Related papers: Powers of the Phantom Ideal

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra $V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of $V_{\mathcal{E}}(\ell)$ equipped with…

Given a finite group scheme $\cG$ over an algebraically closed field $k$ of characteristic $\Char(k)=p>0$, we introduce new invariants for a $\cG$-module $M$ by associating certain morphisms $\deg^j_M : U_M \lra \Gr_d(M) \ \…

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$…

Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\bar{\rho}_f$ attached to $f$ and its twist $\bar{\rho}_{f, D}$ by the quadratic character $\chi_D$ corresponding to a…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if…

We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in…

Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree bigger than one in the class group of $K$. In particular, we explore such extensions…

Let $G$ be the automorphism group of an extension $F|k$ of algebraically closed fields of characteristic zero and of transcendence degree $n$, $1\le n\le\infty$. In this paper we (i) construct some maximal closed non-open subgroups $G_v$,…

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…

Let $\mathfrak F$ be a locally compact nonarchimedean field of positive residue characteristic $p$ and $k$ a field of characteristic $p$. Let $G$ be the group of $\mathfrak{F}$-rational points of a connected reductive group over…

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among…

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit…

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the connected part of the…

Given a finite root system $\Phi$, we show that there is an integer $c=c(\Phi)$ such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with root system $\Phi$ and any irreducible rational $G$-modules $L,L'$. There also is…

Throughout this paper, $R$ is an associative ring (not necessarily commutative) with identity and $M$ is a right $R$-module with unitary. In this paper, we introduce a new concept of $\phi$-prime submodule over an associative ring with…

Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. We introduce \emph{$F$-special subgroups} and \emph{$F$-special elements} of $G$. In the case where $F$ contains a $p$th primitive root of unity for each odd…

Let $T$ be a right exact functor from an abelian category $\mathscr{B}$ into another abelian category $\mathscr{A}$. Then there exists a functor ${\bf p}$ from the product category $\mathscr{A}\times\mathscr{B}$ to the comma category…

Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$ and $I$ is an ideal in $A$. Then M. P. Murthy conjectured that $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. Recently, Fasel \cite{F}…

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed…