and M ⊗ Ω 1 Q [ t ] are naturally W ( R )- mo dules by [10]. Theorem 0. 1 . L et R b e a c ommutative ring c ontaining Q . Then ther e is a W ( R ) - mo dule isomorph ism N 2 K n ( R ) ∼ = ( N K n ( R ) ⊗ t Q [ t ]) ⊕  N K n − 1 ( R ) ⊗ Ω 1 Q [ t ]  . Thus K n ( R ) = K n ( R [ t 1 , t 2 ]) iff N K n ( R ) = N K n − 1 ( R ) = 0 iff N 2 K n ( R ) = 0 . In addition, the fol lowing ar e e quivalent for al l p > 0 : a) K n ( R ) = K n ( R [ t 1 , ..., t p ]) b) N K n ( R ) = 0 and K n − 1 ( R ) = K n − 1 ( R [ t 1 , ..., t p − 1 ]) . c) N K q ( R ) = 0 for al l q such that n − p < q ≤ n . Date : Octob er 29, 2018. Corti ˜ nas’ r esearc h w as partially supported by Conicet and partially supported b y gran ts PICT 2006-00836, UBACyT X051, PIP 112-200801-00900, and M TM2007-64704. Haesemey er’s research was partially supported by NSF grant DMS-0652860. W al ker’s research was partially supported by NSF grant DMS-0601666. W eib el’s researc h was supported by NSA grant MSPF-04G-184 and the Oswald V eblen F und. 1 2 G. COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL The equiv alence of (a ), (b) and (c) is immediate by induction, using the formula for N 2 K n , a nd is included for its historical imp or tance; see [25]. Theorem 0.1 also holds for the K -theory of schemes of finite type ov er a field; see 4.2 b elow. Theorem 0 .1 allows us to refor m ulate Bass’ question as follows: Do e s N K n ( R ) = 0 imply that N K n − 1 ( R ) = 0? Theorem 0.2. a) F or any field F algebr aic over Q , the 2-dimensional normal algebr a R = F [ x, y , z ] / ( z 2 + y 3 + x 10 + x 7 y ) has K 0 ( R ) = K 0 ( R [ t ]) but K 0 ( R ) 6 = K 0 ( R [ t 1 , t 2 ]) . b) Supp ose R is essent ial ly of finite typ e over a field of infinite tr ansc endenc e de gr e e over Q . Then N K n ( R ) = 0 implies that R is K n -r e gular and, in p articular, that K n ( R ) = K n ( R [ t 1 , t 2 ]) . Part (a) is proven in the companion pap er [6], using Theore m 0.1, while pa rt (b) is proven b elow as Corolla ry 6 .7. The pro o f of Theorem 0 .1 relies on methods develope d in [5 ] and [7], which allow us to compute the gro ups N K n and N p K n in ter ms of the Ho chsc hild homolog y of R , and of the cdh - c ohomolog y of the higher K ¨ ahler differentials Ω p , bo th rela tive to Q . The g roups N K n ( R ) have a na tural bigraded s tructure when Q ⊂ R , and it is conv enien t to take adv a nt age of this bigrading in stating our r e sults. The bigrading comes from the eigenspaces N K ( i ) n ( R ) of the Adams op era tions ψ k (arising from the λ -filtration) and the eigenspace s of the homothety op er ations [r] ( i.e. , ba se change for t 7→ rt ). This bigra ding will b e ex pla ined in Sectio ns 1 and 5; the gener al decomp osition fo r Adams weight i has the form: (0.3) N K ( i ) n ( R ) ∼ = T K ( i ) n ( R ) ⊗ Q t Q [ t ] . Here T K ( i ) n denotes th e typic al pie c e of N K ( i ) n ( R ), de fined as the simultaneous eigenspace { x ∈ N K ( i ) n ( R ) : [ r ] x = r x, r ∈ R } . (See Exa mple 1.6.) W e provide a concrete descr iptio n of the typical pieces in Theor em 5.1, repro duced here: Theorem 0.4. If R is a c ommutative Q -algebr a, then N K ( i ) n ( R ) is determine d by its typic al pie c es T K ( i ) n ( R ) and (0.3) . F or i 6 = n, n + 1 we have: T K ( i ) n ( R ) ∼ = ( H H ( i − 1) n − 1 ( R ) if i < n, H i − n − 1 cdh ( R, Ω i − 1 ) if i ≥ n + 2 . F or i = n, n + 1 , we have an exact se quenc e: 0 → T K ( n +1) n +1 ( R ) → Ω n R → H 0 cdh ( R, Ω n ) → T K ( n +1) n ( R ) → 0 . The special case N K 0 = ⊕ N K ( i ) 0 of Theorem 0.4 is that for R essentially of finite type ov er a field o f characteris tic zer o, with d = dim( R ), (0.5) N K 0 ( R ) ∼ =  ( R + /R red ) ⊕ M d − 1 p =1 H p cdh ( R, Ω p )  ⊗ Q t Q [ t ] . Here R + is the seminormaliza tion o f R red ; we sho w in Propo sition 2.5 that R + = H 0 cdh ( R, O ). The dimensio n z e ro case of Theo rem 0.4 is als o revealing: BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 3 i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 T K ( i ) 3 ( R ) 0 H H (1) 2 ( R ) tors Ω 2 R Ω 3 cdh ( R ) / Ω 3 R H 1 cdh Ω 4 0 T K ( i ) 2 ( R ) 0 tors Ω 1 R Ω 2 cdh ( R ) / Ω 2 R H 1 cdh Ω 3 0 T K ( i ) 1 ( R ) nil( R ) Ω 1 cdh ( R ) / Ω 1 R H 1 cdh Ω 2 0 T K ( i ) 0 ( R ) R + /R H 1 cdh Ω 1 0 T K ( i ) − 1 ( R ) H 1 cdh O 0 T K − 2 ( R ) 0 T able 1. The gr oups T K ( i ) n ( R ) for n ≤ 3, dim( R ) = 2. Example 0.6 . If dim( R ) = 0 then we get N K n ( R ) ∼ = H H n − 1 ( R, I ) ⊗ Q t Q [ t ] for all n , where I is the nilradical o f R . It is illuminating to compare this with Go o dwillie’s Theorem [12], which implies that N K n ( R ) ∼ = N K n ( R, I ) ∼ = N H C n − 1 ( R, I ). The ident ification comes from the standar d o bserv a tion (1 .2) tha t the map H H ∗ → H C ∗ induces N H C ∗ ( R, I ) ∼ = H H ∗ ( R, I ) ⊗ Q t Q [ t ]. The calculations of Theo rem 0.4 for small n are summariz e d in T able 1 when dim( R ) = 2. W e will need the fo llowing cas e s of 0 .4 in [6 ], to pr ove Theo r em 0.2(a). Theorem 0.7 . L et R b e normal d omain of dimensio n 2 which is e ssential ly of finite typ e over an algebr aic ext ension of Q . Then a) N K 0 ( R ) = N K (2) 0 ( R ) ∼ = H 1 cdh ( R, Ω 1 ) ⊗ Q t Q [ t ] and b) N K − 1 ( R ) = N K (1) − 1 ( R ) ∼ = H 1 cdh ( R, O ) ⊗ Q t Q [ t ] . Here is an ov erview of this pa per : Section 1 reviews the bigra ding on the Ho chsc hild and cyclic homo lo gy of R [ t ] (and X × A 1 ), and Section 2 reviews the cdh - fibrant analog ue. Section 3 describ es the sheaf cohomolo g y of the fib ers F H H ( X ), F H C ( X ), etc. of H H ( X ) → H cdh ( X, H H ), etc. In Section 4 we use these fib ers to prov e The o rem 0.1 , by re la ting N K n +1 ( X ) to H − n F H H ( X ). W e also show tha t Bass’ question is neg ative for schemes in Lemma 4.5 . In Sec tion 5, we give the detailed computations of the t ypical pieces T K ( i ) n ( R ) needed to establish (0.5) and T able 1; these computatio ns employ the main r esult of [8]. In Section 6, we pr ov e Theor em 0.2(b), that the answer to Ba s s’ question is po sitive provided we are w orking ov er a sufficiently la rge base field. Finally , Section 7 describ es how Theorem 0.7 changes if R is of finite t yp e ov er a n a rbitrar y field of c haracteristic 0: the ma p N K 0 ( R ) → H 1 cdh ( R, Ω 1 /F ) ⊗ Q t Q [ t ] is on to, and an isomorphism if N K − 1 ( R ) = 0. Notation. All rings considered in this pap er should b e assumed to b e comm utative and noether ian, unless otherwis e s tated. Througho ut this pap er , k denotes a field of characteristic 0 and F is a field co n taining k as a subfield. W e write Sch /k for the category of separated schemes es s ent ially of finite type over k . If F is a presheaf on Sch /k , w e write F cdh for the ass o ciated cdh shea f, and often simply write H ∗ cdh ( X, F ) in plac e of the more for mal H ∗ cdh ( X, F cdh ). If H is a functor on Sch /k and R is an algebr a esse n tially of finite t ype, we o ccasiona lly write H ( R ) for H (Spec R ). F or example, H ∗ cdh ( R, Ω i ) is us ed for H ∗ cdh (Spec R , Ω i ) . No te that, b ecause the cdh site is no ether ian (every cov er ha s 4 G. COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL a finite sub cov ering) H ∗ cdh sends inv erse limits of schemes over diagrams with a ffine transition mo rphisms to direct limits. If H is a co nt rav ariant functor from Sch /k to sp ectra, (co)chain complexes , or ab elian gr oups that takes filtered inv erse limits of schemes ov er diagra ms with affine transition mor phisms to colimits (as for example K , H H , H cdh ( − , H H ), and F H H ), then for any k -algebra R , we abuse notation and write H ( R ) for the direct limit o f the H ( R α ) ta ken ov er all subring s R α of R o f finite type ov er k . (If R is essentially of finite type but not of finite t yp e, the tw o definitions o f H ( R ) agree up to ca no nical is omorphism.) In particula r, we will use expressio ns like H cdh ( R, H H ) for g eneral co mm utative Q -algebr as even though we do not define the cdh -top olo gy for ar bitrary Q -schemes. W e use cohomolo gical indexing for all chain c o mplexes in this pap er; for a com- plex C , C [ p ] q = C p + q . F or example, the Ho chschild, c yclic, perio dic, a nd neg ative cyclic homology of schemes ov er a field k ca n be defined using the Zariski hyper- cohomolog y of certa in presheaves o f co mplexes; see [3 2] and [5, 2.7] for precis e definitions. W e sha ll write these presheav es as H H ( /k ), H C ( / k ), H P ( / k ) and H N ( /k ), r esp ectively , omitting k from the nota tion if it is clear fro m the context. It is well known (see [31, 10 .9 .19]) that ther e is an Eilenberg -Mac Lane functor C 7→ | C | from c hain co mplexes o f ab elian groups to sp ectra , and from presheav es of chain complexes of ab elian groups to presheav es of sp ectra. This functor se nds quasi-isomo rphisms of complexes to weak homotopy equiv alences o f sp e c tra, a nd satisfies π n ( | C | ) = H − n ( C ). F or example, applying π n to the Chern character K → | H N | yields maps K n ( R ) → H − n H N ( R ) = H N n ( R ). In this spirit, w e will use descent terminology for pr esheav es of complexes. 1. The bigrading on N H H and N H C Recall that k denotes a field of characteristic 0 . In this section, we consider the Ho chsc hild and cy clic homo logy of p o lynomial extensio ns of co mm utative k - algebras . No g reat orig inality is claimed. Throug hout, we will use the chain level Ho dge decomp os itions H H = Q i ≥ 0 H H ( i ) and H C = Q i ≥ 0 H C ( i ) . The K ¨ unneth for mula for Hochschild homolo gy yields (1.1) N H H ( i ) n ( R ) ∼ =  H H ( i ) n ( R ) ⊗ t Q [ t ]  ⊕  H H ( i − 1) n − 1 ( R ) ⊗ Ω 1 Q [ t ]  . F ro m the exact SBI sequence 0 → N H C n − 1 B − → N H H n I − → N H C n → 0 (see [31, 9.9.1]), and induction on n , the map I induces ca nonical is omorphisms for each i : (1.2) N H C ( i ) n ( R ) ∼ = H H ( i ) n ( R ) ⊗ t Q [ t ] . R emark 1.3 . Both (1.1) and (1.2) gener alize to non-affine quasi-co mpact schemes X o ver k . Indeed, N H H and N H C satisfy Zar iski descent b ecause H H and H C do and b ecause, for any o p e n cover { U i → X } , the collec tio n { U i × A 1 → X × A 1 } is also a cover. Thus we hav e N H H ( i ) ( X ) ∼ = H Z ar ( X, N H H ( i ) ) ∼ = H Z ar ( X, H H ( i ) ) ⊗ t Q [ t ] ⊕ H Z ar ( X, H H ( i − 1) )[1] ⊗ Ω 1 Q [ t ] ∼ = H H ( i ) ( X ) ⊗ t Q [ t ] ⊕ H H ( i − 1) ( X )[1] ⊗ Ω 1 Q [ t ] , and N H C ( i ) ( X ) = H Z ar ( X, N H C ( i ) ) ∼ = H Z ar ( X, H H ( i ) ) ⊗ t Q [ t ] ∼ = H H ( i ) ( X ) ⊗ t Q [ t ]. BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 5 It is easy to iterate the co nstruction F 7→ N F . F or exa mple, w e se e from (1.1) and (1.2) that (1.4) N 2 H C ( i ) n ( R ) ∼ =  H H ( i ) n ( R ) ⊗ t Q [ t ] ⊗ t Q [ t ]  ⊕  H H ( i − 1) n − 1 ( R ) ⊗ t Q [ t ] ⊗ Ω 1 Q [ t ]  . By induction, w e see that H H ( i − j ) n − j ( R ) ⊗  t Q [ t ]  ⊗ ( p − j ) ⊗  Ω 1 Q [ t ]  ⊗ j will o ccur  p − 1 j  times a s a summand of N p H C ( i ) n ( R ) for all j ≥ 0. W e may wr ite this as the for m ula: (1.5) N p H C ( i ) n ( R ) ∼ = p − 1 M j =0 H H ( i − j ) n − j ( R ) ⊗ k ∧ j k p − 1 ⊗  t Q [ t ]  ⊗ ( p − j ) ⊗  Ω 1 Q [ t ]  ⊗ j . Cartier op erations on N H H and N H C . Le t W ( R ) denote the r ing of big Witt v ectors o ver R ; it is w ell known that in c haracteristic 0 we have W ( R ) ∼ = Q ∞ 1 R . (See [28, p. 468 ] for exa mple.) Cartier showed in [3] that the endomorphism ring Ca rt( R ) of the additive functor under lying W consists of column-finite sums P V m [ r mn ] F n , using the homo theties [ r ] (for r ∈ R ), and the V erschiebung and F ro be nius o per ators V m and F m . Restricting the sum to m ≥ m 0 yields a descending sequence of ideals of Ca rt( R ), making it complete as a top ological r ing ; W ( R ) is the co mplete top ologic al subring o f a ll sums P V m [ r m ] F m ; s e e [3]. W e will be in terested in the in termediate (top olog ical) s ubring C a rf ( R ) of a ll row and column-finite sums P V m [ r mn ] F n . As o bserved in [10, 2 .1 4], there is an equiv alence b etw een the categor y of R - mo dules and the catego r y of contin uous Carf ( R )-mo dules given by the c o nstructions in the following example. (A left mo d- ule M is c ontinuous if the annihilator ideal of each element is an op en left idea l.) Example 1.6 . If M is an y R -mo dule, N = M ⊗ t Q [ t ] is a contin uous Carf ( R )-module (and hence a W ( R )-mo dule) via the formu las: [ r ] t i = r i t i , V m ( t i ) = t mi , F m ( t i ) = ( mt i/m if m | i, 0 else. The ring W ( R ) = Q ∞ 1 R acts on M ⊗ t Q [ t ] by ( r 1 , ..., r n , ... ) ∗ P m i t i = P ( r i m i ) t i . Conv ersely , every cont inuous Carf ( R )-mo dule N has a “typical piece” M , defined as the simultaneous eig e ns pace { x ∈ N : [ r ] x = r x, r ∈ R } , and N ∼ = M ⊗ t Q [ t ]. Recall that we can define op er a tors [ r ] on N H H n ( R ) and N H C n ( R ), as so ciated to the endomor phisms t 7→ rt of R [ t ]. There ar e also op erator s V m and F m , defined via the ring inclusions R [ t m ] ⊂ R [ t ] and their transfer s. Thes e op erations co mm ute with the Hodge decomp osition. The following result follows immediately fro m [10, 4.11] using the obs erv a tio n that everything commutes with Adams op erations. Prop ositio n 1 . 7. The op er ators [ r ] , V m and F m make e ach N H C ( i ) n ( R ) into a c ontinu ous Carf ( R ) -mo dule, and henc e a W ( R ) -mo dule. The R -mo dule H H ( i ) n ( R ) is its typic al pie c e, and the c anonic al isomorphism N H C ( i ) n ( R ) ∼ = H H ( i ) n ( R ) ⊗ t Q [ t ] of (1.2 ) is an isomorphism of Car f ( R ) -mo dules, the mo dule str u ctur e on the right b eing given in Example 1.6. 6 G. COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL A similar structure theorem holds for N H H n ( R ) and its Ho dge comp onents, using (1.1). How ev er, it uses a non-s ta ndard R -mo dule structure o n the t ypical piece H H n ( R ) ⊕ H H n − 1 ( R ); see [10, 3.3] for details. R emark 1.7.1 . The co nclus ions o f Pr op osition 1.7 still hold for N H C ( i ) n ( X ) and H H ( i ) n ( X ) whe n X is a ny scheme, where W ( R ) and Carf ( R ) refer to the ring R = H 0 ( X, O ). That is, H H ( i ) n ( X ) is an R - mo dule and N H C ( i ) n ( X ) is a contin uous Carf ( R )-mo dule, isomorphic to H H ( i ) n ( X ) ⊗ t Q [ t ]. This scheme version of 1.7 is not s tated in [1 0], which was written b efore the cyclic homolo gy of sc hemes was developed in [32]. How ever, the pro of in [1 0] is easily ada pted. Since the o per ators V m , F m and [ r ] are defined on the underlying chain c omplexes in [10, 4.1], they extend to op era tio ns on the Ho chschild a nd cyclic homology of schemes. The ident ities requir ed to obtain contin uous Ca rf ( R )-mo dule structures all come from the K ¨ unneth formula for the shuffle pro duct o n the c hain complexes (see [10, 4.3]), so they a ls o hold for the homology o f schemes. 2. cdh -fibrant H H and N H C Now fix a field F containing k ; all schemes will lie in the categor y Sch /F (es- sentially of finite type ov er F ), in order to use the cdh top ology on Sch /F of [22]. All rings will b e comm utative F -a lgebras ; b ecause they are filtered direct limits of finitely g enerated F -alg ebras, we can co ns ider their cdh -cohomolo gy . If C is any (pr e-)sheaf of co chain complexes o n Sch /F , we can for m the cdh - fibrant repla cement X 7→ H cdh ( X, C ) and write H n cdh ( X, C ) for the n th co homology of this complex. (The fibrant repla cement is taken with resp ect to the loc a l injective mo del s tructure, as in [5, 3.3].) F or example, the cdh -fibr ant repla cement o f a cdh sheaf C (concentrated in degree zero) is just an injectiv e resolution, a nd H n cdh ( X, C ) is the usual coho mology of the cdh sheaf a sso ciated to C . Ho chsc hild a nd cy clic homolo gy , as well a s differential forms, will be taken rela - tive to k . F or C = H H ( i ) , it was shown in [7, Theorem 2.4] tha t (2.1) H cdh ( X, H H ( i ) ) ∼ = H cdh ( X, Ω i )[ i ] . This has the following cons equence for C = N H H ( i ) and N H C ( i ) . Lemma 2.2. L et H ( i ) denote either H H ( i ) or H C ( i ) , taken r elative t o a subfield k of F . Then H cdh ( X × A 1 , H ( i ) ) = H cdh ( X, H ( i ) ) ⊕ H cdh ( X, N H ( i ) ) , and: H cdh ( X, N H H ( i ) ) ∼ =  H cdh ( X, Ω i )[ i ] ⊗ t Q [ t ]  ⊕  H cdh ( X, Ω i − 1 )[ i ] ⊗ Ω 1 Q [ t ]  ; H cdh ( X, N H C ( i ) ) ∼ = H cdh ( X, Ω i )[ i ] ⊗ t Q [ t ] . Pr o of. T he dis played fo rmulas follow from (1.1), (1.2) and (2.1), using the fir st assertion and the fact that − ⊗ t Q [ t ] commutes with H cdh . Thus it suffices to verify the first assertio n. By resolutio n of singula rities, we may assume that X is smo oth. Recall from [5, 3.2 .2] that the res triction of the cdh top ology to Sm /k is ca lled the sc dh -top ology . The pro duct of any scdh cov er of X with A 1 is an scdh cov er of X × A 1 , and b oth H H ( i ) and H C ( i ) satisfy scdh -descent by [7, Thm. 2.4 ]. No w by Thomas on’s Cartan-L e ray Theorem [23, 1.56] w e hav e H cdh ( X × A 1 , H ( i ) ) ∼ = H cdh ( X, H ( i ) ( − × A 1 )) ∼ = H cdh ( X, H ( i ) ) ⊕ H cdh ( X, N H ( i ) ) . BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 7 This gives the first asser tion. Alternativ ely , we may prov e the firs t assertion by induction on dim( X ), using the definition of scdh descent to see that for smo oth X we hav e H ( i ) ( X ) = H cdh ( X, H ( i ) ) a nd H cdh ( X × A 1 , H ( i ) ) = H ( i ) ( X × A 1 ) = H ( i ) ( X ) ⊕ N H ( i ) ( X ) . In particular , the first as s ertion holds when dim( X ) = 0.  R emark 2 .2.1 . If R is any commutativ e F -alge br a, the formulas of Lemma 2.2 hold for X = Sp ec( R ) b y na turality . This is because w e may write R = lim − → R α , where R α ranges ov er subring s of finite type over F , and H cdh ( X, − ) = lim − → H cdh (Spec ( R α ) , − ). Corollary 2.3. If X = Spec( R ) is in Sch /F , the mo dules H n cdh ( X, H H ( i ) ) and H n cdh ( X, N H C ( i ) ) ar e zer o unless 0 ≤ n + i < dim( X ) and i ≥ 0 . If n ≥ dim( X ) and n > 0 then H n cdh ( X, H H ) = 0 . Pr o of. B ecause H n cdh ( X, Ω i )[ i ] = H i + n cdh ( X, Ω i ), this follows from (2.1), Lemma 2.2 and the fact that H n cdh ( X, Ω i ) = 0 for n ≥ dim( X ), n > 0. This b ound is g iven in [5, 6.1 ] for i = 0 , and in [7, 2.6] for gener al i .  Here is a useful b ound on the cohomolo g y groups appear ing in Lemma 2.2 . Given X , let Q denote the total ring of fractions o f X red ; it is a finite pro duct of fields Q j , a nd we let e denote the maximum of the transcendenc e degrees tr . deg ( Q j /k ). Lemma 2.4. L et X b e in Sch /F . If i > e t hen H n cdh ( X, Ω i ) = 0 for al l n . Pr o of. B y [19, 12.24 ], we may a ssume X r e duced. Since we may write X as an inv er se limit o f a sequence o f affine morphisms with the same r ing of tota l factions Q , and cdh -cohomo lo gy s e nds such an inv erse limit to a dir e c t limit, we may als o assume that X is of finite type ov er F . This implies that e = dim( X ) + tr . deg( F /k ). The result is clear if dim( X ) = 0, since H n cdh ( X, − ) = H n Z ar ( X, − ) in that ca se. Pro ceeding by induction on dim( X ), choose a re s olution o f singularities X ′ → X and observe that the s ing ular lo cus Y a nd Y × X X ′ hav e smaller dimension. The hypothesis implies that Ω i = 0 on X ′ Z ar , so H n cdh ( X ′ , Ω i ) = 0 b y [7, 2.5]. The re s ult now follows by induction from the Mayer-Vietoris sequence of [22, 12.1].  If R is a commut ative ring, we write R red and R + for the a sso ciated reduced r ing and the seminor malization o f R red , res pec tively . These constr uctions are natural with resp ect to lo calization, so that w e may form the seminor malization X + of X red for any scheme X . Because X + → X is a universal homeomorphism, we hav e H ∗ cdh ( X, − ) ∼ = H ∗ cdh ( X + , − ) for every X in Sch /k , for any field k o f ar bitrary characteristic. The case n = 0 with co efficients O cdh is o f sp ecial in terest; recall our conv ention that H 0 cdh ( X, O ) denotes H 0 cdh ( X, O cdh ). Prop ositio n 2.5. F or any algebr a R , we have H 0 cdh (Spec R , O ) = R + . Mor e over, for every X in Sch /F we have H 0 cdh ( X, O ) = O ( X + ) . Pr o of. W e may assume R and X ar e reduced. W riting R = lim − → R α as in Remar k 2.2.1, we hav e R + = lim − → R + α and H 0 cdh ( R, O ) = lim − → H 0 cdh ( R α , O ), so we may assume that R is of finite t ype. Thus the second asser tion implies the first. Since H 0 cdh ( − , O ) and O ( − + ) a r e Zar iski sheav es, it suffices to consider the case whe n X is affine. Let X = Sp ec R b e in Sch /F , with R reduced. There is an injection R → Q with Q regular (for example, Q co uld b e the total quotient ring o f R ). By [5, 6 .3], 8 G. COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL H 0 cdh (Spec Q , O ) = Q , so R injects in to H 0 cdh (Spec R, O ). This implies that O red is a sepa r ated presheaf for the cdh top o logy on Sch /F . Thus, the ring H 0 cdh ( X, O ) is the direct limit ov er a ll cdh-covers p : U → X of the ˇ Cech H 0 . (See [SGA4, 3 .2.3].) Fix an elemen t b ∈ H 0 cdh (Spec R , O ) and represe nt it by b ∈ O ( U ) for so me cdh cov er U → X . Now reca ll from [19, 12.2 8] o r [22, 5.9] that we may assume, by refining the cdh cov er U → X , that it factors a s U → X ′ → X where X ′ → X is prop er birational cdh cover and U → X ′ is a Nisnevich cov er. If the images o f b ∈ O ( U ) agre e in U × X U , i.e., b is a ˇ Cech cycle fo r U /X , then its imag es agre e in U × X ′ U , i.e., it is a ˇ Cech cycle for U /X ′ . But by faithfully fla t descent, b descends to an elemen t of O ( X ′ ). Th us we can assume that U is pro p er and bira tional ov er X . Next, w e c a n a ssume that the Nisnevich c ov er p : U → X is finite, surjective and birational. Indeed, since p is pr op er and biratio na l we may consider the Stein factorization U q − → Y r − → X . By [EGAI I I, 4.3] o r [16, I I I.11.5 & pro of ], q ∗ ( O U ) = O Y and r is finite surjective and birational. By [2 2, 5.8], r is also a cdh cover. Because q ∗ ( O U ) = O Y , the canonical map O Y ( Y ) → q ∗ ( O U )( Y ) = O U ( U ) is a n isomorphism. Hence b descends to an element o f O ( Y ). By Lemma 2.6 , b lies in the se mino rmalization of R .  