Automorphism-liftable modules

In this paper, we describe all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings. We also study automorphism-liftable non-torsion modules over not necessarily commutative Dedekind prime rings.

Proper classes generated by τ- closed submodules

The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

The generalized Fermat equation with exponents 2, 3,

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(𝔟⋉V2)

Let 𝔟 be the Borel subalgebra of the Lie algebra 𝔰𝔩2 and V2 be the simple two-dimensional 𝔰𝔩2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product 𝔟⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime…”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(𝔟⋉V2)-modules are classified: the Whittaker modules, the 𝕂[X]-torsion modules and the 𝕂[E]-torsion modules.

Triangulated Matlis equivalence

This paper is a sequel to [L. Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220(12) (2016) 3866–3909, arXiv:1503.05523 [math.CT]; Contraadjusted modules, contramodules, and reduced cotorsion modules, preprint (2016), arXiv:1605.03934 [math.CT]]. We extend the classical Harrison–Matlis module category equivalences to a triangulated equivalence between the derived categories of the abelian categories of torsion modules and contramodules over a Matlis domain. This generalizes to the case of any commutative ring [Formula: see text] with a fixed multiplicative system [Formula: see text] such that the [Formula: see text]-module [Formula: see text] has projective dimension [Formula: see text]. The latter equivalence connects complexes of [Formula: see text]-modules with [Formula: see text]-torsion and [Formula: see text]-contramodule cohomology modules. It takes a nicer form of an equivalence between the derived categories of abelian categories when [Formula: see text] consists of nonzero-divisors or the [Formula: see text]-torsion in [Formula: see text] is bounded.

Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.