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THE SPECTRAL SEQUENCE RELATING ALGEBRAIC KTHEORY TO MOTIVIC COHOMOLOGY
"... The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilins ..."
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Cited by 44 (5 self)
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The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic Ktheory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the AtiyahHirzebruch spectral sequence from the singular cohomology to the topological Ktheory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic Ktheory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [BL]. Our construction depends crucially upon the main result of [BL], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative Ktheory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the Ktheory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ωprespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the BlochLichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of Ktheory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.
The Ktheory of fields in characteristic p
, 1996
"... Abstract. The purpose of this paper is to study the ppart of motivic cohomology and algebraic Ktheory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, Hi (k, Z(n)) is uniquely pdivisible for i ..."
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Cited by 38 (3 self)
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Abstract. The purpose of this paper is to study the ppart of motivic cohomology and algebraic Ktheory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, Hi (k, Z(n)) is uniquely pdivisible for i ̸ = n. This implies that the natural map KM n (k) − → Kn(k) from Milnor Ktheory to Quillen Ktheory is an isomorphism up to uniquely pdivisible groups, and that Kn(k) is ptorsion free. As a consequence, one can calculate the Ktheory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example Kn(X, Z/pr) = 0 for n> dimX. Another consequence is Gersten’s conjecture with mod pcoefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all BeilinsonLichtenbaumMilne axioms for motivic complexes, except the vanishing conjecture. 1.
Tamagawa Numbers for Motives with (NonCommutative) Coefficients
 DOCUMENTA MATH.
, 2001
"... Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes a ..."
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Cited by 37 (11 self)
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Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, PerrinRiou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective Astructure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is nonmaximal. In each such case the conjecture with respect to a nonmaximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.
The homotopy theory of fusion systems
"... The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group con ..."
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Cited by 35 (10 self)
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The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group consists, roughly speaking, of a finite pgroup S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on pcompleted classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig [Pu], [Pu2] of systems of fusion among subgroups of a given pgroup. The pcompletion of a space X is a space X ∧ p which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −− → Y induces a homotopy equivalence
Complexes of graph homomorphisms
 Israel J. Math
"... Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom ..."
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Cited by 33 (11 self)
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Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom (Km, Kn) is homotopy equivalent to a wedge of (n−m)dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph G, and integers m ≥ 2 and k ≥ −1, we have ̟k 1 (Hom (Km, G)) ̸ = 0, then χ(G) ≥ k + m; here Z2action is induced by the swapping of two vertices in Km, and ̟1 is the first StiefelWhitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom (G, H) induces a homotopy equivalence. It then follows that Hom (F, Kn) is homotopy equivalent to a direct product of (n−2)dimensional spheres, whileHom (F, Kn) is homotopy equivalent to a wedge of spheres, where F is an arbitrary forest and F is its complement. 1.1. Definition of the main object. 1.
Cohomology of Algebraic Theories
 J. of Algebra
, 1991
"... this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups ..."
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Cited by 30 (17 self)
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this paper were announced in [11], [18]. Some of them were obtained by the second author only, in particular, the main theorem of Section 2. 2. On the Mac Lane Cohomology In this section we prove Theorem A, which is concerned with the relationship between Mac Lane cohomology of rings and Ext groups in functor categories; related questions are discussed. Let us recall the denition of the Mac Lane cohomology from [13]. Consider the sets Cn with 2 n elements  ntuples ("1 ; :::; "n ), where " i = 0 or 1, for n > 0 and i 6 n, and the 0tuple ( ) for n = 0. For convenience Cn can be visualized as the set of vertices of an ncube, the product of n copies of the 1cube with vertices 0 and 1. Dene maps 0 i ; 1 i : Cn ! Cn+1 , 1 6 i 6 n + 1, by the equalities 0 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 0; " i+1 ; :::; "n ); 1 i ("1 ; :::; "n) = ("1 ; :::; " i 1 ; 1; " i+1 ; :::; "n ): For an abelian group A and a set S, let A[S] denote the sum of S copies of the group A. Since the sets Cn are nite, the group A[Cn ] can be identied with the group of all maps t : Cn ! A: Let Q 0 n (A) be the free abelian group generated by the set A[Cn ], i. e., Q 0 n (A) = Z[A[Cn ]]: Following Mac Lane [13], dene for i = 1; 2; :::; n the homomorphisms R i ; S i ; P i : Q 0 n (A) ! Q 0 n 1 (A) by R i = Z[R i ]; S i = Z[S i ]; P i = Z[P i ]; COHOMOLOGY OF ALGEBRAIC THEORIES 257 where R i ; S i ; P i : A[Cn ] ! A[Cn 1 ] are homomorphisms dened for e 2 Cn 1 and t 2 A[Cn ] by (R i t)(e) = t(0 i e); (S i t)(e) = t(1 i e); (P i t)(e) = t(0 i e) + t(1 i e): In [13], Mac Lane denes the boundary homomorphism @ : Q 0 n (A) ! Q 0 n 1 (A) by the equality @ = n X i=1 ( 1) i (P i R i S i ): A generator t : Cn ! A of the group Q 0 n (A) is called a slab when t( ) = 0, for n...
Coefficients for the FarrellJones conjecture
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 402
"... Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with ..."
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Cited by 29 (12 self)
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Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the setup with coefficients we obtain new results about the original FarrellJones Conjecture. The conjecture with coefficients implies the fibered version of the FarrellJones Conjecture. 1.
Isomorphism conjecture for homotopy Ktheory and groups acting on trees
, 2008
"... We discuss an analogon to the FarrellJones Conjecture for homotopy algebraic Ktheory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the asse ..."
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Cited by 28 (13 self)
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We discuss an analogon to the FarrellJones Conjecture for homotopy algebraic Ktheory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the FarrellJones Conjecture in algebraic Ktheory.
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."