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.

The integral geometric Satake equivalence in mixed characteristic.

Abstract: Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^+G}(Gr_G,\Lambda)$ of $L^+G$-equivariant perverse sheaves in $\Lambda$-coefficient on the affine Grassmannian $Gr_G$ where $\Lambda=\mathbb{Z}_{\ell}$ and $\mathbb{F}_{\ell}$ and prove it is equivalent as a tensor category to the category of finitely generated $\Lambda$-representations of the Langlands dual group of $G$.

The resolution of the universal ring for finite length modules of projective dimension two

Abstract: Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0\to \Cal R^{e}\to \Cal R^{f}\to \Cal R^{g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\Cal R\to S$ with $\Bbb V=\Bbb U\otimes_{\Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ by free $\pmb K\otimes_{\Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{\Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2\times 2$ minors of an $e\times g$ matrix.

On the integral solutions of the diophantine equation x4 + y4 = z3

Abstract: This paper is concerned with the existence, types and the cardinality of the integral solutions for diophantine equation 443 xy z += where x , y and z are integers. The aim of this paper was to develop methods to be used in finding all solutions to this equation. Results of the study show the existence of infinitely many solutions to this type of diophantine equation in the ring of integers for both cases, xy = and xy ¹ . For the case when xy = , the form of solutions is given by 334 ( , , ) (4 , 4 ,8 ) xyz n n n = , while for the case when xy ¹ , the form of solutions is given by 31 3 1 4 1 (, , ) ( , , ) k k k x y z un vn n - - = . The main result obtained is a formulation of a generalized method to find