779 research outputs found
Stable and Unstable operations in Algebraic Cobordism
We describe additive (unstable) operations from a theory A^* obtained from
Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any
oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence
between the set of operations, and the set of transformations:
A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying
certain simple properties. This provides an effective tool of constructing such
operations. As an application, we prove that (unstable) additive operations in
Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear
combinations of Landweber-Novikov operations which take integral values on the
products of projective spaces. On our way we obtain that stable operations
there are exactly L-linear combinations of Landweber-Novikov operations. We
also show that multiplicative operations A^* ---> B^* are in 1-to-1
correspondence with the morphisms of the respective formal group laws. We
construct Integral (!) Adams Operations in Algebraic Cobordism, and all
theories obtained from it by change of coefficients, giving classical Adams
operations in the case of K_0. Finally, we construct Symmetric Operations for
all primes p (these operations in Algebraic Cobordism, previously known only
for p=2, are more subtle than the Landweber-Novikov operations, and have
applications to rationality questions), as well as the T.tom Dieck - style
Steenrod operations in Algebraic Cobordism. As a bi-product of the proof of our
main theorem we get the Riemann-Roch Theorem for additive (unstable)
operations.Comment: to appear in Annales Scientifiques de l'Ecole Normale Superieur
Operations and poly-operations in Algebraic Cobordism
We describe all operations from a theory A^* obtained from Algebraic
Cobordism of M.Levine-F.Morel by change of coefficients to any oriented
cohomology theory B^* (in the case of a field of characteristic zero). We prove
that such an operation can be reconstructed out of it's action on the products
of projective spaces. This reduces the construction of operations to algebra
and extends the additive case done earlier, as well as the topological one
obtained by T.Kashiwabara. The key new ingredients which permit us to treat the
non-additive operations are: the use of "poly-operations" and the "Discrete
Taylor expansion". As an application we construct the only missing, the 0-th
(non-additive) Symmetric operation, for arbitrary p, which permits to sharpen
results on the structure of Algebraic Cobordism. We also prove the general
Riemann-Roch theorem for arbitrary (even non-additive) operations (over an
arbitrary field). This extends the multiplicative case proved by I.Panin.Comment: To appear in Advances in Mathematic
Symmetric operations for all primes and Steenrod operations in Algebraic Cobordism
In this article we construct Symmetric operations for all primes (previously
known only for p=2). These unstable operations are more subtle than the
Landweber-Novikov operations, and encode all p-primary divisibilities of
characteristic numbers. Thus, taken together (for all primes) they plug the gap
left by the Hurewitz map L ---> Z[b_1,b_2,...], providing an important
structure on Algebraic Cobordism. Applications include: questions of
rationality of Chow group elements - see [11], and the structure of the Graded
Algebraic Cobordism. We also construct Steenrod operations of T.tom Dieck-style
in Algebraic Cobordism. These unstable multiplicative operations are more
canonical and subtle than Quillen-style operations, and complement the latter.Comment: 21 page
Koszul duality and Galois cohomology
It it shown that the Bloch-Kato conjecture on the norm residue homomorphism
follows from its (partially known) low-degree part
under the assumption that the Milnor K-theory algebra modulo is
Koszul. This conclusion is a case of a general result on the cohomology of
nilpotent (co-)algebras and Koszulity.Comment: AMS-LaTeX v.1.1, 10 pages, no figures. Replaced for tex code
correction (%&amslplain added) by request of www-admi
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