On the Orientability of the Slice Filtration

ON THE ORIENT ABILITY OF THE SLICE FIL TRA T ION P ABLO PELAEZ Abstract. Let X be a No etherian separated scheme of finite Krul l dimension. W e sho w that the lay ers of the slice filtration in the motivic stable homotop y category S H are strict modules ov er V o ev odsky’s algebraic cobordi s m spec- trum. W e als o show that the zero sl i ce of an y comm utativ e ring spectrum in S H is an oriente d ring s pectrum in the sense of Morel, and that its associated formal group law is additive. As a conseque nce, we get t hat wi th rational coefficients the slices are i n fact motives in the s ense of Cisinski -D´ eglise [1], and ha ve transfers if the base sc heme is excellen t. This pr ov es a conjecture of V o evodsky [24, conjecture 11]. 2000 Mathematics Subje ct Classific ation. Pri mary 14, 55. Key wor ds and phr ases. Algebraic C ob ordism, K -theory , Mixed M otiv es, Or ien ted Cohomol- ogy Theories, Rigid Homotopy Groups, Slice Filtration, T ransfers . 1 2 P ABLO PELAEZ 1. Introduction Let X b e a No etheria n separa ted scheme of finite Krull dimension, a nd M X be the categor y of p ointed simplicial presheav es in the smo oth Nisnevich site S m X ov er X equipped with the Quillen mo del struc tur e [16] intro duced b y More l-V o evodsky [7]. W e define T in M X as the p ointed simplicial pr esheaf repr esented by S 1 ∧ G m , where G m is the multiplicativ e group A 1 X − { 0 } p ointed by 1 , and S 1 denotes the simplicial circle. Let S pt ( M X ) denote the category o f symmetric T - spec tra on M X equipp e d with Jar dine’s motivic model structure [4]. The ho mo topy categor y of S pt ( M X ) is a triangulated categor y which w ill b e denoted by S H . Given an integer q ∈ Z , we consider the follo wing family of symmetric T -sp ectra C q ef f = { F n ( S r ∧ G s m ∧ U + ) | n, r , s ≥ 0; s − n ≥ q ; U ∈ S m X } where F n is the left adjoint to the n -ev a luation functor ev n : S pt ( M X ) → M X V o evodsky [2 4] defines the slice filtra tion as the following family of triangula ted sub c ategories of S H · · · ⊆ Σ q +1 T S H ef f ⊆ Σ q T S H ef f ⊆ Σ q − 1 T S H ef f ⊆ · · · where Σ q T S H ef f is the smallest full tria ngulated sub category of S H whic h contains C q ef f and is clos ed under a rbitrary co pro ducts. It follows fro m the work of Neeman [8], [9] that the inclusio n i q : Σ q T S H ef f → S H has a right adjoint r q : S H → Σ q T S H ef f , a nd that the following functors f q : S H → S H s q : S H → S H are triang ulated, whe r e f q is defined as the compo sition i q ◦ r q , a nd s q is c ha racter- ized by the fact that for every E ∈ S pt ( M X ), we hav e the following distinguis hed triangle in S H f q +1 E ρ E q / / f q E π E q / / s q E / / Σ 1 , 0 T f q +1 E W e will refer to f q E as the ( q − 1 )-connective cov er of E , and to s q E as the q -slice of E . It fo llows directly fro m the cons truction that the q -slice of E is right orthog onal with r espec t to Σ q +1 T S H ef f , i.e. Hom S H ( K, s q E ) = 0 for every K in Σ q +1 T S H ef f . ON THE ORIENT ABILITY OF THE SLICE FIL TRA TION 3 2. Strict M GL -modules In this section we will show that all the slices hav e a canonical structure of strict mo dules in S pt ( M X ) over V o evo dsky’s alg ebraic cob ordis m spectr um. Let A b e a cofibrant ring spectrum with unit in S pt ( M X ), and A -mo d b e the category of left A -mo dules in S pt ( M X ). The work of Jardine [4, prop osition 4.19] and Hov ey [3, coro llary 2.2] implies that the adjunction ( A ∧ − , U, ϕ ) : S pt ( M X ) → A -mo d induces a Quillen model structure S pt A ( M X ) in A -mod, this means that a map f : M → N in S pt A ( M X ) is a weak equiv alence o r a fibr a tion if and o nly if U f is a weak equiv alence or a fibra tion in S pt ( M X ). It is ea sy to see tha t the homotopy ca teg ory S H A of S pt A ( M X ) is a triangulated category [13, prop ositio n 3.5.3]. Definition 2. 