Abstract Let k be a number field, and let A P 1 (k) be a finite set of rational points. Deligne and Goncharov have defined the motivic fundamental

Transcription

1 1 bstract Let k be a number field, and let P 1 (k) be a finite set of rational points. Deligne and Goncharov have defined the motivic fundamental group π1 mot (X, x) of X := P 1 \ with base-point x being either a k-point of X or a tangential base-point. We extend the construction of the motivic fundamental group to the setting of a smooth S-scheme p : X S with section x : S X, in case S is itself smooth over a field, X satisfies the Beilinson-Soulé vanishing conjectures and the motive of X in DM(S) Q is a mixed Tate motive. Finally, letting Gal(MT(X)) be the Tannaka group of the Tannakian category of mixed Tate motives over X, we identify π1 mot (X, x) with the kernel of the map p : Gal(MT(X)) Gal(MT(S)). 1

2 Tate motives and the fundamental group Marc Levine Dept. of Math. Northeastern University Boston, M U.S.. marc@neu.edu October 13, 2008 Contents 1 Differential graded algebras dams graded cdgas The bar construction The category of cell modules The derived category Weight filtration Bounded below modules Tor and Ext Change of ring Finiteness conditions Model structure Minimal models t-structure Connection matrices The homotopy category of connections Summary Relative theory of cdgas Definitions and model structure Path objects and the homotopy relation Indecomposables Relative minimal models Relative bar construction The author gratefully acknowledges the support of the Humboldt Foundation and support of the NSF via grant DMS

3 2.6 Base-change Connection matrices Semi-direct products Motives over a base Effective motives over a base T tr -spectra and the category of motives Tensor product in Spt S T tr(s) Motives with Q-coefficients Geometric motives Tate motives Cycle algebras Cubical complexes The cycle cdga in DM eff (S) Products and internal Hom in Sh tr Nis(S) Equi-dimensional cycles N (S)-modules and motives The contravariant motive The dual motive and cycle complexes Cell modules and Tate motives Motives and N S -modules From cycle algebras to motives The cell algebra of an S-scheme Motivic π Cosimplicial constructions The motive of a cosimplicial scheme Motivic π Simplicial constructions The comparison theorem The fundamental exact sequence Introduction In [11], P. Deligne defined the motivic fundamental group of X = P 1 \{0, 1, } over a number field k as an object in the category of systems of realizations. This is a Tannakian category over Q, which he constructed as tuples (Betti, de Rham, l-adic, crystalline), with compatibilities between them, a definition close to the one given by U. Jannsen [24]. The Betti-de Rham component is the mixed Hodge structure, defined by J. Morgan [32], on the nilpotent completion lim Q[π top (X, x)]/i N of the topological fundamental group π top N 1 (X(C), x), for all complex embeddings k C, where the base-point x is either a point in X(k) or a non-trivial tangent vector at x (P 1 \ X)(k). 3

4 . Beilinson [12, proposition 3.4] showed that for any smooth complex variety X, and for base-point x X(C), the ind-system lim Hom Q (Q[π top (X, x)]/i N, Q), N which is a Hopf algebra over Q, arises from the cohomology of a cosimplicial scheme P x (X). s pointed out by Z. Wojtkoviak [39], the Hopf algebra structure on lim (Q[π top (X, x)]/i N ) N similarly arises from operations on P x (X). These key results have many consequences. For instance, one can use P x (X) to define the mixed Hodge structure on lim Q[π top (X, x)]/i N, N cf. [18]. Even more, the cosimplicial scheme P x (X), regardless of the geometry of X, defines an ind-hopf algebra object H gm (P x (X)) in Voevodsky s triangulated category of motives DM gm (k) Q [15, chapter V]; here H gm : Sm/k op DM gm (k) is the cohomological motive functor, dual to the canonical functor M gm : Sm/k DM gm (k). If in addition X is the complement in P 1 k of a finite set of k-rational points, then H gm (P x (X)) Q is actually an ind-hopf algebra in the full triangulated subcategory DMT gm (k) of DM gm (k) Q spanned by the Tate objects Q(n). s explained in [31], if a field k satisfies the Beilinson-Soulé vanishing conjecture, that is, if the motivic cohomology H p (k, Q(q)) vanishes for p 0, q > 0, there is a t-structure defined on DMT gm (k), the heart of which is the abelian category MT(k) of mixed Tate motives over k. MT(k) is a Q-linear, abelian rigid tensor category with the structure of a functorial exact weight filtration W. Taking the associated graded object with respect to W defines a neutral fiber functor gr W, endowing MT(k) with the structure of a Tannakian category over Q. By the work of Borel [6], we know that if k is a number field, then k does satisfy the Beilinson-Soulé conjecture. Thus Beilinson s construction allows one to define the ind-hopf algebra object H 0 (H gm (P x (X)) in MT(k), if k is a number field. In [12, théorème 4.4] P. Deligne and. Goncharov show that the dual π1 mot (X, x) of H 0 (H gm (P x (X)) Q ), which is a pro-group scheme object in MT(k), yields Deligne s original motivic fundamental group upon applying the appropriate realization functors, in case x X(k) and X P 1 k is the complement of a finite set of k-points of P 1. In addition, they show that, even for a tangential base-point x, there is a pro-group scheme object π1 mot (X, x) in MT(k) which maps to Deligne s motivic fundamental group under realization, without, however, making an explicit construction of π1 mot (X, x) in this case. Using this construction as starting point, they go on to construct a motivic fundamental group for any unirational variety over the number field k, as a pro-group scheme over the larger Tannakian category of rtin-tate motives MT(k) (see [12] for details). Using recent work of Cisinski-Déglise [10], one now has available a reasonable candidate for the category of motives over a base X, at least if X is a smooth variety over a perfect field k. The resulting triangulated category DM(X) has Tate objects Z X (n) which properly compute the motivic cohomology of X (defined using Voevodsky s category DM gm (k)). In addition, if X P 1 k is an open defined over a number field k, then the observation made in [31] carries over to the full triangulated subcategory DMT(X) of the category DM(X) Q generated by the Tate objects Q X (n). Thus, assuming k is a number field, there is a heart 4

5 MT(X) DMT(X) which is a Q-linear abelian rigid tensor category, and which receives MT(k) by pull back via the structure morphism p : X Spec k. By Tannaka duality, we therefore have the Tannaka group schemes G(MT(X), gr W ) and G(MT(k), gr W ) over Q, and the functors p : MT(k) MT(X), x : MT(X) MT(k) give a canonical split short exact sequence 1 K G(MT(X), gr W ) G(MT(k), gr W ) 1, where K is defined as the kernel of p. The splitting x also defines an action of the Tannaka group G(MT(k), gr W ) on K, which lifts the Q group-scheme K to a group-scheme object K x in MT(k). In [12, section 4.19], Deligne and Goncharov use the group-scheme π1 mot (X, x) over MT(k) to define MT(X) as the category of MT(k) representations of π1 mot (X, x). In [12, section 4.22] they ask about the relationship between MT(k(P 1 )), defined as above as a subcategory of Voevodsky s category DM(k(P 1 )) Q, and lim MT(X) (this is the formulation for k = Q, X P 1 in general, one needs to use the rtin-tate motives MT). The purpose of this article is to give an answer to this question in the following form: the intrinsic definition of MT(X) mentioned above is equivalent to the category of K x -representations in MT(k), assuming P 1 \ X consists of k-rational points. We now describe our main result for X as above. Theorem 1 Let k be a number field, X(k) a finite (possibly empty) set of k-points of P 1, let X := P 1 \ and take x X(k). Then the pro-group scheme objects K x and π1 mot (X, x) are isomorphic as pro-group-schemes in MT(k). The equivalence of MT(X) with the category of K x -representations in MT(k) follows directly from this. In fact, we have a more general result. Let S be a smooth k-scheme, and let X S be a smooth S-scheme with a section x : S X. One can easily extend the construction of π1 mot (X, x) to this setting, if we assume that X satisfies the Beilinson-Soulé vanishing conjectures, and, in addition, that the motive of X in DM(S) Q is in the Tate subcategory DMT(S) (see definition for details). Note that S also satisfies the Beilinson-Soulé vanishing conjectures, as the section x identifies H (S, Q( )) with a summand of H (X, Q( )). Replacing MT(k) with MT(S), we have the split exact sequence as above 1 K G(MT(X), gr W ) G(MT(S), gr W ) 1. defining the pro-group scheme object K x of MT(S). Our main result in this more general setting is Theorem 2 Suppose X satisfies the Beilinson-Soulé vanishing conjectures, and suppose that the motive of X in DM(S) Q is in the Tate subcategory DMT(S) of DM(S) Q. Then the progroup scheme objects K x and π1 mot (X, x) are isomorphic as pro-group-schemes in MT(S). We now explain the ideas that go into the proof. In [2] S. Bloch and I. Kriz construct a group-scheme G BK (k) over Q, by applying the bar construction to the cycle algebra N k := 5 p x p x

6 Q r 1 N k (r). The rth component N k (r) of N k is a shifted, alternating version of Bloch cycle complex, N m k (r) = z r (k, 2r m) lt ; the alternation makes the product on N k strictly graded-commutative. The additional grading r is the dams grading. The reduced bar construction gives us the dams graded Hopf algebra H 0 ( B(N k )) and G BK (k) is the pro group scheme Spec H 0 ( B(N k )). Bloch-Kriz define the category of Bloch-Kriz mixed Tate motives over k, MT BK (k), as the finite dimensional graded representations of G BK (k) in Q-vector spaces. In [26], I. Kriz and P. May consider, for an dams graded commutative differential graded algebra (cdga) = Q id r 1 (r) over Q, the bounded derived category D f of dams graded dg modules. D f admits a functorial exact weight filtration, arising from the dams grading; in case is cohomologically connected, D f has a t-structure, defined by pulling back the usual t-structure on D f Q = n D b (Q) via the functor M M L Q from Df to Df Q. In particular, they define the heart H f. Next, assuming cohomologically connected, they construct an exact functor ρ : D b( co-rep f Q (H0 ( B())) ) D f where co-rep f Q (H0 ( B())) is the category of graded co-representations of H 0 ( B()) in finitedimension Q-vector spaces. Furthermore, they show that ρ identifies the categories H f and co-rep f Q (H0 ( B())) (although ρ is not in general an equivalence). For those who prefer group-schemes to Hopf algebras, let G := Spec H 0 ( B()). Then G is a pro-affine group scheme over Q with G m action, and co-rep f Q (H0 ( B())) is equivalent to the category of graded representions of G in finite dimensional Q-vector spaces. Taking = N k, and noting that the Beilinson-Soulé vanishing conjectures for k are equivalent to the cohomological connectedness of, this gives an equivalence of the heart H f N k with the Bloch-Kriz mixed Tate motives MT BK (k). M. Spitzweck [37] (see [29, section 5] for a detailed account) defines an equivalence θ k : D f N k DMT(k) DM gm (k) Q for k an arbitrary field. In addition, under the assumption that k satisfies the Beilinson- Soulé conjectures, or equivalently, that N k is cohomologically connected, θ k restricts to an equivalence θ k : H f N k MT(k). From the discussion above, this gives an equivalence of co-rep f Q (H0 ( B(N k ))) with MT(k), and in fact identifies G BK (k) G m as the Tannaka group of (MT(k), gr W ). Our first task is to extend this picture from k to X. To this aim, one defines the cycle algebra N (X) by replacing k with X in the definition of N k and modifying the construction further by using complexes of cycles which are equi-dimensional over X. This yields an dams graded cdga over Q together with a map of dams graded cdgas p : N (k) N (X) arising from the structure morphism p : X Spec k. Essentially the same construction as for k gives an equivalence θ X : D f N (X) DMT(X) DM(X) Q ( ) 6

7 and if X satisfies the Beilinson-Soulé vanishing conjectures, θ X restricts to an equivalence H f N (X) MT(X). Defining the Q pro-group scheme G BK(X) as above, G BK (X) := G N (X) = Spec (H 0 ( B(N X ))), we also have the equivalence of MT(X) with the graded representations of G BK (X) in finite dimensional Q-vector spaces, giving the identification of G BK (X) G m with the Tannaka group of (MT(X), gr W ), and identifying p : G(MT(X), gr W ) G(MT(k), gr W ) with the map p id : G BK (X) G m G BK (k) G m induced from p : N (k) N (X). k-point x of X gives an augmentation ɛ x : N (X) N (k). We discuss the general theory of augmented cdgas in section 2, leading to the relative bar construction HN 0 ( B N (, ɛ)) of a cdg N algebra with augmentation ɛ : N, as an ind-hopf algebra in H f N. Let G /N (ɛ) = Spec HN 0 ( B N (, ɛ)) and let G /N (ɛ) Q be the pro-group scheme over Q gotten from G /N (ɛ) by applying the fiber functor gr W : H f N Vec Q. Note that Tannaka duality gives a canonical action of G N on G /N (ɛ) Q. Of course, in order to make a reasonable relative bar construction, one needs to use a good model for as an N -algebra. This is provided by using the relative minimal model { } N of over N, for which the derived tensor product is just the usual tensor product. In section 2.8, especially theorem 2.8.3, we show that 1. G /N (ɛ) Q = Spec H 0 ( B({ } N N Q)). 2. There is an exact sequence of pro-group schemes over Q: p 1 G /N (ɛ) Q G GN 1 The splitting ɛ to p defines a splitting ɛ : G N G to p. 3. The conjugation action of G N on G /N (ɛ) Q given by the splitting ɛ is the same as the canonical action. To do this, we use an alternate description of dg modules over an dams graded cdga N, that of flat dg connections. Kriz and May describe dg modules M over N as N + := r>0 N (r)-valued connections over M N Q (for the canonical augmentation N Q). Writing { } + N as N + I, with this decomposition coming from the augmentation { } N N, the absolute (i.e. { } + N -valued) connection on H0 ( B()) = H 0 ( B({ } N )) induces a N + -valued connection on H 0 ( B({ } N N Q)). Similarly, the structure of H 0 N ( B N (, ɛ)) as an ind-hopf algebra in H f N gives an N + -valued connection on H 0 N ( B N (, ɛ)) N Q = H 0 ( B({ } N N Q)). Using this description, it is easy to make the identifications necessary for proving (1)-(3) above. H. Esnault has interpreted this argument as saying that G /N (ɛ) is the Gauß-Manin connection of G associated to /N. pplying this theory to the splitting ɛ x : N (X) N (k), the Q pro-group scheme K, and the lifting K x to a MT(k) pro-group scheme, gives us the isomorphism of pro-group schemes K = Spec H 0 ( B(N (X){ } N (k) N (k) Q)) 7

8 and the isomorphism of pro-group scheme objects in H f N (k) K x = Spec H 0 N ( B N (k) (N (X), ɛ x )). ( ) One can make the dg N (k)-module H 0 N ( B N (k) (N (X), ɛ x )) explicit as an object in MT(k) via Spitzweck s theorem. This relies on a crucial property of the transformation from dg N (k) modules to motives (see theorem for a more general statement): Take X Sm/k. If the motive of X in DM(k) Q is in DMT(k) and X satisfies the Beilinson- Soulé vanishing conjectures, then the motive of N X { } N (k) is canonically isomorphic to H gm (X) Q. The explicit decription of the Beilinson simplicial scheme underlying the Deligne-Goncharov construction, together with this essential fact, allows one to conclude that K x with its MT(k) structure induced by the Gauß-Manin connection is precisely π1 mot (X, x), when x comes from a rational point x X(k) (see sections 6.5 and 6.6). In other words, we have the isomorphism of pro-group schemes over MT(k): π mot 1 (X, x) = Spec H 0 N ( B Nk (N X, ɛ x )). Combining this with our identification ( ) proves theorem 1. Replacing k with a more general base-scheme S Sm/k, the program outlined above proves theorem 2. In this article, we do not consider the case of the base-point x being a non-trivial tangent vector at some point x P 1 \ X. s mentioned above, Deligne-Goncharov [12] show in this case as well that the motivic π 1, defined by Deligne [11] as a system of realizations, comes from MT(k). This defines π1 mot (X, x) as an object in MT(k), but does not give a direct construction in MT(k). However, the results of [28] give a section ɛ x to p : N (k) N (X) (in the homotopy category of cdgas) for tangential base-points x as well as for k-points, so we do have a relative bar construction available even for tangential base-points. In order to extend our main theorem to this case, one should define realization functors on the categories of Tate motives, described as dg modules over the cycle algebra, and check that the realization of Spec HN 0 ( B Nk (N X, ɛ x )) agrees with Deligne s motivic π 1. Outline: The paper is organized as follows: We begin in section 1 with a review of the theory of dg modules over an dams-graded cdga, following for the most part the discussion of Kriz-May [26], but adding some new material dealing with the category of weightbounded modules. In section 2 we describe an extension of the classical model structure on cdgas over a field of characteristic zero (cf. [7]) to the category of cdgas over a cdga. This enables us to extend the theory of minimal models and the bar construction to the relative case. We conclude this section with our main result on the relative bar construction, theorem In fact, the reader who is moderately familiar with the Kriz-May theory of dg modules over a cdga could simply skim the first two sections to absorb our notation, and accept theorem on faith for the first reading. We then proceed to a review of the recently available theory of motives over a basescheme, due to Cisinski-Déglise [9, 10], in section 3. Next, in section 4, we take a look at generalizations of the Bloch-Kriz cycle algebra to a functorial construction for smooth 8

9 schemes over k, modifying a construction of Joshua [25]. In section 5, we describe the cohomological motive functor to the Cisinski-Déglise category and show how a Q-version of this functor can be described using the cycle algebra. This section is the technical heart of the paper. In it, we prove our main results relating motives and cycle algebras: our generalization of Spitzweck s representation theorem, theorem 5.3.2, identifying the derived category of dg modules over the cycle algebra N (S) to the triangulated category of Tate motives over S, and our two main results relating the cycle complex of a smooth S-scheme X to the geometric motive of X, theorem and theorem We put everything together in section 6, giving our generalization of the Deligne-Goncharov motivic π 1 and proving our main results, theorem 1 and theorem 2 (these are corollary and theorem 6.6.1, respectively). cknowledgements: Together with H. Esnault, we gave a seminar in the winter at the university of Duisburg-Essen on [12], to try to understand the constructions and results of Deligne-Goncharov, as well as the various constructions of mixed Tate motives and the relationships between them, as developed in the works of Bloch, Bloch-Kriz, Kriz-May and Spitzweck, and summarized in [29]; this paper is to a large extent a product of that seminar. We thank all the seminar participants for their willingness to give talks. In particular we thank Phùng Hô Hai for various discussions on Tannakian categories. Most importantly, this paper is a revision of a joint work with Hélène Esnault [14]. This joint work also contained a proof of theorem 1, with proof along the same lines as the one given here. It was Esnault who had originally suggested relating the Deligne-Goncharov motivic π 1 to the Bloch-Kriz cycle Hopf algebras as a way of answering the question of Deligne and Goncharov on the relation of MT(k(t)) to π 1 (X, x)-representations in MT(k). This paper would never have existed had it not been for the many fruitful discussions and numerous insights Esnault has shared with us; we take this opportunity to thank her for her crucial contribution to this work. Finally, we would like to thank the referee for making a number of useful suggestions. 1 Differential graded algebras We fix notation and recall some basic facts on commutative differential graded algebras (cdgas) over Q. This material is taken mainly from [26], with some refinements and additions. In what follows a cdga will always mean a cdga over Q. 1.1 dams graded cdgas Definition (1) cdga (, d, ) (over Q) consists of a unital, graded-commutative Q- algebra ( := n Z n, ) together with a graded homomorphism d = n d n, d n : n n+1, such that 1. d n+1 d n = d n+m (a b) = d n a b + ( 1) n a d m b; a n, b m. is called connected if n = 0 for n < 0 and 0 = Q 1, cohomologically connected if H n ( ) = 0 for n < 0 and H 0 ( ) = Q 1. 9

10 (2) n dams graded cdga is a cdga together with a direct sum decomposition into subcomplexes := r 0 (r) such that (r) (s) (r + s). In addition, we require that (0) = Q id. n dams graded cdga is said to be (cohomologically) connected if the underlying cdga is (cohomologically) connected. For x n (r), we call n the cohomological degree of x, n := deg x, and r the dams degree of x, r := x. Note that an dams graded cdga has a canonical augmentation Q with augmentation ideal + := r>0 (r). 1.2 The bar construction We let Ord denote the category with objects the sets [n] := {0,..., n}, n = 0, 1,..., and morphisms the non-decreasing maps of sets. The morphisms in Ord are generated by the coface maps δi n : [n] [n + 1] and the codegeneracy maps σi n : [n] [n 1], where δi n is the strictly increasing map omitting i from its image and σi n is the non-decreasing surjective map sending i and i + 1 to i. For a category C, we have the categories of cosimplicial objects in C and simplicial objects in C, namely, the categories of functors Ord C and Ord op C, respectively. For a cosimplicial object X : Ord C, we often write δi n and σi n for the coface maps X(δi n ) and X(σi n ), and for a simplicial object S : Ord op C, we often write d n i and s n i for the face and degeneracy maps S(δi n ) and S(σi n ). Let be a cdga. We begin by defining the simplicial cdga B () as follows: Tensor product (over Q) is the coproduct in the category of cdgas, so for a finite set S, we have S, giving the functor? from finite sets to cdgas. Thus, if we have a simplicial set S such that S[n] is a finite set for all n, we may form the simplicial cdga S, n S[n]. We have the representable simplicial sets [n] := Hom Ord (, [n]); setting [0, 1] := [1] gives us the simplicial cdga B () := [0,1]. The two inclusions [0] [1] define the maps i 0, i 1 : [0] [1]. Letting {0, 1} denote the constant simplicial set with two elements, the maps i 0, i 1 give rise to the map of simplicial sets i 0 i 1 : {0, 1} [0, 1], which makes B () into a simplicial = {0,1} algebra. Suppose we have augmentations ɛ 1, ɛ 2 : Q. Define B (, ɛ 1, ɛ 2 ) by B (, ɛ 1, ɛ 2 ) := B () Q using ɛ 1 ɛ 2 : Q as structure map. Since B n (, ɛ 1, ɛ 2 ) is a complex for each n, we can form a double complex by using the usual alternating sum of the face maps d n i : Bn+1 (, ɛ 1, ɛ 2 ) B n (, ɛ 1, ɛ 2 ) as the second differential, and let B(, ɛ 1, ɛ 2 ) denote the total complex of this double complex. We use cohomological grading throughout, so B n (, ɛ 1, ɛ 2 ) m has total degree m n. For ɛ 1 = ɛ 2 = ɛ, we write B(, ɛ) or simply B(); this is the reduced bar construction for (, ɛ). s is usual, we denote a decomposable element x 1... x n of B() by [x 1,... x n ]. Note that deg([x 1... x m ]) = m + i deg(x i ). 10

11 The bar construction B := B() has the following structures: a differential d : B B of degree +1 coming from the differential in, a product : B B B [x 1... x p ] [x p+1... x p+q ] = σ sgn(σ)[x σ(1)... x σ(p+q) ] where the sum is over all (p, q) shuffles σ S p+q (and the sign is the graded sign of σ, taking into account the degrees of the x i ), a co-product and an involution ι : B B, δ : B B B δ([x 1... x n ]) := n ( 1) i deg([x i+1... x n]) [x 1... x i ] [x i+1... x n ] i=0 ι([x 1 x 2... x n 1 x n ]) := ( 1) m [x n x n 1... x 2 x 1 ]; m = 1 i<j n deg(x i ) deg(x j ), making ( B(), d,, δ, ι) a differential graded Hopf algebra over Q, which is graded-commutative with respect to the product. The cohomology H ( B()) is thus a graded Hopf algebra over Q, in particular, H 0 ( B()) is a commutative Hopf algebra over Q. Let I() be the kernel of the augmentation H 0 ( B()) Q induced by ɛ. The coproduct δ on H 0 ( B()) induces the structure of a co-lie algebra on γ := I()/I() 2. From the formula for the coproduct, we see that, modulo tensors of degree < m, we have δ([x 1... x m ]) = 1 [x 1... x m ] + [x 1... x m ] 1 This implies that the pro-affine Q-algebraic group G := Spec H 0 ( B()) is pro-unipotent. In addition, in case is cohomologically connected, H 0 ( B()) is, as a Q-algebra, a polynomial algebra with indecomposables γ (see, e.g., [2, theorem 2.30, corollary 2.31]). Suppose = r 0 (r) is an dams graded cdga, with canonical augmentation ɛ : Q. The dams grading on induces an dams grading on B () and thus on B(); explicitly B() has the dams grading B() = r 0 B()(r) where the dams degree of [x 1... x m ] is [x 1... x m ] := x j. j Thus H 0 ( B()) = r 0 H 0 ( B()(r)) becomes an dams graded Hopf algebra over Q, commutative as a Q-algebra. We also have the dams graded co-lie algebra γ = r>0 γ (r). Remark Let be a cohomologically connected dams graded cdga. The dams grading equips the pro-unipotent affine Q group scheme G := Spec H 0 ( B()) with a grading, or, equivalently, with a G m -action. Thus γ is a positively graded nilpotent co-lie algebra, and there is an equivalence of categories between the continuous graded co-representations of H 0 ( B()) in finite dimensional graded Q-vector spaces, co-rep f Q (H0 ( B())), and the continuous graded co-representations of γ in finite dimensional graded Q-vector spaces, co-rep f Q (γ ). 11

