Let p be a prime and let R=ℤ_p[C_p^2] , where ℤ_p is the ring of p -adic integers and C_p^2 is the cyclic group of order p^2 . We study the equivalence relation on finite R -modules introduced by Greither and Kataoka through the projective-equivalence classes of first syzygies. We show that the Fitting class of a module—its Fitting ideal up to a principal fractional factor—is the determinant class of its syzygy. Using Ernst Dieterich’s classification of indecomposable R -lattices, we determine the unique minimal generating system of the syzygy-vector monoid by an independent minimal-zero-sum argument. This corrects the previously stated count p^2+p+1 to p^2+2p-1 : exactly 3p generators have rank one and p^2-p-1 have rank two, and none has larger rank. Every reducible rank-two element has a unique factorization, up to order, into two rank-one elements. Consequently, for p≥ 5 , there exist non-equivalent finite R -modules whose Fitting ideals are isomorphic as R -modules.
Let G be a finite group and h_m(G) be the harmonic mean of element orders of G. In this short note, we prove that if h_m(G)<60/23 , then G is solvable.
First we construct a cubic 4-fold whose singularities are 11 cusps and which has an action of the Mathieu group M_11 , all over the ternary field 𝔽_3 . We next consider a certain moduli space of bundles on a supersingular K3 surface of Artin invariant one in characteristic 3. We show that it has 275 (- 2) Mukai vectors which form the McLaughlin graph, and ask questions on it and on its relation with our M_11 -cubic 4-fold.
Let k be a characteristic zero field. Let 𝒞 be an integral affine plane k-curve. In this article, we show that the dual morphism Ψ ^3_𝒪(𝒞) of the canonical morphism of 𝒪(𝒞) -modules (introduced in Le Dréau and Sebag in Osaka J Math 61(3):381–390, 2024) Φ ^3_𝒪(𝒞):Ω _𝒪(𝒞)/k^3→𝒲^3_𝒪(𝒞) , defined from the 𝒪(𝒞) -module of the third-order differential forms on 𝒞 to the third-degree component of the weight grading of the k-algebra 𝒪(ℒ_∞(𝒞)) of the associated arc scheme, is injective. We describe its image and prove that this morphism Ψ ^3_𝒪(𝒞) is not surjective in general. However, we show that the surjectivity can occur.
A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using techniques from commutative algebra. Our approach is to obtain a free resolution of the Milnor algebra through the Koszul complex. The desired formula is then obtained from a Hilbert series calculation.
In this paper, we characterize all fields of the form 𝕂:=ℚ(√(pr), √(pq)) and 𝕃:=ℚ(√(pr), √(pq), √(2)) whose 2-class groups are cyclic, where p≡ 5 8, q≡ 3 4 and r≡ 3 8 are pairwise different primes. Furthermore, as application we study the cyclicity of the unramified Iwasawa module of 𝕂 .
Tarski’s language for plane absolute geometry has points, equidistance, and betweenness as the only primitive notions. We present an axiom system for plane absolute geometry in Tarski’s language that takes Euclid’s Fourth Postulate as an axiom, but has no congruence axioms of the Side-Angle-Side type for arbitrary triangles. The axiom system presented here validates a statement made by Beppo Levi in 1947.
Given a complete Riemannian n-manifold with sectional curvature bounded between two positive constants, the Dirichlet eigenvalue problem on a domain in this manifold is investigated. The gaps between consecutive eigenvalues are estimated to confirm the Chen–Zheng–Yang conjecture (Pac J Math 282(2):301–315, 2016) for n ≥ 3 .
We provide a free resolution of the symmetric algebra of the ideal of an (n+1) -generated ideal of depth n over a Noetherian ring. The resolution is given in terms of the resolution of the ideal itself and of a Buchsbaum-Rim complex associated to an explicit map.
Let G be a finite group. The aim of this paper is to study the number of solutions S⊆ G of the equation ℧^{n}(S)=L, where L is a non-empty subset of G, n is a positive integer and ℧^{n}(S)={ s^n | s∈ S}. Besides our findings obtained in this general frame, we also outline some results which hold for some particular cases such as: i) L is a normal subset of G; ii) G is abelian; iii) G is an extraspecial p-group.
