Collectanea Mathematica最新文献

An incomplete argument for Theorem 1.4 of Stockdale et al. (Collect Math 73(3):535–563, 2022) is corrected. The validity of Stockdale et al. (Collect Math 73(3):535–563, 2022, Theorem 2.7) remains open.

We give characterizations of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over the group algebra for large families of infinite groups and show that every weak Gorenstein projective, weak Gorenstein flat and weak Gorenstein injective module is Gorenstein projective, Gorenstein flat and Gorenstein injective, respectively. These characterizations provide Gorenstein analogues of Benson’s cofibrant modules. We deduce that, over a commutative ring of finite Gorenstein weak global dimension, every Gorenstein projective module is Gorenstein flat. Moreover, we study cases where the tensor product and the group of homomorphisms between modules over the group algebra is a Gorenstein module. Finally, we determine the Gorenstein homological dimension of an ({{textbf {LH}}}mathfrak {F})-group over a commutative ring of finite Gorenstein weak global dimension.

We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by inert maps. We show that left Kan extension along the inclusion takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of (infty )-categories . Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani and Kiani on the Castelnuovo–Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.

In this paper, we give formulas for ({textrm{v}})-number of edge ideals of some graphs like path, cycle, 1-clique sum of a path and a cycle, 1-clique sum of two cycles and join of two graphs. For an ({mathfrak {m}})-primary monomial ideal (Isubset S=K[x_1,ldots ,x_t]), we provide an explicit expression of ({textrm{v}})-number of I, denoted by ({textrm{v}}(I)), and give an upper bound of ({textrm{v}}(I)) in terms of the degree of its generators. We show that for a monomial ideal I, ({textrm{v}}(I^{n+1})) is bounded above by a linear polynomial for large n and for certain classes of monomial ideals, the upper bound is achieved for all (nge 1). For ({mathfrak {m}})-primary monomial ideal I we prove that ({textrm{v}}(I)le {text {reg}}(S/I)) and their difference can be arbitrarily large.

This is a study of the sequences of Betti numbers of finitely generated modules over a complete intersection local ring, R. The subsequences ((beta ^R_i(M))) with even, respectively, odd i are known to be eventually given by polynomials in i with equal leading terms. We show that these polynomials coincide if ({{I}{}^{scriptscriptstyle square }}), the ideal generated by the quadratic relations of the associated graded ring of R, satisfies ({text {height}}{{I}{}^{scriptscriptstyle square }} ge {text {codim}}R -1), and that the converse holds if R is homogeneous or ({text {codim}}R le 4). Subsequently Avramov, Packauskas, and Walker proved that the terms of degree (j > {text {codim}}R -{text {height}}{{I}{}^{scriptscriptstyle square }}) of the even and odd Betti polynomials are equal. We give a new proof of that result, based on an intrinsic characterization of residue rings of c.i. local rings of minimal multiplicity obtained in this paper. We also show that that bound is optimal.

In this paper, we introduce a class of infinite Lie conformal algebras ({mathfrak {B}}(alpha ,beta ,p)), which are the semi-direct sums of Block type Lie conformal algebra ({mathfrak {B}}(p)) and its non-trivial conformal modules of ({mathbb {Z}})-graded free intermediate series. The annihilation algebras are a class of infinite-dimensional Lie algebras, which include a lot of interesting subalgebras: Virasoro algebra, Block type Lie algebra, twisted Heisenberg–Virasoro algebra and so on. We give a complete classification of all finite non-trivial irreducible conformal modules of ({mathfrak {B}}(alpha ,beta ,p)) for (alpha ,beta in {mathbb {C}}, pin {mathbb {C}}^*). As an application, the classifications of finite irreducible conformal modules over a series of finite Lie conformal algebras ({mathfrak {b}}(n)) for (nge 1) are given.

In this paper we show the existence of cones over a 8-dimensional rational sphere at the boundary of the Mori cone of the blow-up of the plane at (sge 13) very general points. This gives evidence for De Fernex’s strong (Delta )-conjecture, which is known to imply Nagata’s conjecture. This also implies the existence of a multitude of good and wonderful rays as defined in Ciliberto et al. (Clay Math Proc 18:185–203, 2013).

This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces exhibit. Our method is a detailed analysis of a general purpose formula by Ranestad and Sturmfels in the case of smooth real algebraic surfaces of low degree (that are rational over the complex numbers). We study both the complex and the real features of the algebraic boundary of Veronese and Del Pezzo surfaces. The main difficulties and the possible approaches to the case of general surfaces are discussed for and complemented by the example of Bordiga surfaces.

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index ( 10,000 ).