Journal of Pure and Applied Algebra最新文献

In this paper, we classify several subcategories of the category of coherent sheaves on a divisorial noetherian scheme (e.g. a quasi-projective scheme over a commutative noetherian ring). More precisely, we classify the torsionfree (resp. torsion) classes closed under tensoring with line bundles by the subsets (resp. specialization-closed subsets) of the scheme, which generalizes the classification of torsionfree (resp. torsion) classes of the category of finitely generated modules over a commutative noetherian ring by Takahashi (resp. Stanley–Wang).

Furthermore, we classify the Serre subcategories of a torsionfree class (in the sense of Quillen's exact categories) by using the above classifications, which gives a certain generalization of Gabriel's classification of Serre subcategories. As explicit applications, we classify the Serre subcategories of the category of maximal pure sheaves, which are a natural generalization of vector bundles for reducible schemes, on a reduced projective curve over a field, and the category of maximal Cohen-Macaulay modules over a one-dimensional Cohen-Macaulay ring.

We consider the restricted Jordan plane in characteristic 2, a finite-dimensional Nichols algebra quotient of the Jordan plane that was introduced by Cibils, Lauve and Witherspoon. We extend results from arXiv:2002.02514 on the analogous object in odd characteristic. We show that the Drinfeld double of the restricted Jordan plane fits into an exact sequence of Hopf algebras whose kernel is a normal local commutative Hopf subalgebra and the cokernel is the restricted enveloping algebra of a restricted Lie algebra $m$ of dimension 5. We show that $u\left(m\right)$ is tame and compute explicitly the indecomposable modules. An infinite-dimensional Hopf algebra covering the Drinfeld double of the restricted Jordan plane is introduced. Various quantum Frobenius maps are described.

The aim of the paper is to investigate the only two families $F_G^A$ of Calabi-Yau 3-folds $A/G$ with A an abelian 3-fold and $G\le \text{Aut}\left(A\right)$ a finite group acting freely: one is constructed in [11] and the other is presented here. We provide a complete classification of the automorphism group of $X\in F_G^A$. Additionally, we construct and classify the quotients $X/\Upsilon$ for any $\Upsilon \le \text{Aut}\left(X\right)$. Specifically, for those groups ϒ that preserve the volume form of X then $X/\Upsilon$ admits a desingularization Y which is a Calabi-Yau 3-fold: we compute the Hodge numbers and the fundamental group of these Y, thereby determining all topological in-equivalent Calabi-Yau 3-folds obtained in this way.

The Nakayama permutations of two derived equivalent, self-injective Artin algebras are conjugate. A different but elementary approach is given to showing that the weak symmetry and self-injectivity of finite-dimensional algebras over an arbitrary field are preserved under derived equivalences.

Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon “filtered colimit elimination,” and study a sufficient condition for it. We show that if a locally finitely presentable category $A$ satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to $A$.

Let G be a finite group and p be a prime. In this paper, we get the sharp bound for $|G/O_p(G){|}_p$ under the assumption that either $p^2\nmid \chi \left(1\right)$ for all $\chi \in \mathrm{Irr}\left(G\right)$ or $p^2\nmid \varphi \left(1\right)$ for all $\varphi \in {\mathrm{IBr}}_p\left(G\right)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet in [7].

In our paper [1], several maps in Section 4 were incorrect. We correct them here. This oversight does not affect any of the results in the paper.

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ^⁎torsionfreeness and ^⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

We prove that an exotic fusion system described by Grazian on a subgroup of the Monster group is block-exotic, thus proving that exotic and block-exotic fusion systems are the same for all p-groups with sectional rank 3, where $p\ge 5$.