12 1.3 The category of cell modules Kriz and May [26] define a triangulated category directly from an dams graded cdga without passing to the bar construction or using a co-lie algebra. We recall some of their work here, with some extensions. Let be a graded algebra over Q. We let [n] be the left -module which is m+n in degree m, with the -action given by left multiplication. If ( ) = n 0,r 0 n (r) is a bi-graded Q-algebra, we let <r>[n] be the left ( )-module which is m+n (r + s) in bi-degree (m, s), with action given by left multiplication. Definition Let be a cdga. (1) dg -module (M, d) consists of a complex M = n M n of Q-vector spaces with differential d, together with a graded, degree zero map Q M M, a m a m, which makes M into a graded -module, and satisfies the Leibniz rule d(a m) = da m + ( 1) deg a a dm; a, m M. (2) If = r 0 (r) is an dams graded cdga, an dams graded dg -module is a dg -module M together with a decomposition into subcomplexes M = s M (s) such that (r) M (s) M (r + s). We say x M has dams degree s if x M (s), and write this as x = s. (3) n dams graded dg -module M is a cell module if (a) M is free as a bi-graded -module, where we forget the differential structure. That is, there is a set J and elements b j M n j (r j ), j J, such that the maps a a b j induces an isomorphism of bi-graded -modules j J < r j >[ n j ] M. (b) There is a filtration on the index set J: such that J = n=0j n and for j J n, J 1 = J 0 J 1... J n... J db j = i J n 1 a ij b i. finite cell module is a cell module with finite index set J. We denote the category of dg -modules by M, the -cell modules by CM and the finite cell modules by CM f. 12

13 1.4 The derived category Let be an dams graded cdga and let M and N be dams graded dg -modules. Let Hom (M, N) be the dams graded dg -module with Hom (M, N) n (r) the -module consisting of maps f : M N with f(m a (s)) N a+n (s + r), f(am) = ( 1) np af(m) for a p and m M, and with differential d defined by df(m) = d(f(m))( 1) n f(dm) for f Hom(M, N) n (r). Similarly, let M N be the dams graded dg -module with underlying module M N and with differential d(m n) = dm n + ( 1) deg m m dn. For f : M N a morphism of dams graded dg -modules, we let Cone(f) be the dams graded dg -module with Cone(f) n (r) := N n (r) M n+1 (r) and differential d(n, m) = (dn + f(m), dm). Let M[1] be the dams graded dg -module with M[1] n (r) := M n+1 (r) and differential d, where d is the differential of M. sequence of the form M f N i Cone(f) j M[1] where i and j are the evident inclusion and projection, is called a cone sequence. Definition Let be an dams graded cdga over Q. We let M denote the category of dams graded dg -modules, K the homotopy category, i.e. the objects of K are the objects of M and Hom K (M, N) = H 0 (Hom (M, N)(0)). The category K is a triangulated category, with distinguished triangles those triangles which are isomorphic in K to a cone sequence. Definition The derived category D of dg -modules is the localization of K with respect to morphisms M N which are quasi-isomorphisms on the underlying complexes of Q-vector spaces. For M in D, we denote the nth cohomology of M, as a complex of Q-vector spaces, by H n (M). We define the homotopy category of -cell modules, resp. finite cell modules, as the full subcategory of K with objects in CM, resp. in CM f, KCM f KCM K. Note that for = Q, M Q is just the category of complexes of graded Q-vector spaces, and D Q is the unbounded derived category of graded Q-vector spaces. Proposition ([26, construction 2.7]) Let be an dams graded cdga. Then the evident functor KCM D is an equivalence of triangulated categories. Explicitly, let f : M M be a quasi-isomorphism in M, N CM. Then the induced map is an isomorphism. f : Hom K (N, M ) Hom K (N, M) 13

14 We let D f D be the full subcategory with objects those M isomorphic in D to a finite cell module. s an immediate consequence of proposition 1.4.3, we have Proposition KCM f Df is an equivalence of triangulated categories. Example (Tate objects) For n Z, let Q(n) be the object of CM f which is the free rank one -module with generator b n having dams degree n, cohomological degree 0 and db n = 0, i.e., Q(n) = <n>. We sometimes write Q (n) for Q(n); Q(n) is called a Tate object. 1.5 Weight filtration Let M be an dams graded dg -module which is free as a bi-graded -module. Choose a basis B := {b j j J}, M = j b j. Write db j = i a ij b i ; a ij. Since a ij 0 and d has dams degree 0, it follows that Thus, we have the subcomplex b i b j if a ij 0. W B n M = {j, bj n} b j of M. The subcomplex Wn B M is independent of the choice of basis: if B = {b j} is another basis and if b j = n, then as b j = i e ijb i and e ij 0, it follows that b j Wn B M and hence Wn B M Wn B M. By symmetry, Wn B M Wn B M. We may thus write W n M for Wn B M. This gives us the increasing filtration as an dams graded dg -module W M :... W n M W n+1 M... M with M = n W n M. Similarly, for n n, define W n/n M as the cokernel of the inclusion W n M W n M, i.e., W n/n M is the dams graded dg -module with basis the b j having n < b j n and with differential induced by the differential in W n M. We write gr W n for W n/n 1 and W >n for W /n. It is not hard to see that W n M is functorial in M. In particular, if f : M M is a homotopy equivalence of cell modules with homotopy inverse g : M M, then f and g restricted to W n M and W n M give inverse homotopy equivalences W n f : W n M W n M, W n g : W n M W n M. Thus the W filtration in CM defines a functorial tower of endofunctors on KCM :... W n W n+1... id (1.5.1) Lemma The endo-functor W n is exact for each n. 2. For n n, the sequence of endo-functors W n W n W n/n canonically extends to a distinguished triangle of endo-functors. 14

