3,540 research outputs found
A theory of nice triples and a theorem due to O.Gabber
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and
better structured proof of the Grothendieck--Serre's conjecture for semi-local
regular rings containing a finite field. The outline of the proof is the same
as in [P1], [P2], [P3]. If the semi-local regular ring contains an infinite
field, then the conjecture is proved in [FP]. Thus the conjecture is true for
regular local rings containing a field.
The present paper is the one [Pan0] in that series. Theorem 1.2 is one of the
main result of the paper. The proof of the latter theorem is completely
geometric. It is based on a theory of nice triples from [PSV] and on its
extension from [P]. The theory of nice triples is inspired by the Voevodsky
theory of standart triples [V].
Theorem 1.2 yields an unpublished result due to O.Gabber (see Theorem
1.1=Theorem 3.1). It states that the Grothendieck--Serre's conjecture for
semi-local regular rings containing a finite field is true providing that the
group is simply-connected reductive and is extended from the base field
Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field
Let R be a regular local ring, containing a finite field. Let G be a
reductive group scheme over R. We prove that a principal G-bundle over R is
trivial, if it is trivial over the fraction field of R. In other words, if K is
the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to
H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel.
Certain arguments used in the present preprint do not work if the ring R
contains a characteristic zero field. In that case and, more generally, in the
case when the regular local ring R contains an infinite field this result is
proved in joint work due to R.Fedorov and I.Panin (see [FP]). Thus the
Grothendieck--Serre conjecture holds for regular local rings containing a
field.Comment: arXiv admin note: substantial text overlap with arXiv:1211.267
Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and
better structured proof of the Grothendieck--Serre's conjecture for semi-local
regular rings containing a finite field. The outline of the proof is the same
as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite
field, then the conjecture is proved in [FP]. Thus the conjecture is true for
regular local rings containing a field.
The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of
the main result of the paper. It is also one of the key steps in the proof of
the Grothendieck--Serre's conjecture for semi-local regular rings containing a
field (see [Pan3]). The proof of Theorem 1.1 is completely geometric.Comment: arXiv admin note: text overlap with arXiv:1406.024
Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and
better structured proof of the Grothendieck--Serre's conjecture for semi-local
regular rings containing a finite field. The outline of the proof is the same
as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite
field, then the conjecture is proved in [FP]. Thus the conjecture is true for
regular local rings containing a field.
A proof of Grothendieck--Serre conjecture on principal bundles over a
semi-local regular ring containing an arbitrary field is given in [Pan3]. That
proof is heavily based on Theorem 1.3 stated below in the Introduction and
proven in the present paper.Comment: arXiv admin note: text overlap with arXiv:1406.1129, arXiv:0905.142
On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II
In three preprints [Pan1], [Pan3] and the present one we prove
Grothendieck-Serre's conjecture concerning principal G-bundles over regular
semi-local domains R containing a finite field (here is a reductive group
scheme). The preprint [Pan1] contains main geometric presentation theorems
which are necessary for that. The present preprint contains reduction of the
Grothendieck--Serre's conjecture to the case of semi-simple simply-connected
group schemes (see Theorem 1.0.1). The preprint [Pan3] contains a proof of that
conjecture for regular semi-local domains R containing a finite field. The
Grothendieck--Serre conjecture for the case of regular semi-local domains
containing an infinite field is proven in joint work due to R.Fedorov and
I.Panin (see [FP]). Thus the conjecture holds for regular semi-local domains
containing a field. The reduction is based on two purity results (Theorem 1.0.2
and Theorem 10.0.29).Comment: arXiv admin note: substantial text overlap with arXiv:0905.142
Proof of Grothendieck--Serre conjecture on principal bundles over regular local rings containing a finite field
Let R be a regular local ring, containing a finite field. Let G be a
reductive group scheme over R. We prove that a principal G-bundle over R is
trivial, if it is trivial over the fraction field of R.
If the regular local ring R contains an infinite field this result is proved
in [FP]. Thus the conjecture is true for regular local rings containing a
field.Comment: arXiv admin note: text overlap with arXiv:1406.024
Purity for Similarity Factors
Two Azumaya algebras with involutions are considered over a regular local
ring. It is proved that if they are isomorphic over the quotient field, then
they are isomorphic too. In particular, if two quadratic spaces over such a
ring are similar over its quotient field, then these two spaces are similar
already over the ring. The result is a consequence of a purity theorem for
similarity factors proved in this text and the known fact that rationally
isomorphic hermitian spases are locally isomorphic.Comment: 22 page
Nice triples and a moving lemma for motivic spaces
It is proved that for any cohomology theory A in the sense of [PS] and any
essentially k-smooth semi-local X the Cousin complex is exact. As a consequence
we prove that for any integer n the Nisnevich sheaf A^n_Nis, associated with
the presheaf U |--> A^n(U), is strictly homotopy invariant.
Particularly, for any presheaf of S^1-spectra E on the category of k-smooth
schemes its Nisnevich sheves of stable A1-homotopy groups are strictly homotopy
invariant.
The ground field k is arbitrary. We do not use Gabber's presentation lemma.
Instead, we use the machinery of nice triples as invented in [PSV] and
developed further in [P3]. This recovers a known inaccuracy in Morel's
arguments in [M].
The machinery of nice triples is inspired by the Voevodsky machinery of
standard triples.Comment: arXiv admin note: text overlap with arXiv:1406.024
Approximate Solution of the Representability Problem
Approximate solution of the ensemble representability problem for density
operators of arbitrary order is obtained. This solution is closely related to
the ``Q condition'' of A.J.Coleman. The representability conditions are
formulated in orbital representation and are easy to use. They are tested
numerically on the base of CI calculation of simple atomic and molecular
systems. General scheme of construction of the contraction operator right
inverses is proposed and the explicit expression for the right inverse
associated with the expansion operator is derived as an example. Two algorithms
for direct 2-density matrix determination are described.Comment: LaTeX2e, 45 pages; significantly revise
(p,q)-Sheaves and the Representability Problem
General properties of new models of the electronic Fock spaces based on the
notion of (p,q)-sheaves are studied. Interrelation between simple sheaves and
density operators is established. Explicit expressions for the transformed
reduced Hamiltonians in terms of the standard creation-annihilation operators
are presented. General scheme of parametrization of p-electron states by
k-electron means (k=2,3,...) is described and studied in detail for the case of
sheaves induced by k-electron wavefunctions. It is demonstrated that under
certain conditions p-electron problem may be reformulated as the eigenvalue
problem in k-electron space equipped with certain p-electron metric. Simple
numerical examples are given to illustrate our approach.Comment: 41 pages, latex, no figures, submitted to IJQ
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