group schemes
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Author(s):  
Johannes Anschütz

Abstract We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.


2021 ◽  
Vol 73 (4) ◽  
Author(s):  
Indranil Biswas ◽  
Phùng Hô Hai ◽  
João Pedro Dos Santos

2021 ◽  
Vol 8 (31) ◽  
pp. 971-998
Author(s):  
Dave Benson ◽  
Srikanth Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ⩾ 3 p\geqslant 3 , starting with a definition of a π \pi -point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π \pi -points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k [ t , τ ] / ( t p − τ 2 ) k[t,\tau ]/(t^p-\tau ^2) , where t t has even degree and τ \tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.


Obiter ◽  
2021 ◽  
Vol 33 (1) ◽  
Author(s):  
N Whitear-Nel ◽  
Matthew Rudling

The concept of constructive dismissal is flexible because the circumstances that may give rise to it are “so infinitely various” (Minister of Home Affairs v Hambidge 1999 20 ILJ 2632 (LC) par 12). As such, there are no clear rules defining precisely when a constructive dismissal has taken place. The facts of each case must be established, interpreted and measured against general principles to determine whether the requirements for constructive dismissal have been met. The Labour Appeal Court (LAC), in the case of Jordaan v CCMA (2010 31 ILJ 2331 (LAC) 2335), made the point that the law has attained more certainty since Hambidge’s case. This is partially true. However, this case note shows that it remains difficult to set down hard and fast rules to determine the existence of a constructive dismissal. The Supreme Court of Appeal (SCA) has held that very strict proof of constructive dismissal is required, and it has not readily found that circumstances complained of by employees constitute such a dismissal. In the case of Old Mutual Group Schemes v Dreyer (1999 20 ILJ 2030 (LAC)) Conradie JA cautioned that constructive dismissal is not for the asking. He held that generally it will be difficult for an employee who resigns to show that he has actually been constructively dismissed, because the onus of proof on the employee in this regard is a heavy one. Jordaan’s case highlights just how hard it is to establish a viable claim of constructive dismissal. It shows that even where an employee experiences a loss of job security as a result of attempts by the employer to protect his business, and this leads to the employee’s resignation, it will not rise to the standard of constructive dismissal. The LAC saw Jordaan’s case as an attempt to “stretch the law relating to constructive dismissal” and held that this was not only inappropriate but that such an attempt “should not be contemplated” by future courts.


Author(s):  
Heer Zhao

Abstract We compare the Kummer flat (resp., Kummer étale) cohomology with the flat (resp., étale) cohomology with coefficients in smooth commutative group schemes, finite flat group schemes, and Kato’s logarithmic multiplicative group. We are particularly interested in the case of algebraic tori in the Kummer flat topology. We also make some computations for certain special cases of the base log scheme.


Author(s):  
Kay Rülling ◽  
Shuji Saito

Abstract We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.


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