15 Proof For (1), it follows directly from the definition that W n transforms a cone sequence into a cone sequence. For (2), take M CM. The sequence 0 W n M W n M W n/n M 0 is split exact as a sequence of bi-graded -modules. Thus (2) follows from the general fact that a sequence in CM 0 N i N p N 0 that is split exact as a sequence of bi-graded -modules extends canonically to a distinguished triangle in KCM. To see this, choose a splitting s to p (as bi-graded -modules), and define t : N N [1] by i t = s d N d N s. It is then easy to check that t is a map of complexes and (s, t) : N N N [1] defines the map of complexes (s, t) : N Cone(i) making the diagram N i N p N t N [1] (s,t) N i N Cone(i) N [1] commute. In particular, (s, t) is an isomorphism in KCM. One sees similarly that another choice s of splitting leads to a homotopic map (s, t ). Note that it is not necessary for M to be a cell module to define W n M; being free as a bi-graded -module suffices. However, it is not clear that W n M is a quasi-isomorphism invariant in general. To side-step this issue, we use instead Definition Define the tower of exact endo-functors on D... W n W n+1... id using (1.5.1) and the equivalence of categories in proposition We define W n/n, gr W n and W >n on D similarly. Remark Since KCM D is an equivalence of triangulated categories, the natural distinguished triangles W n W n W n/n W n [1] in KCM give us natural distinguished triangles in D. W n W n W n/n W n [1] One uses the weight filtration for inductive arguments, for example: Lemma Let M be a finite -cell module. Suppose N is a summand of M in D. Then there is a finite -cell module M with N = M in D. 15

16 Proof By proposition there is an isomorphism N = N in D with with N an object in CM. Thus we may assume that N is a cell module. Since KCM D is an equivalence, N is a summand of M in KCM. Write M = N N in KCM and let p : M M be the projection M N followed by the inclusion N M. Since M is finite, there is a minimal n with W n M 0. Thus W n 1 N is homotopy equivalent to zero and N = W /n 1 N in KCM. Hence, we may assume that W n 1 N = 0 in CM. Similarly, we may assume that M = W n+r M and N = W n+r N in CM for some r 0. We proceed by induction on r. s (0) = Q id, it follows that W n M = Q M 0 for a finite complex of finite dimensional graded Q-vector spaces M 0. Indeed, choose a finite bi-graded -basis {b j } for W n M and let M 0 be the finite dimensional Q-vector space spanned by the b j. Since W n 1 M = 0, all the b j have dams degree n. Writing db j = i a ijb i and noting that the differential has dams degree 0, it follows that a ij = 0 for all i, j, i.e., a ij Q id. Consequently M 0 is a subcomplex of M and W n M = Q M 0 as an dams graded dg module. But such an M 0 is homotopy equivalent to the direct sum of its cohomologies; replacing M 0 with n H n (M 0 )[ n] and changing notation, we may assume that d M0 = 0. Thus W n M = Q M 0 for M 0 a finite dimensional bi-graded Q-vector space; using again the fact that (r) = 0 for r < 0 and (0) = Q id, we see that W n p = id q with q : M 0 M 0 an idempotent endomorphism of the bi-graded Q-vector space M 0. Thus W n N = im(q), hence W n N is homotopy equivalent to a finite -cell module. This also takes care of the case r = 0. Using the distinguished triangle W n N N W n+r/n N W n N[1] we may replace N with the shifted cone of the map W n+r/n N im(q)[1]. Since W n+r/n N is a summand of W n+r/n M, it follows by induction on r that W n+r/n N is homotopy equivalent to a finite cell module, hence the cone of W n+r/n N im(q) is homotopy equivalent to a finite cell module as well. Definition Let D +w W n M = 0 for some n. Similarly, let CM +w such that W n M = 0 for some n and let KCM +w D be the full subcategory of D with objects M such that CM be the full subcategory with objects M be the homotopy category of CM+w Lemma The natural map KCM +w KCM is an equivalence of KCM +w with the full subcategory of KCM with objects the M such that W n M = 0 in KCM for n << The equivalence KCM D induces an equivalence KCM +w D+w. Proof Since KCM +w is the homotopy category of the full subcategory CM+w of CM, the functor KCM +w KCM is a full embedding. Suppose that W n M = 0 in KCM. We have the cell module W >n M and the distinguished triangle W n M M W >n M W n M[1] in KCM. Thus the map M W >n M is an isomorphism in KCM ; since W >n M is in, the essential image of KCM+w in KCM is as described. CM +w 16

17 For (2), following definition 1.5.2, W n M is defined by choosing an isomorphism P M in D with P CM and taking W n M := W n P. Since W n P = W n M = 0 in D, it follows that W n P = 0 in KCM, so P is isomorphic to an object in KCM +w. Thus D+w is the essential image of KCM +w in D. Since KCM D is an equivalence, this proves (2). Remark Take M D +w. Then there is an n 0 such that W n M = 0 for all n n 0. Indeed, by definition, W n0 M = 0 for some n 0. Thus M W >n 0 M is an isomorphism in D. If n < n 0, then W n M W n W >n 0 M = 0 is an isomorphism in D. nother result using induction on the weight filtration is Lemma Let M be an dams graded dg -module. 1. M is a finite -cell module if and only if M is free and finitely generated as a bi-graded -module. 2. M is in CM +w if and only if M is free as a bi-graded -module and there is an integer r 0 such that m r 0 for all m M. Proof We first prove (1). Clearly a finite -cell module is free and finitely generated as a bi-graded -module. Conversely, suppose M is free and finitely generated over ; choose a basis B for M. Clearly Wn B M = 0 for n << 0; let N be the minimum integer n such that Wn B M 0 and let B N be the set of basis elements b of dams degree N. Since (0) = Q id, it follows that B N forms a Q basis for W N M and the differential on B N is given by de α = β a αβ e β with a αβ Q and e β B N. Changing the Q basis for WN B M, we may assume that the subset BN 0 of B N of e α such that de α = 0 forms an Q basis for the kernel of d on the Q-span of B N. Since d 2 = 0, the two-step filtration BN 0 B N exhibits W N M as a finite cell module. The result follows by induction on the length of the weight filtration: By induction W B >NM := M/W n B M is a finite cell module with basis say {b j j J} for some filtration on J. Since M = WN BM W B >N M as an -module, we just take the union of the two bases, and the concatenation of the filtrations, to present M as a finite cell module. The proof of (2) is similar. In fact, the same proof as for (1) shows that the sub-dg -module Wn B M of M is in CM +w for all n and that we may find an basis B n for Wn B M and a filtration = B r 0 1 n B r 0 n... Bn 2n 1 Bn 2n = B n that exhibits Wn B M as a cell module. In addition, we may assume that B i with its filtration is just Bn 2i with the induced filtration, for all i n. Thus, taking the union of the B n gives the desired basis for M, showing that M is in CM +w. 17

18 1.6 Bounded below modules Definition Let D + D be the full subcategory with objects the dams graded dg -modules M having H n (M) = 0 for n << 0, as usual, we call such an M bounded below. Lemma Suppose that is cohomologically connected, and M is an dams graded dg -module with H n (M) = 0 for n < n 0. Then there is a quasi-isomorphism P M with P an -cell module having basis {e α } with deg(e α ) n 0 for all α. If in addition there is an r 0 such that H n (M)(r) = 0 for all (r, n) with r < r 0, we may find P M as above satisfying the additional condition e α r 0 for all α. Proof We first note the following elementary facts: Let V = n,r V n (r) be a bi-graded Q-vector space, which we consider as a complex with zero differential. Then the complex Q V is a cell-module, since a bi-graded Q basis for V gives a bi-graded basis with 0 differential. In addition, the map v 1 v gives a map V n := r V n (r) H n ( V ). Finally, suppose there is an n 0 such that V n0 H n () = 0 for n < 0 and H 0 () = Q, the map 0 but V n = 0 for all n < n 0. Then as V n H n ( Q V ) is an isomorphism for all n n 0. We begin the construction of P M by taking V to be a bi-graded Q subspace of n n0 M n representing n H n (M), giving the map of dams graded dg modules φ n0 : P 0 := n n0 H n (M)[ n] M. From the discussion above, we see that φ n0 is an isomorphism on H n for n n 0 and a surjection on H n for n > n 0. If in addition there is an r 0 such that H n (M)(r) = 0 for r < r 0 and all n, then P 0 has a bi-graded -basis S 0 with v r 0 for each v S 0. Suppose by induction we have constructed a sequence of inclusions of -cell modules and maps of dams graded dg -modules with the following properties: P 0 P 1... P r 1 φ n0 +i : P i M 1. The P i have -bases S(i) := S 0... S i. In addition, for all i 1, all the elements in S i are of cohomological degree n 0 + i 1, and for v S i, dv is in P i The map P i P i+1 is the one induced by the inclusion S(i) S(i + 1). 3. φ n0 +i : P i M induces an isomorphism on H n for n n 0 + i and a surjection for all n. 18

19 4. If H n (M)(r) = 0 for r < r 0 and all n, then v S(i) has dams degree v r 0. We now show how to continue the induction. For this, let n r = n 0 +r and let V P nr r 1 be a bigraded Q-subspace of representatives for the kernel of the surjection H nr (P r 1 ) H nr (M). Let P r := P r 1 Q V as bi-graded -module, where the differential is given by using the differential on P r 1, setting d((0, 1 v)) = (v, 0) P nr r 1 for v V and extending by the Leibniz rule. Note that, for v V P nr r 1, there is an m v M nr 1 with dm v = φ r 1 (v); chosing a bi-graded Q-basis S r for V and extending the assignment s m s from S r to all of V by Q-linearity, we have a Q-linear map f : V M nr 1 with d(f(v)) = φ r 1 (v) for all v V. Thus, we may define the map of dg -modules φ r : P r M by using φ r 1 on P r, f on 1 V and extending by -linearity. Clearly P r is an -cell module with -basis S(r) := S(r 1) S r. In case H n (M)(r) = 0 for r < r 0 and all n, clearly all bi-homogeneous elements of V have dams degree r 0, so v r 0 for all v S r. We can compute the cohomology of P r by using the sequence of -cell modules 0 P r 1 P r Q V 0, where we consider V as a complex with zero differential, which is split exact as a sequence of bi-graded -modules. The resulting long exact cohomology sequence shows that P r 1 P r induces an isomorphism in cohomology H n for n < n r 1 and we have the exact sequence 0 H nr 1 (P r 1 ) H nr 1 (P r ) V H nr (P r 1 ) H nr (P r ) 0. In addition, one can compute the coboundary by lifting the element 1 v ( Q V ) nr 1 to the element (0, 1 v) Pr nr 1 and applying the differential d Pr. From this, we see that the sequence 0 V H nr (P r 1 ) H nr (P r ) 0 is exact, hence H nr 1 (P r 1 ) H nr 1 (P r ) is an isomorphism. This also shows that φ r : P r M induces an isomorphism on H n for n n r and the induction continues. If we now take P to be the direct limit of the P r, it follows that P is an -cell module with basis elements all in cohomological degree n 0, and that the map φ : P M induced from the φ r is a quasi-isomorphism. If there is an r 0 such that H (M)(r) = 0 for r < r 0, then by (4) above, the basis S := r S(r) clearly has e r 0 for all e S. This completes the proof. 19