We tour several Euclidean properties of Poncelet triangles inscribed in an ellipse and circumscribing the incircle, including loci of triangle centers and envelopes of key objects. We also show that a number of degenerate behaviors are triggered by the presence of an equilateral triangle in the family.
Let F be a field and let C be the algebraic closure of F. Let w ∈ M_n(F) be a matrix whose minimal polynomial is of the form g(x)=(x-a_1)(x-a_2) ∈ C[x] , where a_1 a_2 . Let d(x):=wx-xw for all x ∈ M_n(F) . Then S_2n-1(d(x_1),...,d(x_2n-1)) is an identity for M_n(F) .
We describe Minkowski planes in which each reflection about a generator occurs as an automorphism of the Minkowski plane and completely characterize these Minkowski planes by properties of their coordinatizing KT-quasifields. We further investigate topological Minkowski planes with this property and provide examples of Minkowski planes where certain combinations of all symmetries about circles or all reflections about generators exist.
Let ℱ = (…→ ^∂ _n+1ℱ_n → ^∂ _nℱ_n-1→ ^∂ _n-1……→ ^∂ _1ℱ_0 →ℜ→ 0) be a free resolution over the group ring ℜ[Φ ] where ℜ is commutative and Φ is finite. The n^th syzygy Ω _n^ℜ[Φ ] is the stable class of Im(∂ _n) and has a tree structure with roots which do not extend infinitely downwards. We show that Ω _3^ℜ[Q_8p] has infinitely many isomorphically distinct modules at the minimal level when ℜ = ℤ[C_∞ ] is the integral group ring of the infinite cyclic group and Q_8p is the quaternion group of order 8p where p ≥ 3 is prime. This poses severe difficulties in attempting to solve the D(2) problem of CTC Wall for the groups C_∞× Q_8p
Let G be a finite group and p be a prime divisor of |G|. An irreducible p-Brauer character φ of G is called super-monomial if every primitive p-Brauer character inducing φ is linear. The group G is said to be a super M_p-group if every irreducible p-Brauer character of G is super-monomial. In this note, we investigate the conditions under which a finite group G qualifies as a super M_p-group. We demonstrate that every normal subgroup of a super M_p-group of odd order is an M_p-group.
We prove that a projective deformation of a Lagrangian fibration on a projective primitive symplectic variety is unobstructed.
In this paper, we study the homological properties of edge rings of some complete split-like graphs, namely, extended complete split-like graph ℰ𝒞𝒮^a_b and multiple complete split-like graph ℳ𝒞𝒮^a_b,n encoded in the minimal graded free resolution of edge rings of these graphs. In particular, we derive combinatorial formulae for their graded Betti numbers, Castelnuovo-Mumford regularity and the projective dimension. In addition to this, we also discuss, when such graphs are well-covered, vertex decomposable, shellable, Cohen-Macaulay, sequentially Cohen-Macaulay etc. As a consequence, we also obtain the formulae for graded Betti numbers of various families of graphs, namely, complete split graph 𝒞𝒮^a_b , windmill graph Wd(b+1,n) , friendship graph F_n and star graph S_n as a special case.
Complex tight frames can be canonically viewed as elements of a complex Stiefel manifold. We present a class of spaces of such frames which are simply connected relative to the subspace topology. To this class belongs the space of finite unit-norm tight frames.
Let p>0 be a prime, a field of characteristic p and G an elementary abelian p-group of order q=p^n . Let W be an indecomposable G -module of dimension 2 and define V_i = S^i-1(W)^* for each i=1, … , q. We show that V_2 ⊗ V_i ≅ V_i+1⊕ V_i-1 provided i is not divisible by p. Our results generalise results of Almkvist and Fossum (1978) for representations of cyclic groups of order p. We show how our results give formulae for the direct sum decomposition of V_i ⊗ V_j for all i
