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 Theorem 1.2 yields an unpublished result due to O.Gabber (see Theorem 1.1=Theorem 3.1).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

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

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

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 $G$ 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

On the Lower Garland of Certain Subgroup Lattices in Linear Groups

We describe here the lower garland of some lattices of intermediate subgroups in linear groups. The results are applied to the case of subgroup lattices in general and special linear groups over a class of rings, containing the group of rational points T of a maximal non-split torus in the corresponding algebraic group. It turns out that these garlands coincide with the interval of the whole lattice, consisting of subgroups between T and its normalizer.Comment: AmsTeX, 12 pages; minor correction