1. L et E b e a sp e ct rum in S H . We say that E is effectiv e if E b elongs t o the triangulate d c ate gory Σ 0 T S H ef f define d ab ove. Theorem 2.2. Le t A b e an effe ctive c ofibr ant ring sp e ctrum with unit u A : 1 → A in S pt ( M X ) . If s 0 ( u A ) is an isomorphism in S H , t hen for every q ∈ Z the functor s q : S H → S H factors (up to a c anonic al isomorphism) t hr ough S H A S H s q / / ˜ s q # # F F F F S H S H A U R A O O wher e R A denotes a fibr ant r eplac ement functor in S pt A ( M X ) . Pr o of. This follows directly fro m [13, theor em 3.6 .20 and lemma 3.6 .21] or [11, theorem 2 .1(vi)].  The fo llowing pr opo sition was pr ov ed by V o evodsky [24, p. 10 section 3.4] and Spitzw eck [19, co r ollaries 3.2 a nd 3 .3]. Prop osition 2. 3 (V o evo dsky) . L et M GL denote V o evo dsky’s algebr aic c ob or dism sp e ctrum [23] . We have t hat M GL is effe ctive and its unit map u M GL : 1 → M GL induc es an isomorphism on the zer o slic es in S H s 0 ( u M GL ) : s 0 1 ∼ = / / s 0 M GL Now w e ca n state the main r esult of this se c tio n. Theorem 2.4. Le t q ∈ Z denote an arbitr ary int e ger and E denote an arbi tr ary symmetric T -s p e ctrum in S p t ( M X ) . We have that the q -slic e s q E of E is e quipp e d with a c anonic al stru ctur e of M GL -mo dule in S pt ( M X ) . This implies t hat over any b ase scheme, t he s lic es ar e always oriente d c ohomolo gy the ories in t he sense of D´ eglise [2, example 2.12 (2)] . Pr o of. This follows immediately fr om theorem 2.2 and pro po sition 2.3.  4 P ABLO PELAEZ Remark 2.5. One of the inter esting c onse quenc es of the or em 2. 4 is t he fact tha t over any No etherian sep ar ate d b ase scheme of finite Krul l dimension, onc e we p ass to the slic es it is p ossible to apply al l t he formalism develop e d by D´ eglise in [2] , e.g. Chern classes and the Gysin triangle. ON THE ORIENT ABILITY OF THE SLICE FIL TRA TION 5 3. Oriented R ing Spectra an d Formal G r oup La ws In this section we will show that given a c o mm utative ring s pectr um E in S H , its zero slice s 0 E is an oriented r ing sp ectrum (in the sense of More l [21 , definition 3.1]) with additiv e formal g roup law in S H . T o simplify the notation we will denote by P n the trivial pro jective bundle of rank n ov er our bas e scheme X . Definition 3.1. L et E b e a c ommutative ring sp e ct rum in S H with u nit u E : 1 → E We say that E is an oriented ring sp ectrum if ther e exists an element x E in Hom S H ( F 0 ( P ∞ ) , S 1 ∧ G m ∧ E ) , wher e P ∞ is the c olimit of the diagr am P 1 → P 2 → · · · → P n → · · · given by the inclusions of the r esp e ct ive hyp erplanes a t infinity, such that x E pul ls b ack to the fol lowing c omp osition F 0 ( P 1 ) ∼ = F 0 ( S 1 ∧ G m ) id ∧ u E / / F 0 ( S 1 ∧ G m ) ∧ E ∼ = S 1 ∧ G m ∧ E The fo llowing pro po sition is classical. Prop osition 3.2 (cf. [14 ], [15], [21]) . L et ( E , x E ) b e an oriente d ring sp e ct ru m in S H , and let m : P ∞ × P ∞ → P ∞ b e the map induc e d by the c orr esp onding Se gr e emb e ddings. The pul lb ack of x E along m , is a formal gr oup law F E F E = X i + j ≥ 1 c ij x i y j wher e the c o efficients c ij ar e elements in the ab elian gr oup Hom S H ( F 0 ( S i + j − 1 ∧ G i + j − 1 m ) , E ) and x (r esp. y ) is the pul lb ack of x E along the pr oje ction in the first factor p 1 : P ∞ × P ∞ → P ∞ (r esp. se c ond factor). Lemma 3.3. L et E b e a c ommu t ative ring sp e ctrum in S H with u nit u E : 1 → E , then its zer o slic e s 0 E is also a c ommu t ative ring sp e ctrum in S H , and the induc e d map s 0 ( u E ) : s 0 1 → s 0 E is a map of ring sp e ctra in S H . Pr o of. The fact that s 0 E is a ring s pectr um in S H follows from [1 3, theor em 3.6.13 ], on the o ther hand the naturality of the pa ir ings constructed in [13, theorem 3 .6.9] implies that s 0 E is also commut ative in S H and that s 0 ( u E ) is a map o f ring sp ectra in S H .  Lemma 3.4. The natur al map π M GL 0 : M GL ∼ = f 0 M GL → s 0 M GL is a map of ring sp e ctra in S H . Pr o of. By pr opo sition 2.3 we hav e that M GL is naturally iso mo rphic to f 0 M GL in S H . On the other hand, theo rem 3.6.10(3 ) in [13] implies that π M GL 0 is a map of ring sp ectra in S H .  6 P ABLO PELAEZ Theorem 3.5. L et E b e a c ommutative ring sp e ct rum in S H . Then its zer o slic e s 0 E is an oriente d ring sp e ctrum in the sense of Mor el and it is e quipp e d with a c anonic al orientation given by the fol lowing c omp osition M GL π M GL 0 / / s 0 M GL ( s 0 ( u M GL )) − 1 / / s 0 1 s 0 ( u E ) / / s 0 E F ur t hermor e, the asso ciate d formal gr oup law F s 0 E of s 0 E is additive. Pr o of. The universalit y of M GL (cf. [21, theor em 4.3], [10, theor em 2.7] and [20, prop osition A.2]) implies that in order to show that the map defined ab ov e g ives an orientation for s 0 E , it is enough to see that a ll the maps are in fact maps of ring sp ectra in S H . B ut this follows directly fr om prop ositio n 2.3 together with lemmas 3.3 and 3.4. On the o ther hand, the fo rmal gr oup law of s 0 E F s 0 E = X i + j ≥ 1 c ij x i y j has co efficients c ij which by construction are in the ab elian g roup Hom S H ( F 0 ( S i + j − 1 ∧ G i + j − 1 m ) , s 0 E ) How ever, if i + j > 1 then F 0 ( S i + j − 1 ∧ G i + j − 1 m ) is automatically in Σ 1 T S H ef f ; hence Hom S H ( F 0 ( S i + j − 1 ∧ G i + j − 1 m ) , s 0 E ) = 0 since s 0 E is right orthogonal with re s pect to Σ 1 T S H ef f . Therefore, the formal gr o up law of s 0 E F s 0 E = X i + j ≥ 1 c ij x i y j = x + y is additive, as we wan ted.  ON THE ORIENT ABILITY OF THE SLICE FIL TRA TION 7 4. Applica tion s In this section we will show that with r a tional co efficients all the slices s q ( E ) ⊗ 1 Q are in a natural w ay motives in the sense o f Cisinski-D´ eglise [1]; as a conseq ue nce w e will get that over an excellent ba se s c heme the presheaves of ratio nal rigid homotop y groups have tra ns fers, this proves a c onjecture of V o evo dsky [24, conjecture 11]. 4.1. Slices and the Cisins ki-D´ eglise category of m o tiv es. Let H B ,X = K GL (0) X ∈ S pt ( M X ) denote the Beilinso n motivic cohomolog y spe c trum constructed by Riou in [17]. The work of Cisinski-D´ eglise shows in particular tha t H B ,X is a commutativ e cofi- brant ring spectr um in S pt ( M X ) (cf. [1 , corollar y 13.2.6]); and tha t the homo top y category o f H B ,X -mo dules S H H B ,X is na turally equiv a le n t to the Cisinski-D´ eglise category of motives D M B ,X (cf. [1, theorem 13.2.9]). Theorem 4.1. If we c onsider r ational c o