Motivic strict ring models for K-theory

Motivic strict ring m o dels for K -theory Oliver R¨ ondigs, Markus Spitzw ec k, Paul Arne Østvær July 23, 2009 Abstract It is shown that the K -theory of ev ery no etherian base sc heme of fin ite Krull dimension is represen ted by a strict ring ob ject in the setting o f motivic stable homotop y theory . The adjectiv e ‘strict’ is used to distinguish b et w een the t yp e of ring structure w e co nstruct and o ne whic h is v alid only up to homotop y . Both the catego ries of motivic functors and motivic symmetric sp ectra furnish con venien t framew orks for constru cting the r in g mo dels. Analogo us top ological resu lts follo w b y runn in g th e same type of argument s as in the motivic setting. Con t e n ts 1 In t ro duction 2 2 A strict m o del 4 3 The homotopy t yp e 10 4 Multiplicativ e structure 14 1 1 In tro ducti o n Motivic homotopy theory can b e view ed as an expansion of classical homotopy theory to an algebro-geometric setting. This has enabled the in t r o duction of homotop y theoretic tec hniques in the study of gen eralized ring (co)homolo gy theories for s c hemes, and as in classical alg ebra one studies these via mo dules and alg ebras. F rom this p ersp ectiv e, motiv es are simply mo dules o ver the motivic Eilen b erg-MacLane ring sp ectrum [8], [9]. The main purp ose of this pap er is to show t hat the K -theory of ev ery no etherian base sc heme of finite Kr ull dimension a cquires strict ring ob ject mo dels in motivic homotop y theory , and thereby pav e the w a y tow ards a classification of mo dules o v er K -theory . An example of a base sc heme of particular interest is the integers . W orking with some flabby smash pro duct whic h only b ecome asso ciative , comm utativ e and unita l after passage t o the motivic stable homotopy cat ego ry is inadequate f o r our purp oses. The whole pa p er is therefore couche d in terms of motivic functors [1] and motivic symmetric spectra [3]. Throughout the pap er the t erm ‘ K -theory’ is short for homotopy algebraic K -theory . Fix a no etherian ba se sc heme S of finite Krull dimension with m ultiplicativ e group sc heme G m . Denote b y K GL the ordinary motivic sp ectrum represen ting K -theory [13 ]. As sho wn in [11 ], see also [2] a nd [6], inv erting a homoto py class β ∈ π 2 , 1 Σ ∞ B G m + in the motivic susp ension sp ectrum of the classifying space of the multiplicativ e group sc heme yields a natural isomorphism in the motivic stable homot op y catego r y Σ ∞ B G m + [ β − 1 ] ∼ = / / K GL . W e shall turn the Bott inv erted mo del f o r K -theory into a comm utative monoid KGL β in the category of mot ivic symmetric sp ectra. T o start with, the m ultiplicative structure of G m induces a comm utat ive monoid structure on the motivic symmetric susp ension sp ectrum of the classifying space B G m + . A fa r more inv olv ed a nalysis dealing with an actual map rather than some homotopy class a llo ws us to define KGL β and ev en tually v erify that it is a comm utativ e monoid with the same homotopy type as K -theory . Sev eral of the ma in tec hniques emplo y ed in the pro o f are of in terest in their o wn right, and can b e traced bac k to constructions for symmetric sp ectra, cf. [10]. It also turns out that there exists a strict ring mo del for K -theory in the cat ego ry of motivic functors. Here the motivic functor mo del is constructed in a leisurely w a y b y transp o rting K GL β via the strict symmetric monoidal functor relating motivic symmetric sp ectra to motivic functors. 