Article
Mathematics
Yuichiro Hoshi
Summary: This study demonstrates that for a given principally quasi-polarizable p-torsion finite flat commutative group scheme over a perfect field of characteristic p, the deformation to the ring of Witt vectors is unique to W if and only if the group scheme is superspecial.
JOURNAL OF NUMBER THEORY
(2021)
Article
Mathematics
Roghayeh Bagherian, Esmaeil Hosseini
Summary: This article introduces the big (or small) finitistic flat dimension of schemes and their properties, proving the necessary and sufficient conditions for the flat dimensions of affine schemes to be finite are finite projective dimensions. It also identifies the minimum requirements for finite flat dimensions, and demonstrates that under certain conditions, the little finitistic projective dimension is finite.
BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Stefano D'Alesio
Summary: This article introduces a derived representation scheme associated with a quiver and shows its applications and some corollaries. On the technical side, the theory related to reductive subgroups of the general linear group and an equivariant version of the derived representation functor for algebras are also discussed.
SELECTA MATHEMATICA-NEW SERIES
(2022)
Article
Mathematics
Giulio Bresciani
Summary: The passage discusses Grothendieck's section conjecture and introduces a variant of essential dimension. It proves related conclusions in the context of pro-finite etale group schemes.
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE
(2021)
Article
Mathematics
Chia-Fu Yu
Summary: This paper investigates type D moduli spaces in positive characteristic p (p≠2), allowing the prime p to ramify in the defining datum. The isogeny classes of p-divisible groups with additional structures are explicitly classified, and the reduction of type D moduli spaces of minimal rank is also studied.
ANNALES DE L INSTITUT FOURIER
(2021)
Article
Mathematics
Paola Frediani, Gian Paolo Grosselli, Abolfazl Mohajer
Summary: In this paper, we construct Shimura subvarieties of dimension bigger than one in the moduli space of delta-polarized abelian varieties of dimension p. We adapt the techniques used to construct Shimura curves to the higher dimensional case by using families of Galois covers of P1. The case of abelian covers is treated in detail, and explicit computations are used to verify the condition for the family to yield a Shimura subvariety of Ap delta${\mathsf {A}}<^>\delta _{p}$.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics
Phung Ho Hai, Joao Pedro dos Santos
Summary: This work studies affine group schemes over a discrete valuation ring using Neron blowups, focusing on a certain class of infinite-type affine group schemes known as Neron blowups of formal subgroups. The study shows how these group schemes naturally appear in Tannakian categories of D-modules, and introduces a Tannakian property named prudence for affine group schemes, which aids in verifying the underlying ring of functions. This property is then successfully applied to derive a general result on the structure of differential Galois groups over complete DVRs.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics
Stefan Reppen
Summary: This article studies the geometric special fiber of a Hilbert modular variety associated to a totally real field, at a prime unramified in the field. The authors show that the order of vanishing of the Hasse invariant on the fiber is equal to the largest integer m such that the smallest piece of the conjugate filtration lies in the mth piece of the Hodge filtration, which is analogous to Ogus' result on families of Calabi-Yau varieties in positive characteristic. They also demonstrate that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Yair Hartman, Bryna Kra, Scott Schmieding
Summary: In this study, we introduce the stabilization of the automorphism group for a mixing shift of finite type and investigate its algebraic properties. We are able to distinguish many of the stabilized automorphism groups using these properties. Additionally, for a full shift, we show that the subgroup generated by elements of finite order in the stabilized automorphism group is simple.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics
Lars Winther Christensen, Sergio Estrada, Li Liang, Peder Thompson, Dejun Wu, Gang Yang
Summary: We introduce a refinement of the Gorenstein flat dimension for complexes over associative rings, called the Gorenstein flat-cotorsion dimension, which behaves like a homological dimension without extra assumptions. It coincides with the Gorenstein flat dimension for finite complexes and for complexes over right coherent rings.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics
Indranil Biswas, Phung Ho Hai, Joao Pedro dos Santos
Summary: This paper proves an analogue of Armstrong's theorem in the setting of F-divided and essentially finite fundamental group schemes.
TOHOKU MATHEMATICAL JOURNAL
(2021)
Article
Mathematics
Najmuddin Fakhruddin, Rijul Saini
Summary: Inspired by recent work, this paper develops a method for obtaining lower bounds for the essential dimension of a cover of a variety using actions of finite group schemes. The method is then applied to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties and make progress towards a conjecture on the p-incompressibility of the multiplication by p map of an abelian variety.
DOCUMENTA MATHEMATICA
(2022)
Article
Mathematics, Applied
Rostislav A. Devyatov, Nikita A. Karpenko, Alexander S. Merkurjev
Summary: This study establishes the sharp upper bounds on the indexes for most of the twisted flag varieties under the spin groups Spin(n).
FORUM OF MATHEMATICS SIGMA
(2021)
Article
Mathematics
Haining Wang
Summary: This paper explains the stratifications of Shimura varieties using the axioms of He and Rapoport, and discusses a result by Gortz, He, and Nie that the EKOR strata within the basic loci can be described as a disjoint union of Deligne-Lusztig varieties. In the special case of Siegelmodular varieties, comparisons are made with descriptions by Gortz and Yu for supersingular Kottwitz-Rapoport strata, as well as descriptions by Harashita and Hoeve for supersingular Ekedahl-Oort strata.
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
(2021)
Article
Mathematics
Qiaoling Guo, Tingting Shan, Bingliang Shen, Tao Yang
Summary: This paper investigates the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B in a right H-Galois extension A/B over a semisimple Hopf algebra H. It is found that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. The problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension is then studied. Finally, more relations for the crossed products and smash products are obtained as applications.