efficients, the zer o slic e of t he spher e sp e c- trum s 0 ( 1 ) ⊗ 1 Q is e qu ipp e d with a u nique st ructur e of H B ,X -algebr a in S pt ( M X ) . In p articular, ther e exists a unique map η s 0 ( 1 ) ⊗ 1 Q of ring sp e ctr a in S H such that the fol lowing diagr am is c ommut ative M GL ⊗ 1 Q π M GL 0 ⊗ id / / η H   s 0 ( M GL ) ⊗ 1 Q ( s 0 ( u M GL )) − 1 ⊗ id / / s 0 ( 1 ) ⊗ 1 Q H B ,X η s 0 ( 1 ) ⊗ 1 Q 1 1 d d d d d d d d d d d d d d d d d d d d d d d Pr o of. By theorem 3.5 we hav e that s 0 ( 1 ) is or ien table. Therefore, the r e s ult follows directly fro m coro llary 13.2.15 (Ri),(Rii),(Riii) in [1 ].  Theorem 4.2. Le t q ∈ Z denote an arbitr ary int e ger and E denote an arbi tr ary symmetric T -sp e ctrum in S pt ( M X ) . We have that the q -slic e of E with r ational c o efficients s q ( E ) ⊗ 1 Q is e qu ipp e d with a c anonic al structur e of H B ,X -mo dule in S pt ( M X ) . This implie s that over any b ase sch eme, the slic es with r ational c o effi- cients s q ( E ) ⊗ 1 Q ar e motives in the sense of Cisinski-D´ eglise. Pr o of. Using coro llary 13.2.15 (i),(iv),(v) in [1], we get that it is enough to s how that s q ( E ) ⊗ 1 Q is a H B ,X -mo dule in S H . On the other hand, by theorem 4 .1 we just need to chec k that s q ( E ) ⊗ 1 Q is a s 0 ( 1 ) ⊗ 1 Q -mo dule in S H . Finally , this follows dir ectly fr om theorem 3.6.14 (6) in [1 3].  4.2. Rational R igid Ho motop y groups. Given a symmetric T -sp ectrum E in S pt ( M X ), V o evodsk y defines the pre sheav es of rigid homoto p y gro ups π r ig p,q ( E ) on S m X as follows: π r ig p,q ( E ) : S m X / / Abelia n Groups U  / / Hom S H ( F 0 ( S p ∧ G q m ∧ U + ) , s q E ) Conjecture 1 1 in [2 4] claims that these presheaves hav e transfer s. Theorem 4.3 . L et p, q ∈ Z denote arbitr ary inte gers and E denote an arbitr ary symmetric T -sp e ctrum in S pt ( M X ) . F urthermor e, assume that t he b ase scheme 8 P ABLO PELAEZ X is exc el lent . Th en t he pr eshe aves of rigid homotopy gr oups of E with r ational c o efficients π r ig p,q ( E ) ⊗ Q have tr ansfers. Pr o of. Clearly , it suffices to s how that s q ( E ) ⊗ 1 Q has transfers . Now, theor em 4 .2 implies tha t s q ( E ) ⊗ 1 Q is in D M B ,X . Since we ar e assuming that X is excellent, theorem 1 5.1.2 in [1] implies that DM B ,X is naturally equiv a len t to the catego ry of motives D M qfh ,X constructed using q f h -sheav e s with rational coefficients. Th us, the result follows from theorem 3.3.8 in [22] (see a lso prop osition 9 .5.5 in [1]) whic h implies than every q f h -sheaf is ca nonically eq uipped with transfer s.  Remark 4.4. The or em 4.3 was pr ove d using the functoriality of the slic e fi ltr ation in [12, theorem 4.4] for schemes define d o ver a fi eld of char acteristic zer o; on the other hand, if t he b ase scheme X is smo oth over a p erfe ct field k , t hen the or em 4.3 holds even with int e gr al c o efficients (cf. [1 3, theorem 3 .6.22] ). Both pr o ofs r ely on the c omput ation of L evine [5] and V o evo dsky [25] for the zer o slic e of the spher e sp e ctrum, as wel l as on the work of R¨ ondigs-Østv æ r [18] . The analo gue of t his question for the c ate gory of S 1 -sp e ctr a is st udie d by L evine in [6] . ON THE ORIENT ABILITY OF THE SLICE FIL TRA TION 9 Ackno wledgements The a uthor would like to warmly thank F r ´ ed´ eric D´ e g lise for several useful co n- versations and suggestions, a s well as for putting in our hands the technical to ols from [2] and [1 ]; and also thank Denis-Charles Cisinski for bringing to our atten tion the argument which allow ed us to extend theorem 4.3 from geometric ally unibranch base schemes to arbitra ry excellent schemes. References [1] D.