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 Theorem 1.2 yields an unpublished result due to O.Gabber (see Theorem 1.1=Theorem 3.1).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

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 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

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

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

Intermediate Semigroups are Groups

We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group.Comment: Plain TeX, 6 pages; final version accepted for publication in Semigroup Foru

A Note on the Arrangement of Subgroups in the Automorphism Groups of Submodule Lattices of Free Modules

A complete description of subgroups in the general linear group over a semilocal ring containing the group of diagonal matrices was obtained by Z.I.Borewicz and N.A.Vavilov. It is shown in the present paper that a similar description holds for the intermediate subgroups of the group of all automorphisms of the lattice of right submodules of a free finite rank R-module over a simple Artinian ring containing the group consisting of those automorphisms which leave invariant an appropriate sublattice.Comment: AmsTeX, 6 pages, compile twice; revised version of the Bielefeld preprint 99-00

Galois Theory for a Class of Complete Modular Lattices

We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over a semilocal ring containing the group of diagonal matrices, due to Z.I.Borewicz and N.A.Vavilov, can be obtained as a consequence of this theory.Comment: AmsTeX, 4 pages; Translation into English from Zap. Nauchn. Semin. POMI 236 (1997), 129-132, by A. Pani

Electronic Fock space as associative superalgebra

New algebraic structure on electronic Fock space is studied in detail. This structure is defined in terms of a certain multiplication of many electron wave functions and has close interrelation with coupled cluster and similar approaches. Its study clarifies and simplifies the mathematical backgrounds of these approaches. And even more, it leads to many relations that would be very difficult to derive using conventional technique. Formulas for action of the creation-annihilation operators on products of state vectors are derived. Explicit expressions for action of simplest particle-conserving products of the creation-annihilation operators on powers of state vectors are given. General scheme of parametrization of representable density operators of arbitrary order is presented.Comment: LaTex, 26 pages; one equation added, misprints remove