20 1.7 Tor and Ext The Hom functor Hom (M, N) and tensor product functor M N define bi-exact bifunctors Hom : KCM op KCM D : KCM KCM KCM. Via proposition 1.4.3, these give well-defined derived functors of Hom and : RHom : D op D D L : D D D. Restricting to KCM f, we have the derived functors for the finite categories RHom : D fop Df Df L : D f Df Df. In both settings, these bi-functors are adjoint: RHom (M L N, K) = RHom (M, RHom (N, K)). We have as well the restriction of L to D +w : L : D +w D+w D+w. The derived tensor product makes D into a triangulated tensor category with unit 1 := and D +w, D+ and Df are triangulated tensor subcategories. By lemma 1.5.4, Df is closed under taking summands in D ; this property is obvious for D +w. Define M := RHom (M, ) and call M strongly dualizable if the canonical map M M is an isomorphism in D. Note that M is strongly dualizable if M is rigid, i.e., there exists an N D and morphisms δ : M L N and ɛ : N L M such that We have (id M ɛ) (δ id M ) = id M (id N δ) (ɛ id N ) = id N Proposition ([26, theorem 5.7]) M D is rigid if and only if M is in D f, i.e., M = N in D for some finite -cell module N. The precise statement found in [26, theorem 5.7] is that M is rigid if and only if M is a summand in D of some finite cell module, so the proposition follows from this and lemma 1.5.4; Kriz and May are working in a more general setting in which lemma does not hold. Example For n 0, Q(±n) = (Q(±1)) n and for all n, Q(n) = Q( n). 20

21 1.8 Change of ring If φ : is a homomorphism of dams graded cdgas, we have the functor : M M which induces a functor on cell modules and the homotopy category φ : KCM KCM. Via proposition 1.4.3, we have the change of rings functor φ : D D on the derived category. By proposition and lemma 1.5.6, the respective restrictions of φ define exact tensor functors φ : D +w D+w φ : D f Df. From [26] we have Theorem ([26, proposition 4.2]) If φ is a quasi-isomorphism, then φ : D D is an equivalence of triangulated tensor categories. Noting the φ is compatible with the weight filtrations, the theorem immediately yields Corollary If φ is a quasi-isomorphism, then φ : D +w D+w is an equivalence of triangulated tensor categories. In addition, we have Corollary If φ is a quasi-isomorphism, then φ : D f Df is an equivalence of triangulated tensor categories. Proof Since an equivalence of tensor triangulated categories induces an equivalence on the subcategories of rigid objects, the result follows from theorem and proposition Proposition Let φ : B be a map of cdgas. Then φ : D +w D+w B is conservative, i.e., φ (M) = 0 implies M = 0, or equivalently, if φ (f) is an isomorphism then f is an isomorphism. 21

22 Proof Take M D +w, and let S := {n M = W >n M}. Then S ; we claim that either M = 0 or S has a maximal element. Indeed, if S has no maximum then W n M = 0 for all n. But since lim W n M M n is an isomorphism, this implies that M is acyclic, hence M = 0 in D. Thus, we may find a cell module P and quasi-isomorphism P M such that W n 1 P = 0, but W n P is not acyclic. In particular P has a basis {e α } with e α n for all α. If e α = n then since there are no basis elements with dams grading < n, we have de α = j a αj e j with a αj = 0, e j = n, i.e., a αj Q = (0). Since W n P is not acyclic, it thus follows that (W n P ) Q is also not acyclic: if (W n P ) Q were acyclic, this complex would be zero in the homotopy category KCM Q, which would make W n P 0 in KCM. s W n (P B) = (W n P ) B and (W n P ) Q = (W n P B) B Q it follows that P B is not isomorphic to zero in KCM B, and thus φ (M) is non-zero in D +w B. Example Each dams graded cdga has a canonical augmentation ɛ : Q, given by projection on 0 (0) = Q id. In particular, we have the functor and the exact tensor functors q := ɛ : CM M Q, qm := M Q q : D D Q, q +w : D +w q f : D f Df Q. D+w Q, Explicitly, q is given on the derived level by qm := M L Q. 1.9 Finiteness conditions M Q is just the category of graded Q-vector spaces, so D Q is equivalent to the product of the unbounded derived categories D Q = D(Q). Similarly n Z D f Q = n Z D b (Q), 22

23 where D b (Q) is the bounded derived category of finite dimensional Q-vector spaces. Finally, D +w Q = D(Q) D(Q). N n N n Z Remark The inclusion Q splits ɛ, identifying D Q, D +w Q, etc., with full subcategories of D, D +w, etc. Under this identification, and the decomposition of D Q into its dams graded pieces described above, the functor q is identified with the functor gr W := n Z grw n. Indeed, if P is an -cell module with basis {e α }, then as (r) = 0 for r < 0 and (0) = Q id, the differential d decomposes as d = d 0 + d + with d 0 e α = β a 0 αβe β, d + e α = β a + αβ e β where a 0 αβ = 0, a+ αβ > 0. Since d has dams degree 0, it follows that e β < e α if a + αβ 0, and e β = e α if a 0 αβ 0. Thus grw P is the complex of graded Q-vector spaces with Q basis {e α } and with d gr W P e α = d 0 e α. s qp has exactly the same description, we have the identification of gr W and q as described. Lemma Let M be in D +w. Then M is in Df 1. gr W n M is in D b (Q) D(Q) for all n. 2. gr W n M = 0 for all but finitely many n. if and only if Proof It is clear that M D f satisfies the conditions (1) and (2). Conversely, suppose M D +w satisfies (1) and (2). If M = 0, there is nothing to prove, so assume M is not isomorphic to 0. By proposition 1.8.4, qm = n grw n M is not isomorphic to zero. Take N minimal such that gr W N M is not isomorphic to zero. By (2), there is a maximal N such that gr W N M is not isomorphic to zero. If N = N, then M = gr W N M is in Db (Q) by (1), hence M = s i=1< N>[m i ], and thus M is in D f. In general, we apply remark 1.5.3, giving the distinguished triangle gr W N M M M >N gr W N M[1]; note that gr W n M >N = 0 for n > N. By induction on N N, M >N is in D f ; since Df is a full triangulated subcategory of D, closed under isomorphism, it follows that M is in D f Model structure Let cdga denote the category of dams graded commutative differential graded algebras over Q. In the non-dams graded case, Bousfield and Guggenheim [7] have defined a model structure on cdgas with weak equivalences the quasi-isomorphisms. s we are interested in possibly non-connected dams graded cdgas, we modify their definitions slightly. Definition morphism φ : B in cdga is a weak equivalence if φ induces an isomorphism φ : H n ((r)) H n (B(r)) 23