arXiv subjects
Publications and source records attributed to A. Stavrova.
Let D be a DVR of mixed characteristic. Let G be a reductive D-group scheme. Then the Grothendieck-Serre conjecture is true for the D-group scheme G and any geometrically regular local D-algebra R. Also we prove a version of Lindel-Ojanguren-Gabber's geometric presentation lemma in the DVR context.
Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the unipotent radicals of two opposite parabolic subgroups P^+ and P^- of G. Assume that 2 is invertible in R if G is of type B_n,C_n,F_4,G_2 and 3 is invertible in R if G is of type G_2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central. In the course of the proof, we construct Steinberg groups associated to isotropic reductive groups and show that they are central extensions of E(R) if R is a local ring.
Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}] containing a maximal R-torus T (equivalently, loop reductive). Assume also that every semisimple normal subgroup of G contains a two-dimensional split torus G_m^2. We show that the natural map of non-stable K_1-functors K_1^G(R)-> K_1^G(k((x_1))...((x_n))) is injective. This complements the surjectivity result for the same map obtained by V. Chernousov, P. Gille and A. Pianzola in arXiv:1109.5236. As a corollary, we provide a way to evaluate the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.
Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme isotropic over R. We prove that for any Noetherian k-algebra A, the map of etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced by the inclusion of R into K, has trivial kernel. This implies the Serre-Grothendieck conjecture for such groups G. The main theorem for A=k and some other results of the present paper are used significantly in arXiv:1211.2678 to prove the Serre-Grothendieck conjecture for all reductive groups over a regular semi-local ring containing an infinite field.
We give a simplified proof of Tits' classification of semisimple algebraic groups that remains valid over semilocal rings. In particular, we provide explicit necessary and sufficient conditions that anisotropic groups of a given type appear as anisotropic kernels of semisimple groups of a given Tits index. We also give a new proof of the existence of all indices of exceptional inner type using the notion of canonical dimension of projective homogeneous varieties.
Assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner adjoint simple algebraic group of type E_6 or E_7 over R, or any twisted form of one of the split groups of classical type O^+_{n,R}, n>=4; PGO_{n,R}, n>=4; PSp_{2n,R}, n>=2; PGL_{n,R}, n>=2. We prove that the kernel of the map H^1_{et}(R,H)-> H^1_{et}(K,H) induced by the inclusion of R into K is trivial. This continues the recent series of papers by the authors and N. Vavilov on the Grothendieck--Serre conjecture.
Earlier an arbitrary poset $P$ was proved to be isomorphic to the collection of subsets of a space $M$ with two closures which are closed in the first closure and open in the other. As a space $M$ for this representation an algebraic dual space $P^*$ was used. Here we extend the theory of algabraic duality for posets generalizing the notion of an ideal. This approach yields a sufficient condition for the collection of clopen subsets of a subset of $P^*$ (with respect to induced closures) to be isomorphic to $P$. Applying this result to certain classes of posets we prove some representation theorems and get a topological characterization of orthocomplementations.