Journal of Pure and Applied Algebra最新文献

We will prove that if $I$ is a nilpotent ideal of an additive category $A$, then an idempotent e of the category $A$ splits iff its image splits in $A/I$. Based on this fact, we give a short proof of a crucial proposition of Le and Chen.

Let R denote a Noetherian ring and an ideal $J\subset R$ with $U=\mathrm{Spec}R\setminus V\left(J\right)$. For an R-module M there is an isomorphism $\Gamma (U,\tilde{M})\cong \underrightarrow{\lim }{\mathrm{Hom}}_R(J^n,M)$ known as Deligne's formula (see [8, p. 217] and Deligne's Appendix in [7]). We extend the isomorphism for any R-module M in the non-Noetherian case of R and $J=(x_1,\dots ,x_k)$ a certain finitely generated ideal. Moreover, we recall a corresponding sheaf construction.

In this paper, we show that the multiplicative derivations on $J\left(\alpha \right)$-axial algebras, with $\alpha \ne 1,0,\frac{1}{2}$, are additive under suitable conditions, which nowadays are called Martindale-type conditions. Besides, with proper assumptions, we proceed to study the additivity of multiplicative isomorphisms and derivations in the context of $M(\alpha ,\beta )$-axial algebras, except for multiplicative derivations when $\beta =\frac{1}{2}$. In this case, we mention a research question at the end.

We give a parametrization of cyclic pointed categories associated to the cyclic group of order n in terms of n-th roots of unity. We also provide a diagramatic description of these categories by generators and relations, and use it to characterize their 2-group of automorphisms.

We give a geometric interpretation of the Stanley–Reisner correspondence, extend it to schemes, and interpret it in terms of the field of one element.

Let k be an algebraically closed field of characteristic $p>0$ and let $F$ be an algebraically closed field of characteristic 0. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks of finite groups and proved that given a p-group D, there is only a finite number of pairs $(G,b)$ of a finite group G and a block b of kG with defect groups isomorphic to D, up to functorial equivalence over $F$. In this paper, we classify the functorial equivalence classes over $F$ of blocks with cyclic defect groups and 2-blocks of defects 2 and 3. In particular, we prove that for all these blocks, the functorial equivalence classes depend only on the fusion system of the block.

We study the end-behavior of integer-valued $\mathrm{FI}$-modules. Our first result describes the high degrees of an $\mathrm{FI}$-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between $\mathrm{FI}$-tails and finitely supported modules for a new category that we call $\mathrm{FJ}$. Objects of $\mathrm{FJ}$ are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.

In the context of categories equipped with a structure of nullhomotopies, we introduce the notion of homotopy torsion theory. As special cases, we recover pretorsion theories as well as torsion theories in multi-pointed categories and in pre-pointed categories. Using the structure of nullhomotopies induced by the canonical string of adjunctions between a category $A$ and the category $\mathrm{Arr}\left(A\right)$ of arrows, we give a new proof of the correspondence between orthogonal factorization systems in $A$ and homotopy torsion theories in $\mathrm{Arr}\left(A\right)$, avoiding the request on the existence of pullbacks and pushouts in $A$. Moreover, such a correspondence is extended to weakly orthogonal factorization systems and weak homotopy torsion theories.

We show that the semi-strictly generated internal homs of Gray-categories ${[A,B]}_{\text{ssg}}$ defined in [19] underlie a closed structure on the category Gray-Cat of Gray-categories and Gray-functors. The morphisms of ${[A,B]}_{\text{ssg}}$ are composites of those trinatural transformations which satisfy the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible 3-cell. Such trinatural transformations leverage three-dimensional strictification [19] while overcoming the challenges posed by failure of middle four interchange to hold in Gray-categories [3]. As a result we obtain a closed structure that is only partially monoidal with respect to [8]. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories [13], which will be improved further in a forthcoming paper [21].

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{\text{sat}}/I$, where I denotes an ideal generated by $n+1$ generic forms, n is the dimension of X and $I^{\text{sat}}$ is the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms.