15,069 papers in this slice of arXiv.
Valentin Havlovec
We give an example of a finite ring for which the set of polynomials inducing the zero function fails to be a two-sided ideal, disproving a conjecture of Werner.
M. H. Shahzamanian
The determinant of a finite semigroup is closely related to the Frobenius property of its corresponding semigroup algebra, and it plays a significant role in coding theory applications. In this paper, we study determinants of twisted contracted semigroup algebras over finite fields. We extend Steinberg's determinant theory to this setting by establishing a Frobenius criterion for twisted contracted semigroup algebras over finite fields. We also develop a structural theory for two-generated twisted contracted monoid algebras. In particular, we identify classes for which the determinant of the associated matrix depends only on its zero--nonzero pattern, and is therefore independent of the values of the cocycle. Consequently, it suffices to verify the determinant condition for the trivial character, from which the corresponding result for all characters follows. Under suitable assumptions, these semigroups are completely determined by a small set of defining identities. Motivated by the character-theoretic construction of homogeneous weights on finite Frobenius rings, we construct homogeneous weights for Frobenius twisted contracted semigroup algebras and investigate the non-Frobenius case. In particular, we characterize the principal ideals on which the character-average weight satisfies the averaging property of homogeneous weights and thereby extend the classical notion of homogeneous weight to a prescribed family of principal ideals.
Sumit Chandra Mishra, Dibyendu Mondal, Pankaj Shukla
In this article, we characterize the isotropy groups of certain special monomial and Jouanolou-type derivations of polynomial rings over fields of characteristic zero. Under suitable conditions, we determine the structure of these isotropy groups.
Sudipta Das, Jonathan Montaño, Aniketh Sivakumar
We give a convex-geometric formula for the multiplicity sequence of a monomial ideal in terms of mixed volumes of polytopes constructed from its Newton polyhedron. We also construct a counterexample to a conjecture of Achilles and Manaresi proposing a different volume formula for the multiplicity sequence. Finally, we derive a mixed-volume formula for the mixed multiplicities of arbitrary monomial ideals.
Viet-Hoang Tran, Dung V. Nguyen, Quang X. Nguyen +2
Let R be a commutative Noetherian ring. We prove that if the one-variable formal power-series ring R[[x]] is a unique factorization domain, then so is the two-variable formal power-series ring R[[x,y]]. This resolves a question raised by Bayart in 1973 for Noetherian coefficient rings. The proof uses the divisor theory of Noetherian normal domains, expressed through finite rank-one reflexive modules.
Janaine Martins, Rosa M. Miró-Roig
In this paper, we prove that the Artinian Gorenstein K-algebra AFs of codimension n, socle degree d and Macaulay dual generator Fs:=ℓ1d+⋯+ℓsd∈K[X1,…,Xn] where ℓ1,⋯,ℓs are general linear forms satisfies the strong Lefschetz property (SLP). This result allows us to study whether the Waring rank of Fs is exactly s. Furthermore, we show that AFs is the doubling of a suitable 0-dimensional scheme ZFs in Pn−1, the so-called tight annihilating scheme of AFs, and we compute the minimal free resolution of AFs in terms of the minimal free R-resolution of I(ZFs). Finally, we determine the linear general Jordan type of AFs.
Mohamed Ali Belabbas
Let A be a finitely generated commutative algebra over a field K of characteristic zero. We prove that every finitely generated Lie subalgebra L⊆DerK(A) whose elements are locally finite on A is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on A, and all its elements are locally finite. A key ingredient is a second theorem of independent interest: every Lie subalgebra of DerK(A) whose elements are locally nilpotent is solvable; when A is reduced, its derived length is at most dimA, and this bound is sharp. For an affine variety X over an algebraically closed field, we deduce that a subgroup of Aut(X) generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this provides an answer to a problem posed by Popov in 2005. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proofs rest on the introduction of a cofinite ideal meeting the closure of every associated point of SpecA: on the subalgebra of derivations vanishing to second order along the corresponding finite subscheme, local finiteness forces local nilpotence. The intersection of that subalgebra with L is therefore solvable by the second theorem, and has finite codimension in L; together with local finiteness of the adjoint action, this yields finite-dimensionality.
Tomasz Kowalczyk
Let (p,q) be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of p is 7 and the highest degree homogenous part is of the form αx7+βx6y for α2+β2=0. We then show that there are no atypical Jacobian pairs such that degp=7 and degq is even and coprime with 7.
Steven Dale Cutkosky
Let R be a d-dimensional Noetherian local ring with maximal ideal mR and I be an ideal of R. The epsilon multiplicity ε(I) of I is shown in a recent paper of Stephen Landsittel to be the limit of Amao multiplicities ε(I)=m→∞limmda(Im,(Im)sat) if the dimension of the nilradical of R^ is less than d. It is shown in this paper that for an arbitrary local ring R which is of finite type over a field, the limit m→∞limmda(Im,(Im)sat) always exists.
Masoud Gharahi
Persistent tensors form a recursively defined class adapted to the substitution method and yield nontrivial lower bounds on tensor rank. Although persistence is not preserved under Kronecker products in general, we prove that it is preserved in the symmetric setting: the Kronecker product of any two symmetric persistent tensors is again persistent. We first establish a global differentiation identity showing that the Hessian matrix of the Kronecker product of arbitrary homogeneous polynomials is the Kronecker product of the Hessian matrices of the factors, up to the normalization dictated by the restitution convention. For persistent factors, we then prove a polarized perfect-power identity for the Hessian determinants of all (n−2)-fold partial polarizations. Combined with the Hessian characterization of symmetric persistence, this yields closure under Kronecker products. As consequences, symmetric persistence is closed under iterated Kronecker products and Kronecker powers.
Likun Xie
We show that any d-dimensional local ring A with a dualizing complex, depthA=d−1, and cyclic deficiency module Kd−1(A) admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection A→Kd−1(A). When A is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module ωA/xA, for any x∈annAKd−1(A) that is regular on A. This recovers a theorem of Tavanfar and Shimomoto in the 3-dimensional quasi-Gorenstein case with K2(A)≅k. We also give examples of section rings satisfying the hypotheses of our theorem.
Zhichao Chen, Yimin Huang
We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all the irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation x2+y2+z2+2yz=kxyz is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if 1≤k≤5 and determine all mutation orbits of these solutions. In particular, multiple orbits occur for each k=1,3, whereas the solutions form a single orbit for each k=2,4,5. We also prove a conjecture proposed by Chen-Li on Laurent mutation invariants of rank 3 cluster algebras with irreducible sign-equivalent exchange matrices, showing that every Laurent mutation invariant is essentially a polynomial in the corresponding basic invariant.
Cristian Anghel, Filip Chindea
Let X be a smooth projective surface with pg=0, and let H be an ample divisor with h0(OX(H))=0, χ(OX(H))≥q, and h1(OX(H))=0. We prove that no rank two bundle E with c1(E)=3H+KX, with the Ulrich value of c2(E), and satisfying h0(E(−H))=0, can arise from an extension 0→OX(H+KX)→E→OX(2H)⊗IZ→0. Thus, for this natural Cayley-Bacharach construction, non-speciality of the polarization is necessary rather than merely convenient. We then study primary Burniat surfaces. We show that every ample and base point free divisor is non-special; consequently, every polarization carries a stable special Ulrich bundle of rank two, and the surface is strictly Ulrich wild with respect to every polarization. We also locate the special ample classes on three numerical rays through KX, compute explicit families on these rays, and analyze a twisted-kernel variant of the construction. The degree bound underlying the non-speciality result overlaps with recent work of Y. Cho, while the global consequences and the no-go theorem are independent.
Jong In Han, Jeyoung Song
In this paper, we show that the tensor rank of the 5×5 permanent tensor is at least 16 over C and its subfields, which matches the known upper bound. The 5×5 permanent tensor is a symmetric tensor corresponding to the monomial x1x2x3x4x5 whose Waring rank is known to be 16. Previously, it was known that its tensor rank is either 15 or 16 by the higher-order Koszul flattening and Glynn's formula.
Bernhard Böhmler, Rene Marczinzik
We give a negative answer to a question of Avramov, Buchweitz and Şega by constructing a commutative local finite-dimensional non-Gorenstein algebra R with ExtR1(D(R),R)=0; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, Şega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra R over F2 and an indecomposable non-projective R-module M such that ExtR1(M,M)=ExtR2(M,M)=0, related to the second Tachikawa conjecture and answering a question of Dao.
Weiqing Li
We prove that all semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. It turns out that the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture hold for virtually Gorenstein Artin algebras. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This allows us to show the validity of the Auslander-Reiten Conjecture for (infinitely generated) modules over Artin algebras such that all Gorenstein projective modules are projective.
Mohsen Asgharzadeh, Shravan Patankar
Let R be an excellent local domain. R is said to be NBIM if ToriR(R+,k)=0 for some i≥d:=dim(R). Bhatt, Iyengar, and Ma ask if equi-characteristic zero NBIM rings are regular. If R is of positive characteristic, Asgharzadeh and Mahdavi conjecture that ExtRi(k,R∞)=0 for some i>d implies that R is regular. It is an open question whether R+ and R∞ are m-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric NBIM rings are regular and solves the conjecture for F-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is R→S finite and flat on the punctured spectrum and S is regular, this uses Cohen-Macaulayness of S+.
Sunil Rampuria, Ruddarraju Amrutha, Pratyusha Chattopadhyay
A. A. Suslin and V. I. Kopeiko proved that the elementary linear, symplectic, and orthogonal groups are normal in the general linear group, symplectic group, and orthogonal group respectively. They also proved relative versions of these normality results with respect to an ideal of a ring. A. Bak, R. Basu, and R. A. Rao proved that the linear, symplectic, and orthogonal transvection groups are normal in the respective automorphism groups. These results are generalizations of the normality results by Suslin and Kopeiko in the setup of modules. In this paper, we prove stronger versions of the normality results proved by Bak, Basu, and Rao, which say that the relative transvection groups are normal in the respective automorphism groups.
Desiree Martin
For any ideal I (whose projective dimension need not be finite) in a local Noetherian ring R, we show that, if the conormal module I/I2 has a free summand given by F, the natural map from ⋀F to Tor∗R(R/I,R/I) splits as algebras. To achieve this, we use dg algebra techniques and remodel results of André and Iyengar, proving them in setting of semi-free extensions, replacing the need for semi-free Γ-extensions. In this new setting, we show that free summands of the conormal module correspond to central elements of the relative homotopy Lie algebra π∗(φ) and, lastly, we provide an explicit splitting morphism in lower degrees.
Dimitra-Dionysia Stergiopoulou
We study the Gorenstein homological dimension GhdkG of groups G which are of type FP∞ over a commutative ring k. Our main technical result shows that, when the Gorenstein weak global dimension of a ring R is finite, every finitely presented Gorenstein flat R-module is projectively coresolved Gorenstein flat. Consequently, for a group G of type FP∞ over k with sflik<∞, the three natural dimensions for G, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel′dman's theorem, a formula for iterated m-fold self-extensions of a group N of type FP∞ over Z, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type FP∞ over a principal ideal domain is realized after passing to a suitable field.