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Let p be a prime. It is fundamental problem to classify the absolute Galois groups G F of fields containing primitive th root unity ξ . In this paper we present several constraints on such , using restrictions cohomology index normal subgroups from N. Lemire, J. Mináč and Swallow module structure partial Euler-Poincaré characteristics, reine angew. Math. 613 (2007), 147–173. section 1 all maximal -elementary abelian-by-order quotients these case > 2, each quotient contains unique closed...

A linear algebraic group G over a field k is called Cayley if it admits map, that is, G-equivariant birational isomorphism between the variety and its Lie algebra. map can be thought of as partial analog exponential map. prototypical example classical "Cayley transform" for special orthogonal SOn defined by Arthur in 1846. stably some r≥0. Here denotes split r-dimensional k-torus. These notions were introduced 2006 Lemire, Popov, Reichstein, who classified simple groups an algebraically...

Let S n denote the symmetric group on letters. We consider n-lattice Anˇ1a fOz1; ... ; znU A Z j P i zia 0g, where acts by permuting coordinates, and its squares n2 nˇ1, Sym 2 3 nˇ1. For odd values of n, we show that nˇ1 is equivalent to Anˇ1 in sense Colliot-Thelene Sansuc (6). Consequently, rationality prob- lem for generic division algebras amounts proving stable multipli- cative invariant field kO3 Anˇ1U (n odd). Furthermore, confirming a conjecture Le Bruyn (16), na are only cases...

Let F be a field containing primitive pth root of unity, and let U an open normal subgroup index p the absolute Galois group GF F. Using Bloch-Kato Conjecture we determine structure cohomology Hn(U, 𝔽p) as 𝔽p[GF/U]-module for all . Previously this was known only n = 1, until recently even H1(U, determined local field, case settled by Borevič Faddeev in 1960s. For when maximal pro-p-quotient T is finitely generated, apply these results to study partial Euler-Poincaré characteristics χn(N)...

Let p>2 be prime, and let n,m positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root unity, we show the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have sparse decomposition. When is additionally subextension cyclic, p^{n+1} extension E'/F, give more refined F_p[Gal(E/F)]-decomposition H^m(G_E,mu_p).

Let ${\mathcal O}$ be a nilpotent orbit in $\mathfrak{so}(p,q)$ under the adjoint action of full orthogonal group ${\rm O}(p,q)$. Then closure (with respect to Euclidean topology) is union and some O}(p,q)$-orbits smaller dimensions. In an earlier work, first author has determined which belong this closure. The same problem for identity component SO}(p,q)^0$ O}(p,q)$ on much harder we propose conjecture describing closures SO}(p,q)^0$-orbits. proved when $\min(p,q)\le7$. Our method indirect...

Let E be a cyclic extension of degree p^n field F characteristic p. Using arithmetic invariants E/F we determine k_mE, the Milnor K-groups K_mE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, show that each indecomposable summand k_mE has Fp-dimension power That powers p^i, i=0,1,...,n, occur suitable examples is shown subsequent paper [MSS2], where additionally main result this becomes an essential induction step determination K_mE/p^sK_mE (Z/p^sZ)[Gal(E/F)]-modules m, s

Abstract Let p be a prime and F field containing primitive -th root of unity. Then for n ∈ N, the cohomological dimension maximal pro- -quotient G absolute Galois group is at most if only corestriction maps are surjective all open subgroups H index . Using this result, we generalize Schreier's formula

Let p be a prime and F field containing primitive pth root of unity. E/F cyclic extension degree G_E &lt; G_F the associated absolute Galois groups. We determine precise conditions for cohomology group H^n(E)=H^n(G_E,Fp) to free or trivial as an Fp[Gal(E/F)]-module. examine when these properties H^n(E) are inherited by H^k(E), k&gt;n, and, analogy with cohomological dimension, we introduce notions freeness triviality. give examples each n in N prescribed dimension.

Let $G$ be $Sl_n, Sp(2n)$ or SO(2n). We consider the moduli space $M$ of semistable principal $G$-bundles over a curve $X$. Our main result is that if $U$ Zariski open subset then there no universal bundle on $U\times X$.

We find upper bounds for the essential dimension of various moduli stacks $\sln$-bundles over a curve. When $n$ is prime power, our calculation computes stack stable bundles exactly and not equal to in this case.

Abstract We find upper bounds on the essential dimension of moduli stack parabolic vector bundles over a curve. When there is no structure, we improve known bound usual stack. Our calculations also give lower semistable locus inside rank r and degree d without structure.

A connected linear algebraic group G is called a Cayley if the Lie algebra of endowed with adjoint G-action and variety conjugation are birationally G-isomorphic. In particular, classical map, X \mapsto (I_n-X)/(I_n+X), between special orthogonal SO_n its so_n, shows that group. an earlier paper (see math.AG/0409004) we classified simple groups defined over algebraically closed field characteristic zero. Here consider new numerical invariant G, degree, which "measures" how far from being...

We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root unity, using restrictions cohomology index p normal subgroups from previous paper by three authors. first classify all maximal p-elementary abelian-by-order quotients such G_F. In case p&gt;2, each quotient contains unique closed elementary abelian subgroup. This seems to be in which one can completely nontrivial characteristic subgroups. then derive analogues theorems Artin-Schreier...

We investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. determine conditions which such sheaf is sheaf. then produce some examples Severi Brauer flag varieties and arithmetic toric in our method produces sheaf, adding to list positive results literature. also counterexamples show that pushfoward need not be

We determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, holds degree n when the cohomological dimension of group maximal p-extension F at most n.