2 When suitably adopted the motivic a rgumen t w orks also in top ological categories. The top ological strict ring mo dels app ear to b e new, ev en in the case of symmetric sp ectra. The Bott eleme n t considered b y V o ev o dsky in [13] is obtained from the virtual v ector bundle [ O P 1 ] − [ O P 1 ( − 1)] . A k ey step in the construction of K GL β is to interpret the same elemen t, view ed in the p oin ted motivic unstable homotop y categor y , as an actual map b et w een motivic spaces. In o r der to mak e this part precise w e shall use a la x symmetric monoidal fibrant replacemen t functor for p ointed motivic spaces. Fibrancy is a constant source for extra fun in abstract homotopy theory . The pro blem resolv ed in this pap er is no exception in that resp ect. It is also w ort hwhile to emphasize the in triguing fact that β do es not pla y a role in the definition of the multiplic ativ e structure of K GL β . How ev er, the Bott elemen t enters in the definition of the unit map 1 → KGL β , which is part of the monoid structure, and in the structure maps. In fact, up to some fibrant replacemen t, KGL β is constructed fa irly directly from Σ ∞ B G m + b y intert wining a map represen ting β with the structure maps. O n the lev el o f homotopy groups this t yp e o f in tert wining has the effect of in v erting the Bo tt elemen t. As a result, w e obtain the desired homotop y type. In [7] it is show n that under a certain normalization a ssumption the ring structure on KGL in the motivic stable homotop y catego ry is unique ov er the ring of in tegers Z . F or any base sc heme S t he m ultiplicative structure pulls bac k to give a distinguished monoidal structure on KGL . W e sho w the m ultiplicativ e structures on K GL β and KGL coincide in the motivic stable ho mo t o p y category . T he pro of o f this result is not forma l. A k ey input is that for K - theory there exists no non trivial phantom maps. In turn, this is a consequence of La ndweber exactness in motivic homoto p y theory [5]. In [2] the setup of ∞ -categories is used to no te the existence of an E ∞ or coheren tly homotop y commutativ e structure on K -theory . W ork in progress suggests there exists a unique suc h structure. Ho wev er, with the construction of K GL β in hand one has strict mo dels for K -theory . And the strictness of a mo del has the pleasing consequenc e that it is amenable to a simpler homotopical study . This has b een the sub ject to m uc h w ork dealing with top ological K -theories. 3 2 A strict mo del The main fo cus of t his section is the construction of a strict mo del for K -theory in the category of motivic symmetric sp ectra. Throughout w e use the ‘closed’ motivic mo del structure in [7] with a view to w ards realizatio n functors. An extensiv e back ground in motivic stable homotopy theory is not assumed. The classifying space B G m has terms G × n m for n ≥ 0 with the conv en tion that its zeroth term is a p oint. Its fa ce and degeneracy maps, whic h are defined in a standard w ay using diagonals and pro ducts, allow to consider B G m as a motivic space ( t hat is, a simplicial presheaf on the Nisnevic h site of the base sc heme S ). Throughout w e use the follo wing standard notation: Let S 2 , 1 denote the motivic sphere defined as the smash pro duct of the simplicial circle S 1 , 0 = ∆ 1 /∂ ∆ 1 with G m p oin ted by its one-section. F or n ≥ 2 w e set S 2 n,n = S 2 n − 2 ,n − 1 ∧ S 2 , 1 . When forming motivic sp ectra, w e shall fo r consistency with [10] b e smashing with S 2 , 1 on the right. In the in tro duction it w as recalled that t he Bott elemen t is a homotop y class β ∈ π 2 , 1 Σ ∞ B G m + . As suc h, it is represen ted b y a map of p ointed motivic spaces S 2 n +2 ,n +1 → ( B G m + ∧ S 2 n,n ) fib for some n , where ( − ) fib denotes a fibra nt replacemen t functor. The construction w e giv e of KGL β w orks fo r any suc h represen tativ e provided the fibran t replacemen t functor is lax symmetric monoidal. By Lemma 2.2 we may c ho ose a fibrant replacemen t functor with the stated prop erties. As shown in the following, the situation at hand allo ws for an explicit construction of a map S 4 , 2 → ( B G m + ∧ S 2 , 1 ) fib that represen ts the Bott elemen t. Before pro ceeding with the construction o f the strict mo dels w e discuss fibrancy of the multiplicativ e group sc heme and its classifying space, p ertaining t o the discussion of a fibran t replacemen t functor in the ab ov e. 4 Example 2.1: The classifying s p ac e of the multiplic ative gr oup scheme is se ctionwise fibr ant b e c ause it takes values in simplicial ab elian gr oups. When S i s r e gular then G m is fibr ant. Howev e r, as the fol lowing discussio n shows, B G m is not fibr ant. The standar d op en c overing of the pr oje ctive line by affin e lines yields an elemen tary distinguishe d squar e: G m / /   A 1   A 1 / / P 1 (1) L et P denote the homotopy pul lb ack of the diagr am B G m ( A 1 ) / / B G m ( G m ) B G m ( A 1 ) o o obtaine d by applying B G m to ( 1 ) . Then the homotopy fib er of the m a p P → B G m ( A 1 ) i s we akly e quivalent to the homotopy fib er F of B G m ( A 1 ) → B G m ( G m ) . F or a ring R , let R × denote its m ultiplic ative gr oup of units. With these defin itions ther e exis t induc e d exact se quenc es of homotopy gr oups 0 / / π 1 P / / O × S = O S [ t − 1 ] × / / π 0 F / / π 0 P / / 0 , and 0 / / O × S = O S [ t ] × / / O S [ t, t − 1 ] × / / π 0 F / / 0 . (Her e t is an indetermin ate.) Henc e O S [ t − 1 ] × → π 0 F c ann ot b e surje ctive. It fol lows that P is not c onne cte d and, in p articular, not we akly e quivalent to B G m ( P 1 ) = B O × S . This shows that B G m do es not satisfy the Nisnevich fib r ancy c ondition, se e [1], [3 ] , [4]. W e write i : S 1 , 0 ∧ G m → B G m for the inclusion of t he 1-ske leton G m in to the classifying space B G m . Let c : S 1 , 0 ∧ G m → B G m denote the constant map. Via t he motivic w eak equiv alences S 1 , 0 ∧ G m ≃ P 1 and B G m ≃ P ∞ the map i can b e iden tified in t he p o inted motivic unstable homo t o p y category with the inclusion P 1 → P ∞ . 5 In homogeneous co ordinat es the inclusion ma p is giv en by [ x : y ] 7→ [ x : y : 0 : · · · ] . Similarly , the map c coincides with the canonical comp osite map P 1 → S → P ∞ giv en b y [ x : y ] 7→ [1 : 0 : · · · ] . Adding a disjoin t base p oin t to the classifying space of G m yields p ointe d ma ps i + , c + : S 2 , 1 → B G m + for the base p oin t of B G m . No w in order to mo v e the base p oint in B G m + w e tak e the unreduced susp ension of b oth these maps. Recall the unreduced susp ension of a motivic space A is defined as the pushout S ( A ) = A × ∆ 1 ∪ A × ∂ ∆ 1 ∂ ∆ 1 . One can view it as a p oin ted motivic space b y t he imag e of 0 ∈ ∂ ∆ 1 . With this definition, the unreduced susp ensions of the maps i + and c + are p oin ted with resp ect to the image of (+ , 0) in their ta r get. If A is p ointed, the canonical map q : S ( A ) → Σ A to the reduced susp ension is a we ak equiv alence. Hence there exists a map o f p o in ted motivic spaces S 4 , 2 → ( S ( S 2 , 1 ) ∧ G m ) fib lifting the inv erse o f the map q ∧ G m in the p ointed motivic unstable homo t o p y category . By comp o sing we end up with the t wo p ointed maps i β , c β : S 4 , 2 ∼ − → ( S ( S 2 , 1 ) ∧ G m ) fib ⇒ ( S ( B G m + ) ∧ G m ) fib ∼ − → ( B G m + ∧ S 2 , 1 ) fib . As a first approx imation of the Bott elemen t β w e consider the analog of the virtual v ector bundle [ O P 1 ] − [ O P 1 ( − 1)] in the p ointed motivic unstable homoto p y category c β − i β : S 4 , 2 → ( B G m + ∧ S 2 , 1 ) fib . (2) 6 In order to form the difference map we use that S 4 , 2 is a (t w o-fo ld) simplicial susp ension, and therefore a cogroup ob ject in the p oin ted motivic unstable homotop y cat ego ry . Note, ho wev er, tha t c β represen ts the trivial map b ecause it factors through the base p oin t. By app ealing to the motivic mo del structure it follow s that (2) lifts to a ‘strict’ motivic Bott map S 4 , 2 → ( B G m + ∧ S 2 , 1 ) fib . b et w een p ointed motivic spaces. Here w e use that the motivic sphere is cofibrant in t he closed motivic mo del structure. The fibrancy ca veat ab ov e requires us to replace the susp ension ob ject Σ ∞ B G m + with a lev elwise fibrant motivic sp ectrum. An arbitrary suc h replacemen t need no t preserv e comm uta tiv e mo no ids. T he f o llo wing lemma will therefore b e of relev ance later in the pap er. Lemma 2.2: Th e r e exists a lax s ymm etric monoidal fibr ant r eplac emen t functor Id → F on the c ate gory of p ointe d motivic sp ac es. Pr o of. The straigh tf orw ar d simplicial presheaf analog of [4 , Theorem 2.1.66] prov ides a lax symmetric monoida l fibra n t replacemen t functor Ex ∞ for the lo cal mo del structure on any site of finite t yp e. Moreo v er, the singular endofunctor Sing ∗ [4] constructed by means of the standar d cosimplicial S -sc heme ∆ • A 1 with terms ∆ n A 1 = S × Spec ( Z ) Sp ec( Z [ x 0 , . . . , x n ] /x 0 + · · · + x i − 1) is strict symmetric monoidal, b ecause it commutes with limits a nd colimits. Th us the lemma fo llows b y using the iterated construction Ex ∞ ◦ (Ex ∞ ◦ Sing ∗ ) ω ◦ Ex ∞ as the fibrant replacemen t functor [4, Lemma 3 .2 .6]. In this definition ω denotes the cardinalit y of the natura l n um b ers. Corollary 2.3: The motivic symmetric sp e ctrum F (Σ ∞ B G m + ) obtaine d by applying the functor F levelwise to Σ ∞ B G m + is a c ommutative monoid. Pr o of. The assertion follows immediately by com bining Lemma 2.2 and the fact that G m is a comm utativ e g roup sc heme. 7 Corollary 2.4: Ther e exists a motivic Bott map b etwe en p ointe d motivic sp ac es b : S 4 , 2 → F ( B G m + ∧ S 2 , 1 ) that r epr esents the differ enc e map c β − i β in the p oin te d motivic unstable homotopy c ate gory. The map b is c entr al i n the sense that the diagr am F ( B G m + ) n ∧ S 4 , 2 id ∧ b / / t wist ∼ =   F ( B G m + ) n ∧ F ( B G m + ) 1 µ n, 1 / / F ( B G m + ) n +1 χ n, 1   S 4 , 2 ∧ F ( B G m + ) n b ∧ id / / F ( B G m + ) 1 ∧ F ( B G m + ) n µ 1 ,n / / F ( B G m + ) 1+ n c ommutes. Her e F ( B G m + ) k is s hort for F ( B G m + ∧ S 2 k, k ) and χ n, 1 denotes the cyclic p ermutation (1 , 2 , . . . , n, n + 1) . Pr o of. This is immediate from Corollary 2.3 and the comm utativit y of G m . With these preliminary r esults in hand w e are ready to construct a strict ring mo del for K -theory . In the follow ing w e shall adopt constructions for symmetric sp ectra giv en in Sc h we de’s manus cript [10] to the setting of motivic symmetric spectra. Let Ω 2 n,n denote the righ t a dj o in t of the susp ension functor − ∧ S 2 n,n on p oin ted motivic spaces. Define KGL β to b e the motivic symmetric sp ectrum with constituen t spaces K GL β n = Ω 4 n, 2 n F ( B G m + ∧ S 4 n, 2 n ) . The g r o up Σ n acts on S 4 n, 2 n and therefore also on F ( B G m + ∧ S 4 n, 2 n ) via restriction along the diagonal em b edding ∆ n : Σ n → Σ 2 n defined f or 1 ≤ j ≤ 2 and 1 ≤ i ≤ n by setting ∆ n ( σ )( j + 2( i − 1)) = j + 2( σ ( i ) − 1) . No w the Σ n -action o n the (4 n, 2 n )-lo op space KGL β n is defined b y conjug a tion. That is, for elemen t s σ ∈ Σ n and φ ∈ K GL β n define σ · φ ( − ) = σ ( φ ( σ − 1 ( − ))) . 8 In this definition, taking sections is implicit in the notat io n. Let µ m,n : F ( B G m + ∧ S 2 m,m ) ∧ F ( B G m + ∧ S 2 n,n ) → F ( B G m + ∧ S 2( m + n ) ,m + n ) denote the maps comprising the multiplic ativ e part o f the monoid structure on F (Σ ∞ B G m + ) . Define the m ultiplication map K GL β m ∧ K GL β n → KGL β m + n (3) b y f ∧ g 7→ µ 2 m, 2 n ◦ ( f ∧ g ) . The m ultiplicatio n map (3) is strictly associative on a ccoun t of the strict asso ciativity of the smash pro duct and the m ultiplicative structure on F (Σ ∞ B G m + ). Moreo ver, (3) is Σ m × Σ n -equiv arian t due to the equiv ariance of the multiplicativ e structure on F (Σ ∞ B G m + ) and the compatibilit y relatio n ∆ m ( σ ) × ∆ n ( σ ′ ) = ∆ m + n ( σ × σ ′ ) for the diagonal em b eddings ∆ k : Σ k → Σ 2 k (in our cases of interes t k = m, n, m + n ). Let F ( B G m + ∧ S 2 n,n ) → KGL β n = Ω 4 n, 2 n F ( B G m + ∧ S 4 n, 2 n ) b e the adjoint of the comp osite map of F ( B G m + ∧ S 2 n,n ) ∧ S 4 n, 2 n id ∧ b ∧ n − − − − → F ( B G m + ∧ S 2 n,n ) ∧ F ( B G m + ∧ S 2 n,n ) µ n,n − − → F ( B G m + ∧ S 4 n, 2 n ) and F ( B G m + ∧ S 4 n, 2 n ) → F ( B G m + ∧ S 4 n, 2 n ) giv en b y t he p erm utation σ ∈ Σ 2 n defined by σ ( i ) = ( 1 + 2( i − 1) 1 ≤ i ≤ n 2 + 2( k − 1) i = n + k , 1 ≤ k ≤ n. 9 These maps assem ble into a map o f motivic symmetric ring sp ectra F (Σ ∞ B G m + ) → KGL β . In fact the structure maps K GL β n ∧ S 2 , 1 → K GL β n +1 and the unit map of K GL β are obtained from the ab ov e and the unit map of F (Σ ∞ B G m + ). Lemma 2.5: The motivic symmetric sp e ctrum KGL β is a c ommutative monoid. Pr o of. The equation for commutativit y µ 2 m, 2 n ◦ ( f ∧ g ) = µ 2 n, 2 m ◦ ( g ∧ f ) holds b ecause G m is a comm utativ e g roup sc heme. 3 The homotop y t yp e In this section w e finish the pro of of our main result and elab o rate further o n some closely related results. First w e prepare for the comparison o f KGL β with the homotop y colimit o f the Bott t o wer introduced in [11]. Let sh ( − ) denote the shifted motivic symmetric sp ectrum functor defined by sh ( E ) n = E 1+ n . Its structure maps are induced from the ones for E b y reindexing. The Σ n -action on the n th term of sh ( E ) is determined b y the injection (1 × − ) : Σ n → Σ 1+ n giv en b y (1 × σ )( i ) = ( 1 i = 1 σ ( i − 1) + 1 i 6 = 1 . F or our purp oses the main a pplicatio n of the shift functor is to introduce the notion o f a semistable motivic symmetric sp ectrum. There exists a natural map φ ( E ) : S 2 , 1 ∧ E → sh ( E ) . (4) 10 In lev el n it is defined as the comp osite map S 2 , 1 ∧ E n ∼ = − → E n ∧ S 2 , 1 → E n +1 → E 1+ n of the t wist isomorphism, the n th structure ma p of E and the cyclic p erm uta t io n χ n, 1 = (1 , 2 , . . . , n, n + 1) . Using only the structure maps of E would not giv e a map of motivic symmetric sp ectra. The map (4) is not a stable w eak equiv a lence in g eneral. Definition 3.1: A motivic symmetric sp e ctrum E is c al le d semistable if (4) is a stable we ak e quivalenc e of underlying (non-symm etric) motivic sp e ctr a. Prop osition 3.2: L et E b e a motivic symmetric sp e ctrum such that for every n and every p ermutation σ ∈ Σ n with sign sign( σ ) = 1 the action of σ on E n c oincides with the identity in the p ointe d motivic uns tabl e homotopy c ate gory. Then E is semistable. Pr o of. W e may a ssume E is lev elwise fibrant. Then the standard natur a l stabilization construction Q giv es a stably fibrant replacemen t of E . Recall t hat in lev el n Q ( E ) n = colim k ( E n → Ω 2 , 1 E n +1 → · · · → Ω 2 k, k E n + k → · · · ) , (5) where the colimit is take n ov er the structure maps. It suffices to sho w that Q ( φ ( E )) is a lev elwise w eak equiv alence. The assumption on E implies the comp o site map Ω 2 k, k F ( S 2 , 1 ∧ E n + k ) Ω 2 k,k F ( φ ( E ) n ) − − − − − − − − → Ω 2 k, k F ( sh ( E ) n + k ) can − − → Ω 2 k + 4 , k +2 F ( S 2 , 1 ∧ sh ( E ) n + k +1 ) coincides with the canonical map Ω 2 k, k F ( S 2 , 1 ∧ E n + k ) can − − → Ω 2 k + 4 , k +2 F ( S 2 , 1 ∧ E n + k +2 ) in the p o in ted motivic unstable homoto p y category . The canonical maps denoted in the ab ov e b y ‘can ’ app ear implicitly in (5). Th us φ ( E ) induces a w eak equiv alence on colimits Q ( F ( S 2 , 1 ∧ E )) n → Q ( F ( sh ( E ))) n for eve ry n . 11 Example 3.3: The motivic symm etric sp e ctrum Σ ∞ B G m + is semistable. This fol lows fr om Pr op osition 3.2 b e c ause the even p ermutations ar e homotopic to the iden tity map on the motivic sph er es. F or the same r e ason, the m otivic symmetric sp e ctrum K GL β is semistable. Theorem 3.4: L et E b e a semistable motivic symmetric s p e ctrum, and l e t U denote the right Quil len functor to motivic sp e ctr a that for gets the symmetric gr oup action s . Then the value of the total right derive d functor of U at E is U ( E ) . Pr o of. W e may assume E is cofibran t and lev elwise fibrant. Let R ∞ E denote the colimit of the sequence E φ ( E ) ⋆ − − − → Ω 2 , 1 ( sh ( E )) Ω 2 , 1 ( sh ( φ ( E ) ⋆ )) − − − − − − − − − → · · · in the catego ry of motivic symmetric sp ectra. Here φ ( E ) ⋆ is the adjoin t of the map φ ( E ) defined in (4). By assumption U ( φ ( E )) is a stable w eak equiv alence of motivic sp ectra. Since U commu tes with the functors S 2 , 1 ∧ − and Ω 2 , 1 forming a Quillen equiv alence o n motivic sp ectra, the deriv ed adjoin t U ( E ) → Ω 2 , 1 ( U ( sh ( E ))) → Ω 2 , 1 (( U sh ( E )) fib ) is a stable w eak equiv alence. The stably fibran t r eplacemen t functor Q comm utes with Ω 2 , 1 due to the finiteness o f S 2 , 1 . Th us the map Ω 2 , 1 ( U ( sh ( E ))) → Ω 2 , 1 Q ( U ( sh ( E ))) is a stable we ak equiv alence. It follo ws that U ( φ ( E ) ⋆ ) is also a stable we ak equiv alence of motivic sp ectra. Hence the canonical map E → R ∞ E is a stable w eak equiv alence of underlying motivic sp ectra. By [9, Theorem 18] the same map is also a stable w eak equiv alence of motivic symmetric sp ectra. Since R ∞ E is stably fibran t in the category of motivic symmetric sp ectra by construction, it giv es a fibrant replacemen t of E . The res ult follo ws no w, since U ( E ) → U ( R ∞ E ) is a stable weak equiv alence. Define a : F (Σ ∞ B G m + ) → Ω 4 , 2 sh ( F (Σ ∞ B G m + )) 12 to b e the adjoin t of the map S 4 , 2 ∧ F (Σ ∞ B G m + ) → sh ( F (Σ ∞ B G m + )) . In lev el n the latter is the comp osite map S 4 , 2 ∧ F (Σ ∞ B G m + ) n b ∧ id − − → F (Σ ∞ B G m + ) 1 ∧ F (Σ ∞ B G m + ) n µ 1 ,n − − → F (Σ ∞ B G m + ) 1+ n . Corollary 3.5: In the motivic stable homotopy c ate gory ther e ex i s ts an isomorphism b etwe en the ho motopy c olimit of the di a g r am of motivic symmetric sp e ctr a F (Σ ∞ B G m + ) a − → Ω 4 , 2 sh ( F (Σ ∞ B G m + )) Ω 4 , 2 sh ( a ) − − − − − → · · · and the homotopy c olimit Σ ∞ B G m + [ β − 1 ] of the B ott tower Σ ∞ B G m + β − → Σ − 2 , − 1 Σ ∞ B G m + Σ − 2 , − 1 β − − − − − → · · · . (6) Pr o of. Due to semistabilit y of F (Σ ∞ B G m + ), established in Example 3 .3, w e may iden tify Ω 4 , 2 sh ( F (Σ ∞ B G m + )) with Ω 4 , 2 F ( S 2 , 1 ∧ F (Σ ∞ B G m + )) and th us with Ω 2 , 1 F (Σ ∞ B G m + ) up to stable w eak equiv alence. The result follows since a lifts the multiplication by the Bott elemen t map (b y construction). Theorem 3.6: The motivic symm e tric sp e ctrum K GL β has the homotopy typ e of the Bott inverte d m otivic sp e ctrum Σ ∞ B G m + [ β − 1 ] . Pr o of. Corollar y 3.5 iden tifies the Bott in verted motivic sp ectrum Σ ∞ B G m + [ β − 1 ] with the homot op y colimit of the diagram F (Σ ∞ B G m + ) a − → Ω 4 , 2 sh ( F (Σ ∞ B G m + )) Ω 4 , 2 sh ( a ) − − − − − → · · · . (7) Since the lo op and shift functors app earing in (7) preserv e semistabilit y , it f ollo ws that the terms a re semistable. Next we shall identify the homotopy colimit of (7) with K GL β . In effect, note that lea ving the symmetric groups actions aside, K GL β is the diag onal of the diagram of mot ivic symmetric sp ectra in (7). Example 3.3 and Theorem 3.4 sho w t hat the v alue of the right derive d functor of U a t K GL β is giv en by forgetting the group actions on K GL β . Hence there exists an abstract isomorphism b et w een K GL β and Σ ∞ B G m + [ β − 1 ] in the motivic stable homotopy category . 