Topic
Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: In this article, the number of orbits of pairs in a finitely generated torsion module over a discrete valuation ring was investigated and the answer was a polynomial in the cardinality of the residue field whose coefficients are integers which depend only on the elementary divisors of the module, and not on the ring in question.
Abstract: We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring The answer is found to be a polynomial in the cardinality of the residue field whose coefficients are integers which depend only on the elementary divisors of the module, and not on the ring in question The coefficients of these polynomials are conjectured to be non-negative integers
5 citations
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TL;DR: In this paper, a triangulation of an R-linear triangulated category was studied for equivariant bivariant K-theory with torsion coefficients.
Abstract: The localisation of an R-linear triangulated category
$\mathcal{T}$
at S
−1
R for a multiplicatively closed subset S is again triangulated, and related to the original category by a long exact sequence involving a version of
$\mathcal{T}$
with coefficients in S
−1
R/R. We examine these theories and, under some assumptions, write the latter as an inductive limit of
$\mathcal{T}$
with torsion coefficients. Our main application is the case where
$\mathcal{T}$
is equivariant bivariant K-theory and R the ring of integers.
5 citations
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TL;DR: In this paper, a Dennis trace map mod n, from K_1(A,Z/n) to the Hochschild homology group with coefficients HH_ 1(A; Z/n).
Abstract: Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and the number of roots of Mirimanoff polynomials.
5 citations
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TL;DR: In this paper, it was shown that compatible systems of -adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification.
Abstract: We show that compatible systems of -adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on -independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.
5 citations
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TL;DR: In this article, it was shown that the Newton polygon lies above the Hodge polygon, itself lying above a certain polygon depending on the datum, and that the total Hasse invariant is non-zero if and only if the $p$-divisible group is $\mu$-ordinary.
Abstract: We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $\mu$ ; this condition consists in a filtration on the sheaf of differentials $\omega_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying above a certain polygon depending on the datum $\mu$. We then construct Hasse invariants for such $p$-divisible groups over an arbitrary base scheme of characteristic $p$. We prove that the total Hasse invariant is non-zero if and only if the $p$-divisible group is $\mu$-ordinary, i.e. if its Newton polygon is minimal. Finally, we study the properties of $\mu$-ordinary $p$-divisible groups. The construction of the Hasse invariants can in particular be applied to special fibers of PEL Shimura varieties models as constructed by Pappas and Rapoport.
5 citations