Lemma 2.6. L et A b e a seminormal ring and B a ring b et we en A and its normal- ization. Then t he ˇ Ce ch c omplex A → B → B ⊗ A B is exact. Pr o of. W e us e T raverso’s descriptio n of the seminormalization (see [24, p. 585]): the se mino rmalization of a ring A inside a r ing B is A + = { b ∈ B | ( ∀ P ∈ Sp ec A ) b ∈ A P + rad( B P ) } . Let b ∈ B such that 1 ⊗ b = b ⊗ 1. W e hav e to s how that b ∈ A P + rad( B P ), for all pr imes P of A . Let J = r ad( B P ); since B P /J is faithfully flat ov er the field A P /P , the image of b in B P /J lies in A P /P by fla t desc ent . That is, b ∈ A P + J , as r e quired.  R emark 2.7 . E ven if X is affine seminormal, it can happ en that H i cdh ( X, O ) 6 = 0 for so me i > 0. F or exa mple, if R denotes the subring F [ x, g , y g ] of F [ x, y ] for g = x 3 − y 2 then it is eas y to show that R is seminormal and that H 1 cdh (Spec ( R ) , O ) = F , bec ause the normalizatio n of R is F [ x, y ] and the conductor ideal is g F [ x, y ]. F or another ex ample, the normal ring of Theo rem 0 .2 has H 1 cdh ( X, O ) 6 = 0, by Theorems 0.1 and 0.7(b). 3. The fibers F H H and F H C If C is a presheaf of complexes on Sc h /F , w e write F C for the shifted mapping cone of C → H cdh ( − , C ), so that we hav e a distinguished triangle: (3.1) H cdh ( X, C )[ − 1] → F C ( X ) → C ( X ) → H cdh ( X, C ) Example 3.1.1 . When C is concentrated in deg ree 0 we hav e H n F C = 0 for all n < 0 . F or C = O and X = Spe c( R ), we see fro m Pr op osition 2 .5 tha t H 0 F O ( X ) = nil( R ), H 1 F O ( X ) = R + /R , and H n F O ( X ) = H n − 1 cdh ( X, O ) fo r n ≥ 2. Note that, if X = Sp ec R ∈ Sch /F , then H n F O ( X ) = 0 for n > dim( X ) by [5, 6.1]. W e now co nsider the Ho chsc hild and cyclic homology complexes, taken relative to a subfield k of F . F or legibility , we write F ( i ) H H for F H H ( i ) , etc. By the usual homologica l y oga, F H H is the direct sum o f the F ( i ) H H , i ≥ 0, and s imilarly for F H C . BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 9 Example 3.1.2 . If X is smoo th ov er F then F H H ( X ) ≃ 0 by [7 , 2.4]. Lemma 2.2 and Remarks 2.2.1 and 1.3 imply the following analogue for N F . Lemma 3. 2. If X is in Sch /F , or if X = Spec( R ) for an F -algebr a R , we have quasi-isomorphi sms: N F ( i ) H H ( X ) ∼ =  F ( i ) H H ( X ) ⊗ t Q [ t ]  ⊕  F ( i − 1) H H ( X )[1] ⊗ Ω 1 Q [ t ]  ; N F ( i ) H C ( X ) ∼ = F ( i ) H H ( X ) ⊗ t Q [ t ] . Mimicking the argument that establishes (1.4 ) and (1.5) yields: Corollary 3 .3. If X is in Sch /F , or if X = Spe c ( R ) for an F -algebr a R , N 2 F ( i ) H C ( X ) ∼ =  F ( i ) H H ( X ) ⊗ t Q [ t ] ⊗ t Q [ t ]  ⊕  F ( i − 1) H H ( X )[1] ⊗ t Q [ t ] ⊗ Ω 1 Q [ t ]  and N p F ( i ) H C ( X ) ∼ = p − 1 M j =0 F ( i − j ) H H ( X )[ j ] ⊗ k ∧ j k p − 1 ⊗ t Q [ t ] ⊗ ( p − j ) ⊗  Ω 1 Q [ t ]  ⊗ j . The co homology of the typical pieces F ( i ) H H ( R ) is given as follows. Lemma 3.4. If R is an F - algebr a and i ≥ 0 , then ther e is an ex act se qu enc e: 0 → H − i F ( i ) H H ( R ) → Ω i R → H 0 cdh ( R, Ω i ) → H 1 − i F ( i ) H H ( R ) → 0 . F or n 6 = i, i − 1 we have: H − n F ( i ) H H ( R ) ∼ = ( H H ( i ) n ( R ) if i < n, H i − n − 1 cdh ( R, Ω i ) if i ≥ n + 2 . Pr o of. As in Remark 2.2.1, we may a s sume R is of finite t yp e. Since H H ( i ) i ( R ) = Ω i R for all i ≥ 0 , and H H ( i ) n ( R ) = 0 when i > n (see [3 1, 9.4 .1 5] or [17, 4.5.10]), it suffices to use (2.1) and to observe that H − n cdh ( R, H H ( i ) ) = H i − n cdh ( R, Ω i ) v anis hes when n > i .  Example 3.5 . Let X = Sp ec ( R ) b e in Sch /F . Since H H (0) = O , F (0) H H ( R ) is describ ed in Example 3.1.1. Applying Corolla ry 2 .3 and Lemma 3.4 for i > 0 , and using [7 , 2.6] to bound the ter ms, we see that if d = dim( R ) then H n F H H ( X ) = 0 for n > d . If d = 1, then the only nonze r o p ositive c o homology o f F H H is H 1 F H H ( R ) = R + /R ; if d > 1, we hav e: H 1 F H H ( R ) ∼ = ( R + /R ) ⊕ H 1 cdh ( X, Ω 1 ) ⊕ · · · ⊕ H d − 1 cdh ( X, Ω d − 1 ) , H 2 F H H ( R ) ∼ = H 1 cdh ( X, O ) ⊕ H 2 cdh ( X, Ω 1 ) ⊕ · · · ⊕ H d − 1 cdh ( X, Ω d − 2 ) , . . . . . . H d F H H ( R ) ∼ = H d − 1 cdh ( X, O ) . Example 3.6 . When R is essentially of finite type ov er F a nd tr . deg ( F /k ) < ∞ , H m F H H ( R ) is Ho chschild homology for large negative m . T o see this, observe that e = tr . deg( R /k ), the maximum trans c endence degree of the residue fields of R at 10 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL its minimal primes, is finite. Using Lemmas 2.4 and 3.4 , we get H − n F ( i ) H H ( R ) = 0 and H − n F ( n ) H H ( R ) = Ω n R for i > n > e , and hence H − n F H H ( R ) ∼ = H H n ( R ) for all n > e. If R = k ⊕ R 1 ⊕ R 2 ⊕ . . . is gr a ded, and g H C ∗ ( R ) = H C ∗ ( R ) /H C ∗ ( k ), it is well known that the map g H C ∗ ( R ) S − → g H C ∗− 2 ( R ) is ze r o. (See [31, 9 .9.1] for ex a mple.) In Lemma 3.8 b elow, w e pr ov e a simila r pr op erty for F H H and F H C , whic h w e derive from Lemma 3 .2 using the following trick. Standard T ri c k 3.7. If R is a non-nega tively g r aded algebr a , there is an algebr a map ν : R → R [ t ] sending r ∈ R n to r t n . The comp osition o f ν with ev a luation at t = 0 fac to rs a s R → R 0 → R , and so if H is a functor on algebras taking v alues in ab elian gro ups , then the comp ositio n H ( R ) ν − → H ( R [ t ]) t =0 − → H ( R ) is zero o n the kernel e H ( R ) of H ( R ) → H ( R 0 ). Similarly , the co mpo sition of ν with ev alua tion at t = 1 is the identit y . That is , ν maps e H ( R ) isomorphically onto a summand of N H ( R ), and e H ( R ) is in the image of ( t = 1) : N H ( R ) → H ( R ). Lemma 3.8. If R = k ⊕ R 1 ⊕ · · · is a gr ade d algebr a, then for e ach m the map π m F H C ( R ) S − → π m − 2 F H C ( R ) is zer o and ther e is a split short exact se quenc e: 0 → π m − 1 F H C ( R ) B − → π m F H H ( R ) I − → π m F H C ( R ) → 0 . Similarly, ther e ar e split short exact se quenc es: 0 → ˜ H m +1 cdh ( R, H C ) B − → ˜ H m cdh ( R, H H ) I − → ˜ H m cdh ( R, H C ) → 0 and 0 → ˜ H m − 1 cdh ( R, Ω 0 . (3.9b) An ana logous exact sequence · · · → π m − 1 F H H ( R ) d − → π m F H H ( R ) d − → π m +1 F H H ( R ) → · · · is obtained by splicing the other sequences in Lemma 3.8. Using the interpretation of their Ho dg e c o mpo nents, describ ed in L e mma 3 .4, pro duces tw o more exa c t BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 11 sequences: 0 → nil( R ) → tors Ω 1 R → tors Ω 2 R → tors Ω 3 R → · · · (3.9c) 0 → ( R + /R ) → Ω 1 cdh ( R ) / Ω 1 R → Ω 2 cdh ( R ) / Ω 2 R → · · · . (3.9d) Here we hav e written Ω i cdh ( R ) for H 0 cdh ( R, Ω i ), a nd tors Ω i R is defined as the kernel of Ω i R → Ω i cdh ( R ); the notation refle c ts the fact that if R is reduced then tors Ω i R is the torsion submodule of Ω i R (see Remark 5.6.1 below). 4. Bass’ groups N K ∗ ( X ) In this sec tio n, we r elate algebra ic K -theory to our Hochsc hild a nd cyclic ho - mology calc ula tions relative to the gro und field k = Q . Consider the trace ma p N K n +1 ( X ) → N H C n ( X ) = N H C n ( X/ Q ) induced by the Cher n character. In the affine case, it is defined in [2 7]; for schemes it is defined using Z ariski des c ent . As expla ined in [27], it arise s from the Chern char- acter fr o m the sp ectrum N K ( X ) to the E ilenberg-Mac Lane sp ectrum | N H C ( X )[1] | asso ciated to the co chain complex N H C ( X )[1]. Note that our indexing co nven tio ns are such that π n +1 | N H C ( X )[1] | = H − n N H C ( X ) = N H C n ( X ) . Prop ositio n 4. 1. Su pp ose that R = Γ( X , O ) for X in Sc h /F , or that X = Sp ec( R ) for an F -algebr a R . Then for al l n , the Chern char acter induc es a natur al isomor- phism N K n +1 ( X ) ∼ = H − n F H H ( X ) ⊗ t Q [ t ] . This is an isomorphi sm of gr ade d R -mo dules, and even Car f ( R ) -mo dules, id enti- fying the op er ations [ r ] , V m and F m on N K ∗ ( X ) with the op er ations on the right side describ e d in Example 1.6. Pr o of. B y Remark 2.2.1 , we may supp o se X ∈ Sch /F . By [7, 1.6], the Chern character K → H N induces weak equiv a lences F K ( X ) ≃ |F H C ( X )[1] | and F K ( X × A 1 ) ≃ |F H C ( X × A 1 )[1] | . Since for any pr esheaf of sp ectra E we have a natural ob ject wise equiv alence E ( − × A 1 ) ≃ E × N E , we obtain a natural weak equiv alence from N K ( X ) to | N F H C ( X )[1] | . No w take homotopy gr oups and a pply Lemma 3.2. As observed in [10, 4.12 ], the Chern character also co mm utes with the ring maps used to define the op er ators [ r ], V m , and with the transfer for R [ t n ] → R [ t ] defining F m . That is, it is a homo morphism of Carf ( R )-mo dules. Since the transfer is defined v ia the ring map R [ t ] → M n ( R [ t n ]), follow ed b y Morita inv arianc e , there is no trouble in pa ssing to schemes.  W e now co me to one of our main results, whic h implies Corolla ry 0.1. Theorem 4.2. F or al l n , N 2 K n ( X ) ∼ =  N K n ( X ) ⊗ t Q [ t ]  ⊕  N K n − 1 ( X ) ⊗ Ω 1 Q [ t ]  , and N p +1 K n ( X ) ∼ = p M j =0 N K n − j ( X ) ⊗ ∧ j Q p ⊗ ( t Q [ t ]) ⊗ ( p − j ) ⊗  Ω 1 Q [ t ]  ⊗ j . This holds for every X in Sch /F , as wel l as for Spec ( R ) wher e R is an arbitr ary c ommut ative F -algebr a. Pr o of. T his is immediate fro m 4.1 and Coro llary 3.3.  12 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL R emark 4.2.1 . J im Davis has p ointed o ut (see [9]) that a computation equiv alent to 4.2 ca n also b e derived — for arbitrar y rings R — from the F arrell- Jones con- jecture for the gr oups Z r . This par ticular cas e is covered by F. Quinn’s pro of of hyperelementary a ssembly for virtually ab elia n gro ups; see [20]. As a n immediate consequence of 4.2 and [1, XII(7 .3 )], we deduce: Corollary 4. 3 . S upp ose that X is in Sch /F , or t hat X = Sp ec( R ) for an F -algebr a R . Then: a) If N K n ( X ) = N K n − 1 ( X ) = 0 then N 2 K n ( X ) = 0 . b) If N K n ( X ) = 0 and K n − 1 ( X ) = K n − 1 ( X × A p ) then K n ( X ) = K n ( X × A p +1 ) . c) K n ( X ) = K n ( X × A p ) if and only if N K q ( X ) = 0 for al l q s uch that n − p < q ≤ n . Recall that X is called K n -r e gular if K n ( X ) = K n ( X × A p ) for all p . Corollary 4. 4 . S upp ose that X is in Sch /F , or t hat X = Sp ec( R ) for an F -algebr a R . Then t he fol lowing c onditions ar e e qu ivalent: a) X is K n -r e gular; b) N K n ( X ) = 0 and X is K n − 1 -r e gular; c) N K q ( X ) = 0 for al l q ≤ n . R emark 4.4.1 . This gives another pro o f o f V ors t’s The o rem [25, 2.1 ] (in character- istic 0) that K n -regular ity implies K n − 1 -regular ity , a nd extends it to schemes. The assumption that the scheme be affine is esse ntial in Bass’ question — here is a non-affine exa mple where the answer is negative. Negativ e answer to Bass’ question fo r nonaffine curves. Le t X be a smo o th pro jective elliptic cur ve over a num b er fie ld k a nd let L b e a nontrivial degre e zero line bundle with L ⊗ 3 trivial. F o r example, if X is the F ermat cubic x 3 + y 3 = z 3 , we may take the line bundle asso c ia ted to the divisor P − Q , where P = (1 : 0 : 1) and Q = (0 : 1 : 1). Lemma 4 .5. Writ e Y for the nonr e duc e d scheme with the same underlying sp ac e as X but with structu r e she af O Y = O X ⊕ L = Sym( L ) / ( L 2 ) , that is, L is r e gar de d as a squar e-zer o ide al. Then N K 7 ( Y ) = 0 but N 2 K 7 ( Y ) ∼ = N K 6 ( Y ) ⊗ Ω 1 Q [ t ] is nonzer o. Pr o of. I n this setting, the relative Ho chsc hild homolog y presheaf H H n ( Y , L ) is the kernel of H H n ( Y ) → H H n ( X ); sheafifying, H H n ( Y , L ) is the kernel o f HH n ( Y ) → HH n ( X ). Since Ω 1 X ∼ = O X we see from Lemma 5.3 o f [7] that HH n ( Y , L ) is: L ⊗ 3 ⊕ L ⊗ 5 if n = 4 ; L ⊗ 5 ⊕ L ⊗ 5 if n = 5 ; a nd L ⊗ 5 ⊕ L ⊗ 7 if n = 6 . By Ser re duality , H ∗ ( X, L ⊗ i ) = 0 if 3 ∤ i (cf. [7, 5.1]). By Za riski descent, this implies that H H 5 ( Y , L ) ∼ = H 1 ( X, H H 4 ) ∼ = H 1 ( X, L ⊗ 3 ) ∼ = k a nd H H 6 ( Y , L ) = 0. Since F H H ( Y ) ∼ = H H ( Y , L ), it follows fro m 4.1 and 4.2 that N K 7 ( Y ) = 0 but N K 6 ( Y ) ∼ = t Q [ t ] and N 2 K 7 ( Y ) ∼ = N K 6 ( Y ) ⊗ Ω 1 Q [ t ] ∼ = t Q [ t ] ⊗ Ω 1 Q [ t ] .  W e conclude this section by r e fining Pro po sition 4.1 a nd Coro llary 4.3 to take account of the Adams/Ho dge/ λ -decomp ositions o n K- theo ry and Ho chschild ho- mology , and by establishing the triviality of K ( i ) ∗ ( X ) for i ≤ 0 . Recall that by definition, K ( i ) n ( X ) = { x ∈ K n ( X ) ⊗ Q : ψ k ( x ) = k i x } . F or n < 0, the Adams op erations cannot b e defined in tegrally . How ever, it is p ossible to define the op era tio ns ψ k on K n ( X ) ⊗ Q for n < 0 us ing descending inductio n o n BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 13 n and the for mula ψ k { x, t } = k { ψ k ( x ) , t } in K n +1 ( X × ( A 1 − 0)) for x ∈ K n ( X ) and O ( A 1 − 0) = F [ t, 1 /t ]. This definitio n was p ointed out in [30, 8 .4]. By [11, 2.3] or [8, 7.2], the Chern character N K n +1 ( X ) → N H C n ( X ) commutes with the Adams o per ations ψ k in the s ense that it sends N K ( i +1) n +1 ( X ) to N H C ( i ) n ( X ) for a ll i ≤ n (a nd to 0 if i > n ). Here is the λ -decomp osition of the isomo rphism in Prop ositio n 4 .1: Prop ositio n 4. 