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We study the category of GL -equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology. [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame G-local systems over X and logarithmic G-Higgs bundles over the Frobenius twist X ′ . To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence. [ABSTRACT FROM AUTHOR]

We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension k ≤ k ′ is a right adjoint on the category of k -coalgebras). [ABSTRACT FROM AUTHOR]

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Nearly 60 years ago, Pfister defined what are now called Pfister forms in quadratic form theory. In addition to their remarkable intrinsic properties, Pfister forms are related to symbols in Galois cohomology and K-theory modulo 2, and are at the heart of the Milnor conjecture. In this paper, we intend to show their importance by means of three explicit examples. They also illustrate the evolution of quadratic form theory from its algebraic aspects to geometry of quadrics—more generally of varieties of isotropic subspaces of a given dimension—and their Grothendieck–Chow motives. [ABSTRACT FROM AUTHOR]

Basing ourselves on the work of Conrad-Gabber-Prasad, we study retract rationality of some pseudo-reductive groups and calculate the Picard groups of some pseudo-semisimple groups. [ABSTRACT FROM AUTHOR]

We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine $\mathbb {Z}[t]$ -groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting $\mathbb {F}_p(t)$ -groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their $\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $ -models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g = Lie(G) . It turns out that G has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for g , which says that the G-invariant ring F [ g ] G is a polynomial ring if g satisfies a certain "positivity" condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras. [ABSTRACT FROM AUTHOR]

For a split maximal torus T of a split spin group G = Spin (n) over an arbitrary field, we consider the restriction homomorphism of the Chow rings of their classifying spaces with W the Weyl group of G. For n ≤ 6 , f is known to be surjective. For n ≥ 7 , an obstruction for an element of to be in the image of f is given by the Steenrod operations on. Using it, we show that several standard generators of , including the defined for even n Euler class , are outside the image of f. This result differs from the analogues topological result. [ABSTRACT FROM AUTHOR]

In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties. [ABSTRACT FROM AUTHOR]

In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle, and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincaré transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov's original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov's theorem depends on the structure of quantities. We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincaré models of Borisov's axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields. As an application, we note that there is a model of Borisov's axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time's arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov's axioms, and neither the assumption of time's arrow, nor the rejection of instantaneous motion, can eliminate them. [ABSTRACT FROM AUTHOR]

We construct explicit Sullivan’s minimal models of the rational polynomial de Rham complexes of the classifying spaces B Γ of torsion-free virtually polycyclic groups Γ with values in certain local systems by proving a Nomizu-type theorem on the simplicial de Rham cohomology of B Γ . [ABSTRACT FROM AUTHOR]

Let k ′ / k be a finite purely inseparable field extension and let G ′ be a reductive k ′ -group. We denote by G = R k ′ / k (G ′) , the Weil restriction of G ′ across k ′ / k , a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical R u (G k ¯ ) in terms of invariants of the extension k ′ / k , starting with the case G ′ = GL n and applying these results to the case where G ′ is a simple group. [ABSTRACT FROM AUTHOR]

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with Z / 2 -coefficients of the Nisnevich classifying space of the spin group S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from I 3 , which are closely related to S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between S p i n 7 and G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971). [ABSTRACT FROM AUTHOR]

For a subring $R$ of the rational numbers, we study $R$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in ${\mathbb {A}}^1$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in ${\mathbb {A}}^1$ -homotopy theory, paying attention to future applications for vector bundles. We show that $R$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $BGL_n$ is ${\mathbb {A}}^1$ -nilpotent when $n$ is odd, and analyze the (more complicated) situation where $n$ is even as well. We establish analogs of various classical results about rationalization in the context of ${\mathbb {A}}^1$ -homotopy theory: if $-1$ is a sum of squares in the base field, ${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we show that if G is a smooth nilpotent group acting on an algebraic variety V defined over an admissible valued field k and v ∈ V (k) , then the Zariski closedness of the geometric orbit G (k ¯). v in V (k ¯) is equivalent to the Hausdorff closedness of the rational orbit G (k). v in V(k). Second, we provide some calculations for the fact that there is a bijection between the set of G(k)-orbits and the kernel of the natural map in flat cohomology. These results are obtained in the framework of studying the rational orbits. [ABSTRACT FROM AUTHOR]

We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group. [ABSTRACT FROM AUTHOR]

By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set H 1 (R , G) of a quasi-connected reductive group G over the field R of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G. [ABSTRACT FROM AUTHOR]

Let K be a local field – i.e. the field of fractions of a complete DVR A whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic K -group G which splits over an unramified extension of K . We study the rational nilpotent orbits of G– i.e. the orbits of G (K) in the nilpotent elements of Lie (G) (K) – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over K is unramified if there is a reductive model M over A for which M = M K . Our main result shows for any nilpotent element X1 ∈Lie(G) that there is an unramified, reductive K -subgroup M which contains a maximal torus of G and for which X1 ∈Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G. [ABSTRACT FROM AUTHOR]