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Register a new userA Witt-Burnside ring attached to a pro-dihedral group
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Authors
Lance Edward Miller
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The ring of classic Witt vectors is a fundamental object in mixed characteristic commutative algebra which has many applications in number theory. There is a significant generalization due to Dress and Siebeneicher which for any profinite group G produces a ring valued functor W_G, where the classic Witt vectors are recovered as the example G = Z_p. This article explores the structure of the image of this functor where G is the pro-2 group formed by taking the inverse limit of 2-power dihedral groups, and the image of W_G is taken on a field of characteristic 2.
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In this paper, we construct a $q$-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where $q$ ranges over the set of integers. When $q=1$, it coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. {70} (1988), 87-132). To achieve our goal we first show that there exists a $q$-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case $q=1$, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199 (1998), 703-732). Finally, we classify $mathbb W_G^q$ up to strict natural isomorphism in case where $G$ is an abelian profinite group.
The rings of $p$-typical Witt vectors are interpreted as spaces of vanishing cycles for some perverse sheaves over a disc. This allows to localize an isomorphism emerging in Drinfelds theory of prismatization [Dr], Prop. 3.5.1, namely to express it as an integral of a standard exact triangle on the disc.
In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.
In this paper we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field of characteristic different from 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. We show that the triviality of the Clifford algebra of a bilinear space over C gives the main relation. The calculation is then completed using classical results for bilinear spaces over fields.
Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodskys h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusies de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.
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