- C. Cisinski and F. D´ eglise. T riangulated categories of mixed m otiv es. pr eprint , 2009. [2] F. D´ eglise. A r ound the Gysin triangle. I I. Do c. M ath. , 13:613– 675, 2008. [3] M. Hov ey . M onoidal model categories. pr eprint , 1998. [4] J. F. Jardine. Motivic symmetric sp ectra. Do c. Math. , 5:445–553 (elect ronic), 2000. [5] M. Levine. The homotop y coniv eau tow er. J. T op ol. , 1(1):217 –267, 2008. [6] M. Levine. Sl ices and transfers. Do c. Math. , Extra volume: Andrei A. Suslin’s Si xtieth Birthda y:393–443, 2010. [7] F. Morel and V. V oevodsky . A 1 -homotop y theory of schemes. Inst. Hautes ´ Etudes Sci. Publ. Math. , (90):45–143 (2001), 1999. [8] A. Neeman. The Grothendiec k duality theorem via Bousfield’s tec hniques and Brown repre- sen tabilit y . J. A mer. Math. So c. , 9(1):205–236, 1996. [9] A. Neeman. T riangulate d c ate gories , volume 148 of Annals of Mathematics Studies . Princeton Unive rsity Press, Princeton, NJ, 2001. [10] I. Panin, K. Pi meno v, and O. R¨ ondigs. A univ ersality theorem f or Vo ev o dsky’s algebraic cobordism s pectrum. Homolo gy, Homotopy Appl. , 10(2):211–226, 2008. [11] P . Pelaez. Mixed motive s and the sl ice filtration. C. R. M ath. A c ad. Sci. Paris , 347(9-10):541– 544, 2009. [12] P . Pe laez. On the f unctorialit y of the slice filtration. pr eprint , 2010. [13] P . Pe laez. M ultiplicativ e prop erties of the sl ice filtration. Ast´ erisque , (335):xvi+291 pp., 2011. [14] D. Quill en. On the f or mal group l a ws of unorient ed an d complex cob ordism theory . Bul l. Am er. Math. So c. , 75:1293–1298, 1969. [15] D. Quillen. E l emen tary pr oofs of some results of cobor dism theory usi ng Steenrod operations. A dvanc es in Math. , 7:29–56 (197 1), 1971. [16] D. G. Quillen. Homotopic al algebr a . Lecture Notes in Mathematics, No. 43. Spri nger-V erl ag, Berlin, 1967. [17] J. Riou. Al gebraic K -theory , A 1 -homotop y and Riemann-Ro ch theorems. J. T op ol. , 3(2):229– 264, 2010. [18] O. R¨ ondigs and P . A. Østvær. Mo dules ov er motivic cohomology . Ad v. Math. , 219(2):68 9–727, 2008. [19] M. Spitzwec k. Relations b et ween slices and quotien ts of the algebraic cobordism sp ectrum. Homolo gy, Homotopy Appl. , 12(2):335–3 51, 2010. [20] M. Spitz we c k and P . A. Østvær. The Bott inv erted infinite pro jectiv e space is homotop y algebraic K -theory . Bul l. L ond. Math. So c. , 41(2):281–292 , 2009. [21] G. V ezzosi. Brown-Peterson spectra in stable A 1 -homotop y theory . R end. Sem. Mat. Univ. Padova , 106:47–64, 2001. [22] V. V oevodsky . Homol ogy of sc hemes. Sele cta Math. (N.S.) , 2(1):111–153, 1996. [23] V. V o evodsky . A 1 -homotop y theory . In Pr o c e edings of the International Congr ess of Mathe- maticians, Vol. I (Berlin, 1998) , num ber Extra V ol. I, pages 579–604 (electronic), 1998. [24] V. V oevodsky . Op en problems in the m otivic s table homotop y t heory . I. In Motives, p oly- lo garithms and Ho dge t he ory, Part I (Irvine, CA, 1998) , volume 3 of Int. Pr ess L e ct. Ser. , pages 3–34. In t. Press, Somerville, MA, 2002. [25] V. V o evodsky . On the zero slice of the sphere sp ectrum. T r. Mat. Inst. Steklova , 246(Algebr. Geom. Metody , Svyazi i Pri lozh.):106–115 , 2004. Universit ¨ at Duisburg-Essen, Ma thema tik, 45117 Essen, Germany E-mail addr e ss : pablo.pelae z@uni-due .de