6 . S u pp ose that X ∈ Sch /F , or that X = Sp ec( R ) for an F -algebr a R . Then for al l n and i , the Chern char acter induc es a natur al isomorph ism: N K ( i ) n ( X ) ∼ = H 1 − n F ( i − 1) H H ( X ) ⊗ t Q [ t ] . In p articular, if i ≤ 0 t hen N K ( i ) n ( X ) = 0 for al l n . Pr o of. B y [8 ], the Chern character K → H N sends K ( i ) ( X ) to H N ( i ) ( X ). Th e pro of in [8] shows that the lift F K ( X ) → F H N ( X ), shown to b e a weak e q uiv alence in [7, 1.6], may b e taken to send F ( i ) K ( X ) to F ( i ) H N ( X ). Since H C → H N sends H C ( i − 1) to H N ( i ) , the weak equiv a le nc e F H C [1] ≃ F H N ident ifies F ( i − 1) H C [1] and F ( i ) H N . Finally F ( i − 1) H H = 0 for i ≤ 0 .  Corollary 4.7 . K ( i ) n ( X ) ∼ = K ( i ) n ( X × A p ) if and only if N K ( i − j ) n − j ( X ) = 0 for all j = 0 , ..., p − 1 . Theorem 4.8. F or X in Sch /F or X = Sp ec( R ) , and al l inte gers n , we have: (1) F or i < 0 , K ( i ) n ( X ) = 0 . (2) F or i = 0 , K (0) n ( X ) ∼ = K H (0) n ( X ) ∼ = H − n cdh ( X, Q ) . Here K H deno tes the homotopy K -theory o f [29]. Theor em 4 .8 answers Question 8.2 of [30]. Pr o of. W e first show that K ( i ) n ( X ) ∼ = K H ( i ) n ( X ) when i ≤ 0. C overing X with affine op ens a nd us ing the May er-Vietoris sequences o f [29, 5 .1], it suffices to consider the case X = Sp ec( R ). Since K ( R ) Q is the pro duct of the eigen-comp onents, the descent s pec tr al se- quence E 1 p,q = N p K q ( R ) Q ⇒ K H p + q ( R ) Q (see [2 9, 1.3]) breaks up into one for each eigen-comp onent. If i ≤ 0, the sp ectral seque nce co lla pses by Prop ositio n 4.6 to yield K ( i ) n ( R ) ∼ = K H ( i ) n ( R ) for all n . T o determine the g roups K H ( i ) n ( R ) when i ≤ 0, we us e the cdh descent sp ectral sequence of [1 5, 1 .1]. If i < 0 , then the cdh sheaf K ( i ) cdh is tr iv ial a s X is lo ca lly smo oth, so we hav e K H ( i ) n ( R ) = 0 for all n . If i = 0 then the cdh shea f K (0) cdh is the sheaf Q cdh ; see [21, 2.8 ]. Hence we hav e K (0) n ( R ) = K H (0) n ( R ) = H − n cdh ( X, Q ).  5. The typical pieces T K ( i ) n ( R ) In this section, R will b e a co mm utative F -a lgebra. The default ground field k for K¨ a hler differentials and Hochschild homolo g y will b e Q . As stated in (0.3), the Adams summands N K ( i ) n ( R ) of N K n ( R ) decomp ose as N K ( i ) n ( R ) = T K ( i ) n ( R ) ⊗ t Q [ t ] for ea ch n and i ; the deco mpo sition is obta ined from an action of finite Car tier op erator s precisely as the co r resp onding one for N H C 14 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL and N H H , expla ined in Section 1. The typical pieces T K ( i ) n ( R ) are describ ed by the following formulas. Theorem 5.1. L et R b e a c ommutative F -algebr a. F or i 6 = n , n + 1 we have: T K ( i ) n ( R ) ∼ = ( H H ( i − 1) n − 1 ( R ) , if i < n, H i − n − 1 cdh ( R, Ω i − 1 ) if i ≥ n + 2 . F or i = n, n + 1 , the typic al pie c e T K ( i ) n ( R ) is given by t he exact se quenc e: 0 → T K ( n +1) n +1 ( R ) → Ω n R → H 0 cdh ( R, Ω n ) → T K ( n +1) n ( R ) → 0 . Pr o of. B y Pr o p osition 4.6, T K ( i ) n = H 1 − n F ( i − 1) H H . The rest is a r estatement of Lemma 3.4 .  R emark 5.1.1 . If R is essentially o f finite type over a field F who se transcendence degree is finite ov er Q , then the T K ( i ) n ( R ) ar e finitely ge ne r ated R -mo dules . This fails if tr . deg( F / Q ) = ∞ be cause then Ω i F / Q is infinite dimensional. F o r instance, Example 0.6 implies that, for R = F [ x ] / ( x 2 ), we hav e T K (2) 2 ( R ) = H H 1 ( R, x ) = F ⊕ Ω 1 F / Q . Corollary 5.2. Supp ose that R is essen t ial ly of finite t yp e over F and has dimen- sion d . If n < 0 then N K ( i ) n ( R ) = 0 u nless 1 ≤ i ≤ d + n , in which c ase N K ( i ) n ( R ) = H i − n − 1 cdh ( R, Ω i − 1 ) ⊗ t Q [ t ] . In p articular, N K n ( R ) = 0 for al l n ≤ − d . If d ≥ 2 then: N K 0 ( R ) ∼ =  ( R + /R ) ⊕ H 1 cdh ( R, Ω 1 ) ⊕ · · · ⊕ H d − 1 cdh ( R, Ω d − 1 )  ⊗ t Q [ t ] , N K − 1 ( R ) ∼ =  H 1 cdh ( R, O ) ⊕ H 2 cdh ( R, Ω 1 ) ⊕ · · · ⊕ H d − 1 cdh ( R, Ω d − 2 )  ⊗ t Q [ t ] , . . . . . . N K 1 − d ( R ) ∼ = H d − 1 cdh ( R, O ) ⊗ t Q [ t ] . If d = 1 then N K 0 ( R ) = ( R + /R ) ⊗ t Q [ t ] and N K n ( R ) = 0 for n < 0 . R emark 5 .2.1 . The d = 1 pa rt of Theo rem 5.2 holds for any 1-dimensional no ether - ian ring by [26, 2.8]. R emark 5.2.2 . Observe that 5.2 and 4.4 imply that R is K − d -regular . This recov ers the a ffine case o f one o f the main results in [5]. Here is a s pec ia l ca se o f the calculations in Theore m 5.1, which proves Theo - rem 0.7. W e will use it to co nstruct the counterexample to Ba ss’ question in the companion pap er [6]. Theorem 5 .3. L et F b e a field of char acteristic 0 and R a normal domain of dimension 2 , essential ly of finite typ e over F . Then a) H 1 F H H ( R/F ) ∼ = H 1 cdh ( R, Ω 1 /F ) , b) H 2 F H H ( R/F ) ∼ = H 1 cdh ( R, O ) , c) N K 0 ( R ) ∼ = H 1 cdh ( R, Ω 1 ) ⊗ t Q [ t ] , and d) N K − 1 ( R ) ∼ = H 1 cdh ( R, O ) ⊗ t Q [ t ] . BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 15 Pr o of. Parts (a) and (b) a re immediate from E xample 3.5 and the fa c t that R is reduced and seminormal. Parts (c) and (d) fo llow from (a) and (b) using Pr op osition 4.1; cf. Corollar y 5.2.  W e introduce some no tation to make the statement of the nex t theore m more readable. The letter e deno tes the maxim um tra nscendence degree of the co mpo nen t fields in the total ring of fractions Q of R red . F or simplicity , we write Ω i cdh ( X ) for H 0 cdh ( X, Ω i ), a nd we have written Ω i cdh ( R ) / Ω i R for the cokernel o f Ω i R → Ω i cdh ( R ). Definition 5.4. F or any co mmu tative r ing R containing Q , we define: E n ( R ) = Ω n cdh ( R ) / Ω n R ⊕ M ∞ p =1 H p cdh ( R, Ω n + p ); g H H n ( R ) = ker  H H n ( R ) → Ω n Q  = ker(Ω n R → Ω n Q ) ⊕ M n − 1 i =1 H H ( i ) n ( R ) . Theorem 5.5. L et R b e a c ommutative ring c ontaining Q . Then for al l n : N K n ( R ) ∼ =  g H H n − 1 ( R ) ⊕ E n ( R )  ⊗ t Q [ t ] . If furthermor e R is ess en tial ly of fi nite typ e over a field, and n ≥ e + 2 , then N K n ( R ) ∼ = H H n − 1 ( R ) ⊗ t Q [ t ] . Pr o of. As s emblin g the descr iptions o f the T K ( i ) n ( R ) in Theorem 5.1 yields the first assertion. The se c o nd part is immediate from this and 3.6.  R emark 5.5.1 . The Chern c haracter N K n ( R ) → N H C n − 1 ( R ) ∼ = H H n − 1 ( R ) ⊗ t Q [ t ] is an isomor phism fo r n ≥ e + 2. If n ≤ e +1 , neither it no r the map H 1 − n F H H ( R ) → H H n − 1 ( R ) of 4.1 need b e a sur jection. In or der to compar e the tor s ion submo dules tors Ω ∗ R with the typical pieces of N K ∗ ( R ), we need the a ffine ca se o f the following lemma. F ollowing tra dition, we write F ( X ) for the total r ing of fractions of X red . That is , F ( X ) is the pro duct of the function fields of the irreducible comp onents of X red . When X = Spec( R ) is affine, we write Q instead of F ( X ). Lemma 5. 6. L et X ∈ Sch /F ; for F ( X ) as ab ove, the map Ω i cdh ( X ) → Ω i F ( X ) is an inje ction. Pr o of. W e may ass ume X reduced, and pro ceed by induction on d = dim( X ), the case d = 0 being trivial. Cho os e a resolutio n o f sing ula rities X ′ → X a nd let Y b e the singular lo cus of X , with Y ′ = Y × X X ′ . By [22, 12.1 ], there is a May er-Vietoris exact seq uence 0 → Ω i cdh ( X ) → Ω i cdh ( X ′ ) ⊕ Ω i cdh ( Y ) → Ω i cdh ( Y ′ ) ∂ − → H 1 cdh ( X, Ω i ) → · · · . Since F ( Y ) ⊆ F ( Y ′ ), Ω i F ( Y ) ⊆ Ω i F ( Y ′ ) . Beca use dim( Y ′ ) < d , the inductive hypoth- esis implies that Ω i cdh ( Y ) → Ω i cdh ( Y ′ ) is an injection. Hence Ω i cdh ( X ) → Ω i cdh ( X ′ ) is an injection. But X ′ is smoo th, so b y scdh descent for Ω i (see [7, 2.5]) we have Ω i cdh ( X ′ ) ∼ = Ω i ( X ′ ) ⊂ Ω i F ( X ′ ) = Ω i F ( X ) .  R emark 5.6.1 . Lemma 5 .6 remains true if, instead o f Ω i , we use Ω i /k for k ⊆ F . In particular, if X = Sp ec( R ) is r educed affine, then Ω i cdh ( R/k ) = H 0 cdh ( R, Ω i /k ) injects into Ω i Q/k . Th us tor s(Ω i R/k ), defined as the kernel o f Ω i R/k → Ω i cdh ( R/k ) in (3.9c), is the tor sion submo dule of Ω i R/k . 16 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL Corollary 5 .7. F or al l n ≥ 1 , T K ( n ) n ( R ) ∼ = ker( Ω n − 1 R → Ω n − 1 Q ) . In p articular if R is r e duc e d, then T K ( n ) n ( R ) is the torsion submo dule of Ω n − 1 R . Pr o of. B y 5.1, T K ( n ) n ( R ) is the kernel of Ω n − 1 R → Ω n − 1 cdh ( R ), so 5.6 applies.  The typical pieces of N K (2) 1 ( R ) and N K (2) 2 ( R ) of 5.1 and 5.7 may b e describ ed as follows. Prop ositio n 5. 8. F or al l re duc e d F -algebr as R , the typic al pie c es T K (2) 1 ( R ) = Ω 1 cdh ( R ) / Ω 1 R and T K (2) 2 ( R ) = tors(Ω 1 R ) fit into an exact se quenc e: 0 → tors(Ω 1 R ) → tors(Ω 1 R/F ) → Ω 1 F ⊗ ( R + /R ) → Ω 1 cdh ( R ) Ω 1 R → Ω 1 cdh ( R/F ) Ω 1 R/F → 0 . Pr o of. W e may a s sume Sp ec R ∈ Sch /F . Reca ll from [7 , 4.2 ] that there is a b ounded second qua drant homolog ical sp ectral sequence for all p (0 ≤ i < p , j ≥ 0): p E 1 − i,i + j = Ω i F /k ⊗ F H H ( p − i ) p − i + j ( R/F ) ⇒ H H ( p ) p + j ( R/k ) . When p = 1, this sp ectral sequenc e degenera tes to y ield exactness of the b ottom row in the following commutativ e diagra m; the top row is the Firs t F undamental Exact Sequence for Ω 1 [31, 9 .2.6]. Ω 1 F ⊗ R   / / Ω 1 R   / / Ω 1 R/F   / / 0 0 / / Ω 1 F ⊗ R + / / Ω 1 cdh ( R ) / / Ω 1 cdh ( R/F ) / / 0 . The upp er left horizo ntal map is a n injection b ecause the left vertical map is an injection. Now apply the snake lemma, using Rema rk 5.6.1 .  6. Bass’ question f or algebras over l arge fields. W e will now show that the answer to B ass’ question is p os itive for alge br as R essentially of finite type ov er a field F of infinite trans cendence degr ee ov er Q . Recall from Pr op osition 4.1 that N K n +1 ( R ) ∼ = H − n F H H ( R/ Q ) ⊗ t Q [ t ] . In light of this identification, the version of B ass’ questio n stated b efore Theor em 0.2 b ecomes the ca se k = Q of the following que s tion: (6.1) D o es H m F H H ( R/k ) = 0 imply that H m +1 F H H ( R/k ) = 0? In Theorem 6.6, we show that the answer to question (6.1 ) is p os itive provided R is o f finite type ov er a field F that has infinite tra nscendence deg ree ov er k . The pro of is essentially a forma l conseq ue nc e o f the K ¨ unneth formula in Lemma 6.3. Lemma 6.2. L et R b e a c ommutative F -algebr a, and supp ose k is a s ubfield of F . Then H −∗ F H H ( R/k ) and H −∗ cdh ( R, H H ( /k )) ar e gr ade d mo dules over the gr ade d ring Ω • F /k . Pr o of. As in 2.2.1 , w e may suppo se that R is of finite t yp e ov er F . Cons ider the functor on F -alge br as tha t ass o ciates to an F -alg ebra A the Ho chsc hild complex H H ( A/k ). The shuffle pro duct makes this into a functor to dg - H H ( F /k )-mo dules. Since the cdh -site has a s e t of points (corresp onding to v aluations by [13, 2.1]), we can use a Go dement res olution to find a mo del for the cdh -h yp ercohomo logy BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 17 H cdh ( − , H H ( /k )) which is also a functor to dg - H H ( F /k )-mo dules. It follows tha t there is a mo del fo r F H H ( R/k ) that is a dg - H H ( F /k )-mo dule, functorially in R . This implies the a ssertion, since Ω • F /k = H − • H H ( F /k ).  Lemma 6.3. (K ¨ u nneth F ormula) Su pp ose that Q ⊆ k ⊆ F 0 ⊆ F ar e fields. L et R 0 b e an F 0 -algebr a, and set R = F ⊗ F 0 R 0 . i) L et T = { t i } b e tra nsc endenc e b asis of F /F 0 ; writing F [ dT ] for t he ex terior algebr a on the set { dt i } , we have Ω • F /F 0 = F [ dT ] and: Ω • F /k ∼ = F [ dT ] ⊗ F 0 Ω • F 0 /k In p articular, the gr ade d algebr a homomorphism Ω • F 0 /k → Ω • F /k is flat. ii) H H ∗ ( R/k ) ∼ = Ω • F /k ⊗ Ω • F 0 /k H H ∗ ( R 0 /k ) ∼ = F [ dT ] ⊗ F 0 H H ∗ ( R 0 /k ) . Pr o of. I t is class ical that F [ dT ] = Ω • F /F 0 . The tensor product decompo sition o f part i) follows from the fact that the fundamental sequence 0 → F ⊗ F 0 Ω 1 F 0 → Ω 1 F → Ω 1 F /F 0 → 0 is split exact. This pro ves i). T o pro ve ii), cho ose a free chain dg - F 0 -algebra Λ and a surjective quasi-iso morphism of dg -algebras Λ ∼ ։ R 0 . Then Λ ′ = F ⊗ F 0 Λ → F ⊗ F 0 R 0 = R is a free chain mo del of R as a k -a lgebra. W rite Ω • Λ for differential forms; consider Ω • Λ as a chain dg -algebra with the different ial δ induced by that of Λ. Note Λ and Λ ′ are homolog ically r egular in the sense of [4], so that Theorem 2.6 of [4] applies. Co mb ining this with part (i), w e obtain H H ∗ ( R ) = H H ∗ (Λ ′ ) = H ∗ (Ω • Λ ′ ) = H ∗ (Ω • F ⊗ Ω • F 0 Ω • Λ ) = Ω • F ⊗ Ω • F 0 H ∗ (Ω • Λ ) = Ω • F ⊗ Ω • F 0 H H ∗ ( R 0 ) .  Here is an ea sy co nsequence o f Lemmas 6.2 and 6.3 . Prop ositio n 6 .4. Supp ose Q ⊆ k ⊆ F 0 ⊆ F ar e fi eld extensions, that R 0 is an F 0 - algebr a and R = F ⊗ F 0 R 0 . Then ther e is an isomorphism of gr ade d Ω • F /k -mo dules F [ dT ] ⊗ F 0 H −∗ F H H ( R 0 /k ) ∼ = H −∗ ( F H H ( R/k )) . W e also need the following lemma to pr ov e the main re s ult of this sec tio n. Lemma 6.5. L et R b e essential ly of finite typ e over F ⊃ Q , and let H n ( R ) denote either H H n ( R ) or H − n F H H ( R ) . Assu me t hat H n i ( R ) = 0 for some finite set { n 1 , . . . , n r } of p ositive inte gers. Then ther e exist an F -algebr a of finite typ e R ′ , and a mu lt iplic atively close d set S such that R ∼ = S − 1 R ′ and H n i ( R ′ ) = 0 for 1 ≤ i ≤ r . Pr o of. B ecause R is ess ent ially of finite t yp e, it is the lo c a lization R = S − 1 R ′′ of some finite t ype F -algebr a R ′′ . It is well known that H H n ( S − 1 R ′′ ) ∼ = S − 1 H H n ( R ′′ ) (see [31, 9.1.8]), and H − n F H H ( S − 1 R ′′ ) ∼ = S − 1 H − n F H H ( R ′′ ) by [7, 2.8–9]. Because R ′′ is o f finite type ov er F , we may wr ite R ′′ = F ⊗ F 0 R 0 for some finitely generated field extens io n F 0 of Q and some finite type F 0 -algebra R 0 . Note R 0 is essentially of finite t yp e ov e r Q , whence H p ( R 0 ) is a finitely gene r ated R 0 -mo dule ( p ≥ 0 ). By 6.3 a nd/or 6.4, H p ( R ′′ ) is isomorphic, as an R ′′ -mo dule, to a direct sum o f copies of R ′′ ⊗ R 0 H q ( R 0 ) with q ≤ p . In particular , M = L r i =1 H n i ( R ′′ ) is 18 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL a finite sum of R ′′ -mo dules, ea ch of which is a — p ossibly infinite — dir e c t sum of copies o f one finitely gener a ted mo dule. Given tha t M ha s this form, the hypothesis that S − 1 M = 0 implies that there exists a no nzero element s ∈ Ann ( M ) ∩ S . Consider the finite type F - a lgebra R ′ = R ′′ [1 /s ]. Then R ∼ = S − 1 R ′ and we hav e L i H − n i ( R ′ ) = M [1 /s ] = 0.  Theorem 6.6. Su pp ose k ⊂ F is an ex tension with tr . deg ( F /k ) = ∞ , and R is essential ly of finite typ e over F . If H n ( F H H ( R/k )) = 0 , t hen H m ( F H H ( R/k )) = 0 for al l m ≥ n . Pr o of. B y Lemma 6 .5, we may ass ume tha t R is of finite t yp e ov er F . There is a finitely genera ted fie ld extension F 0 ⊂ F of k and a finite type F 0 -algebra R 0 such that R = R 0 ⊗ F 0 F . Note that tr . de g ( F /F 0 ) = ∞ . By Lemma 6 .3 and Pro po - sition 6.4, Ω i F /F 0 ⊗ F 0 H n + i ( F H H ( R 0 /k )) is a dir ect summand of H n ( F H H ( R/k )) for each i ≥ 0. Since Ω i F /F 0 6 = 0 for all i , a ll the H n + i ( F H H ( R 0 /k )) v anish as well. Similarly , H m ( F H H ( R/k )) is a direct sum of copies of the gr oups Ω j F /F 0 ⊗ F 0 H m + j ( F H H ( R 0 /k )) for j ≥ 0, a ll of which v anish when m ≥ n , as w e just ob- served.  Corollary 6.7 . L et Q ⊂ F b e a field extension of infinite tr ansc endenc e de gr e e, and supp ose R is essent ial ly of finite t yp e over F . Then N K n ( R ) = 0 implies that R is K n -r e gular. Pr o of. C o mbine Theor e m 6.6 with Pr op osition 4.1 a nd Coro llary 4.4.  Here is another pr o of of Co rollar y 6.7 , which is esse n tially due to Mur th y and Pedrini and given in their 1 972 pap er [18]; they stated the result only for n ≤ 1 bec ause tr ansfer maps for hig her K -theory a nd the W ( R )-mo dule structure ha d not yet b een discov ered. W e a re g rateful to Jose ph Gub eladze [14] fo r p ointing this out to the a uthors. Lemma 6.8. If R is an algebr a over a field k of char acteristic 0, N p K n ( R [ t ]) → N p K n ( R ⊗ k k ( t )) is inje ctive. Pr o of. T he pro of in [18, 1.3 –1.6] go es through, taking into account that the nor m map a nd lo ca lization seq uences us ed there for K 0 , K 1 are now k nown for all K n .  Lemma 6. 9. Supp ose that k is an algebr aic al ly close d field of infinite tr ansc endenc e de gr e e over Q , and that R is a finitely gener ate d k -algebr a. If N K n ( R ) is zer o, t hen K n ( R ) ≃ − → K n ( R [ x 1 , ..., x p ]) for al l p > 0 . Pr o of. Muthy a nd Pedrini prov e this in [18, 2.1.]; although their result is o nly stated for i ≤ 1, their pro of works in g e neral. Note that since N K n ( R ) ha s the form T K n ( R ) ⊗ t Q [ t ] by (0.3 ) (a result which was not known in 197 2), N K n ( R ) is torsionfree, a nd has finite rank if a nd only if it is zer o.  Pr o of of Cor ol lary 6.7. Let Φ denote the functor N p K n . If k ⊂ k 1 is a finite alge- braic field extension a nd R is a k -algebra , then Φ( R ) → Φ( R ⊗ k k 1 ) is a n injection bec ause its comp osition with the transfer Φ( R ⊗ k k 1 ) → Φ( R ) is m ultiplication b y [ k 1 : k ], and Φ( R ) is a tor sionfree group. Since Φ commutes with filter e d colimits of rings, Φ( R ) → Φ( R ⊗ k ¯ k ) is an injection. Th us Le mma 6.9 suffices to pr ov e Corollar y 6.7 when R is of finite type.  BASS’ N K GR OUPS AND cdh -FIBRANT HOCHSCHILD HOMOLOGY. 19 7. N K 0 of surf aces W e conclude with a general description for affine surfaces of the ca no nical map Ω 1 F ⊗ F N K − 1 → N K 0 . This sheds light o n the difference be tw een the ca ses o f small and large base fields, and als o explains some results of [3 3]. If R is a 2-dimensio nal no etherian ring then N K 0 ( R ) is the direct sum o f N K (1) 0 ( R ) = N P ic( R ) and N K (2) 0 ( R ) Theorem 7.1. L et R b e a 2-dimensional normal domai n of finite typ e over a field F of char acteristic 0. Ther e is an exact se quenc e: 0 → N K (2) 1 ( R ) →  H 0 ( R, Ω 1 /F ) / Ω 1 R/F  ⊗ t Q [ t ] → Ω 1 F ⊗ F N K − 1 ( R ) → N K 0 ( R ) → H 1 cdh ( R, Ω 1 /F ) ⊗ t Q [ t ] → 0 . Pr o of. C o nsider the following sho rt exa ct sequence of sheav es in (Sc h /F ) cdh : 0 → Ω 1 F ⊗ F O → Ω 1 → Ω 1 /F → 0 Applying H cdh yields 0 → Ω 1 F ⊗ F R ι → H 0 ( R, Ω 1 ) → H 0 ( R, Ω 1 /F ) ∂ → Ω 1 F ⊗ F H 1 cdh ( R, O ) → H 1 cdh ( R, Ω 1 ) → H 1 cdh ( R, Ω 1 /F ) → 0 Note tha t, b ecause Ω 1 R → Ω 1 R/F is on to, the map ∂ kills the ima ge of Ω 1 R/F . Simi- larly , the image of ι is co ntained in that o f Ω 1 R . Th us we obtain 0 → H 0 ( R, Ω 1 ) / Ω 1 R → H 0 ( R, Ω 1 /F ) / Ω 1 R/F → Ω 1 F ⊗ F H 1 cdh ( R, O ) → H 1 cdh ( R, Ω 1 ) → H 1 cdh ( R, Ω 1 /F ) → 0 Now apply ⊗ t Q [ t ] a nd use 5.1 and parts c) and d) of 5 .3.  Corollary 7.2. L et R b e a 2-dimensional normal domain of finite typ e over a field F of char acteristic 0. If N K − 1 ( R ) = 0 t hen N K 0 ( R ) ∼ = H 1 cdh ( R, Ω 1 /F ) ⊗ t Q [ t ] . Example 7.3 . L et R b e a 2-dimensio na l no rmal domain o f finite type over Q , and put R F = R ⊗ F . By 4.1 and 6.4, (7.4) N K ∗ ( R F ) ∼ = N K ∗ ( R ) ⊗ Ω ∗ F / Q . Keeping track of the λ -decomp osition, as in 5.1, w e see from The o rem 0.7 that T K (2) 1 ( R F ) ∼ = T K (2) 1 ( R ) ⊗ F ∼ = H 0 ( R, Ω 1 ) ⊗ F / Ω 1 R ⊗ F ∼ = H 0 ( R F , Ω 1 /F ) / Ω 1 R F /F . F ro m Theorem 7.1 we get a n exact sequence (7.5) 0 → Ω 1 F / Q ⊗ F N K − 1 ( R F ) → N K 0 ( R F ) → H 1 cdh ( R F , Ω 1 /F ) ⊗ t Q [ t ] → 0 Using (7.4) and 0.7 aga in, we s ee that the sequence (7.5) is isomorphic to the s um (0 → Ω 1 F / Q ⊗ H 1 cdh ( R, O ) ⊗ t Q [ t ] ≃ − → Ω 1 F / Q ⊗ H 1 cdh ( R, O ) ⊗ t Q [ t ] → 0 → 0) ⊕ (0 → 0 → F ⊗ H 1 cdh ( R, Ω 1 ) ⊗ t Q [ t ] ≃ − → F ⊗ H 1 cdh ( R, Ω 1 ) ⊗ t Q [ t ] → 0) 20 G . COR TI ˜ NAS, C. HAE SEMEYE R, MARK E. W ALKER, AND C. W EIBEL F or example, for R F := F [ x, y , z ] / ( z 2 + y 3 + x 10 + x 7 y ) the res ults of [6] show that: N K − 1 ( R F ) = F ⊗ t Q [ t ] N K 0 ( R F ) = Ω 1 F / Q ⊗ t Q [ t ] ∼ = tr . deg( F ) M p =1 F ⊗ t Q [ t ] . In other words, bo th typical pieces T K − 1 ( R F ) and T K 0 ( R F ) are F - vectorspaces, but while dim F T K − 1 ( R F ) = 1 for all F , any cardinal num be r κ can be r ealized a s dim F T K 0 ( R F ) for an appropriate F . Ac kno wl edgements. The author s w ould like to thank M. Schlic h ting, whose co n- tributions go b eyond the co llab oration [5]. W e would also like to thank W. V as- concelos, L. Avramov, E . Sell and J. W a hl for useful discuss ions. References [SGA4] M. Artin, A. Grothendiec k, and J. L. V erdier. Th ´ eorie des top os e t c ohomo lo gie ´ etale des sch´ emas. Tome 2 . Springer-V erl ag, Berli n, 1972. S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois-Marie 1963–1964 (SGA 4), Di rig´ e par M. Ar tin, A. Grothendiec k et J. L. V erdier. 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W eibel , Cyclic homology for schemes, Pr o c. AMS 124 (1996), 1655–1662. [33] C. A. W eibel. The negative K -theory of norm al surf aces. Duke Math. J. 10 8 (2001), 1–35. Dep. Ma tem ´ atica, FCEyN-UBA, Ciudad Universit aria P ab 1 , 1428 Buenos Aires, Ar- gentina E-mail addr e ss : gcorti@gm.uba .ar Dept. of Ma thema tics, University of California, Los An geles CA 90 095, USA E-mail addr e ss : chh@math.ucla .edu Dept. of Ma thema tics, University of Nebraska - Lincoln, Lincoln, NE 68588, USA E-mail addr e ss : mwalker5@math .unl.edu Dept. of Ma thema tics, Rutgers Univ ersity, New Brunswick, NJ 0890 1, USA E-mail addr e ss : weibel@math.r utgers.edu