13 Next w e discuss in bro a d strok es a mo t ivic functor mo del for K -theory . The re exists a strict symmetric monoida l functor fro m motivic symmetric sp ectra to motivic functors MSS → MF for t he base sc heme S , whic h is part o f a Quillen equiv alence prov en in [1 ]. Therefore the image KGL β of the motivic symmetric sp ectrum mo del for K - theory yields a strict motivic functor model for K -theory . In order to make this mo del more explicit, o ne could try to construct it en tirely within the framework of motivic functors by starting out with the mot ivic f unctor B G m + ∧ − and g o through constructions reminiscen t of the ones for motivic symmetric sp ectra in this pap er. Remark 3.7: Applying the ar guments in this p ap er to Kan ’s (lax s ymmetric monoidal) fibr ant r eplac ement functor for simp licial sets and the Bott elemen t in π 2 Σ ∞ B C × yields a c ommutative symmetric ring sp e ctrum with the homo topy typ e of top olo gic al unitary K - the ory. Mor e gener al ly, for A an ab elian c omp act Lie g r oup, the same ar gument appl i e s to the Bott inverte d m o del for A -e quivariant unitary top olo gic al K -the ory in [12]. We le ave further d e tails to the inter este d r e ader. 4 Multiplic ati ve stru c ture Theorem 4.1: The multiplic ative structur es on K GL β and K GL c oincide in the motivic stable homotopy c ate gory. Pr o of. The pro of pro ceeds b y show ing there is a comm utative dia g ram of monoids K GL β / / Σ ∞ B G m + [ β − 1 ]   Σ ∞ B G m + O O 6 6 n n n n n n n n n n n n / / K GL (8) in the motivic stable homoto p y categor y . On the lev el of bigraded homology theories there is a comm utative diagram: (Σ ∞ B G m + [ β − 1 ]) ∗ , ∗ ( )   (Σ ∞ B G m + ) ∗ , ∗ ( ) 4 4 j j j j j j j j j j j j j j j j / / K GL ∗ , ∗ ( ) 14 This diagram lifts uniquely to a commu tativ e diagram o f monoids in the motivic stable homotop y catego ry , as a sserted b y the right hand side of (8), since for K -theory there exist no non trivial phan tom maps according to [5 , Remark 9.8 (ii), (iv)]. On the left hand side o f (8), recall Σ ∞ B G m + → KGL β is a map of motivic symmetric ring sp ectra. F or the discussion of KGL β → Σ ∞ B G m + [ β − 1 ] we shall use the following mo del fo r the homotopy colimit. Let E b e the stably fibra n t replacemen t of Σ ∞ B G m + obtained by fir st applying the functor F lev elwise and second the stabilization functor Q . No w define E [ β − 1 ] as the diagonal sp ectrum of the naturally induced seque nce E → Σ − 2 , − 1 E → · · · lifting the Bot t tow er (6). Here Σ − 2 , − 1 E is realized as a shift, so that E [ β − 1 ] n = E 0 and its structure maps ar e giv en by m ultiplication with the Bott elemen t. In lev el n the map K GL β → E [ β − 1 ] is the canonical map Ω 4 n, 2 n F ( B G m + ∧ S 4 n, 2 n ) → Ω 2 ∞ , ∞ F ( B G m + ∧ S 2 ∞ , ∞ ) = E 0 . When n = 0 the latter map corresp onds via adjo in tness to the diagonal map in (8 ) . The eviden t monoid structure on B G m + induces a monoid structure on E 0 and hence a naive m ultiplication on E [ β − 1 ] giv en b y E [ β − 1 ] m ∧ E [ β − 1 ] n = E 0 ∧ E 0 → E 0 = E [ β − 1 ] m + n . 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In Pr o c e e di n gs of the International Congr ess of Mathema ticia ns, V ol. I (Berlin, 1998) , V ol. I: 579–604 (electronic), 19 9 8. Institut f ¨ ur Mathematik, Univ ersit¨ at Osnabr ¨ uck, German y . e-mail: oro endig@math.uos.de F akult¨ at f ¨ ur Mathematik, Unive rsit¨ at Regensburg, German y . e-mail: Markus.Spitzw ec k@mathematik.uni-regensburg.de Departmen t of Mathematics, Univ ersity o f Oslo, Norw a y . e-mail: paularne@math.uio.no 16