We compute the derived functors associated with the ideal of compact operators on Banach spaces and analyze the resulting “compact” short exact sequences. This leads to new results on the extension and lifting of compact operators.
This paper is the second in a series devoted to describing the integral Chow ring of the moduli stacks RHg of hyperelliptic Prym pairs. For fixed genus g, the stack RHg is the disjoint union of ⌊(g+1)/2⌋ components RHgn for n=1,…,⌊(g+1)/2⌋. In this paper, we give presentations and compute the integral Chow rings of the components RHg(g+1)/2 for odd g. As an application, we also obtain presentations and Chow rings for all irreducible components of the moduli stack of hyperelliptic Spin curves of odd genus. An intermediate result of independent interest is the computation of the integral Chow ring of the moduli stack of unordered pairs of divisors of the same even degree in P1.
In this paper we analyze the properties of tame nodal stacky curves, which include twisted curves and doubly-twisted curves. Our main results are a complete classification of the possible structures of a tame stacky node, along with computations of the Picard and Brauer groups of nodal stacky curves.
We show how to quickly find a non-square in Fpn for all n given a non-square in Fq for some power q of a prime number p. For those q for which it is easy to find a non-square in Fp, e.g. when −1 is a non-square, this produces a non-square in Fpn for all n.
Let f(x) = (x(k) + c)(m)-ax(n) is an element of & Zopf;[x] be an irreducible polynomial over & Qopf;, where k, m, n is an element of & Nopf; with km > n, and let K = & Qopf;(9), where 9 is a root of f (x). We investigate the arithmetic properties of the number fields that arise from this family. We first obtain an explicit formula for the discriminant of f (x). Using this formula, we establish necessary and sufficient conditions for the monogeneity of f (x), expressed in terms of the prime divisors of a and c and the parameters k, m, n. This yields infinite families of monogenic polynomials of arbitrary degree, including families with a non-square-free discriminant. Building on these results, we extend our algebraic characterization to composite polynomials, establishing some explicit conditions for the monogeneity of the composition of f(x) with an arbitrary polynomial g(x). From an analytic point of view, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. We further study non-monogeneity via the field index i(K) and, for each prime p, provide sufficient conditions ensuring nu(p)(i(K)) = 1, yielding partial progress toward a problem of Narkiewicz. We also highlight a connection with a class of differential equations naturally associated with f (x). As an application, we determine the conditions under which the splitting field of f (x) has a full symmetric Galois group. Several explicit examples illustrate our results. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
For a module G over a ring R, the concepts of iso-noetherian and iso-artinian are studied. Particularly it is shown that iso-noetherian modules over perfect rings are noetherian and the rest of paper is devoted to the case where R=Z, so that G is an abelian group. If G is such a group with torsion T and A=G/T, it is shown that G has either property if and only if it splits as T⊕A where both T and A have the corresponding property. The torsion groups satisfying either property are completely characterized, and when A is a Butler group, a complete description of when it is either iso-noetherian or iso-artinian is given.
This paper investigates the special linear groups over polynomial rings over arithmetical rings. Let R be an arithmetical ring. We proved the special linear group over R[t1,…,tn] is equal to the product of the special linear group over R and the group of elementary matrices over R[t1,…,tn], which is K1-analogue of Lequain-Simis Theorem.
The purpose of this paper is to define an effective multiplicity for real algebra limited to a local ring. The multiplicity for a filtration of ideals is defined that generalizes Samuel's multiplicity for an ideal in a local ring. Real filtrations presented include those attributed to Brumfiel, Stengle and the canonical filtration of the semi-algebraic closure of an ideal. The application of multiplicity to measure semi-definiteness in the real plane determines that polynomials due to Motzkin and Robinson have multiplicity 4 at each zero. The best estimate for Stengle's polynomial is that its multiplicity lies between 8 and 32.
Let R be a commutative Noetherian ring and let M be a finitely generated Rmodule. Writing Q := I(M) for the first nonzero Fitting ideal of M, and N for the quotient of M by regular torsion, we study the reflexive defect module D-Q(M) := Coker(delta(N)), where delta(N) : N -> ** is the canonical bidual map. We show that Q acts as a support-theoretic control ideal for M, its regular-torsion submodule T(M), the quotient N, and the defect D-Q(M): outside V(Q) the modules M and N are free and T(M) vanishes, while Supp(T(M)) and Supp(D-Q(M)) are contained in V(Q). Moreover, QT(M) = 0 and some power of Q annihilates D-Q(M). When every prime containing Q has height at least two, M and N are locally free in codimension one; over Noetherian normal domains this yields the criterion N is reflexive double left right arrow DQ(M) = 0 double left right arrow N satisfies (S-2). In the two-dimensional normal local case, the defect admits a local-cohomological interpretation: D-Q(M) congruent to H-m(1) (N). As a main application, for a Noetherian normal domain A essentially of finite type over a perfect field, the first nonzero Fitting ideal of Omega(A)/k is the Jacobian ideal, so the preceding criterion applies to the quotient of Omega(A)/k by regular torsion. We also treat the primary height-one case over a Noetherian UFD. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We determine the complete list of composition factors, with multiplicities, of the monomial Burnside p-biset functor C BC & times; over the complex field K of characteristic 0. The method relies on a reduction to restriction kernels, transforming the problem into the computation of certain C [Aut(G)]-modules attached to finite p-groups G. We describe these modules explicitly for all G with non-trivial restriction kernel, and identify the corresponding simple functors. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We construct a domain that does not embed in a duo ring. (A ring is duo when its left and right ideals are all two-sided). On the other hand, we show that a large class of rings can be embedded in duo rings. This leads to the construction of a duo ring whose polynomial ring is not semicommutative (definition recalled below). (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We begin by giving a derived characterization of rational singularities for pairs in the sense of Schwede-Takagi. This characterization extends a characterization of rational singularities due to Lank-Venkatesh to pairs on normal varieties over fields of characteristic zero. As an application, we introduce a categorical invariant that measures the failure of rationality for pairs on affine varieties that are locally complete intersections. (c) 2026 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
A monomial curve Cis defined by a sequence of coprime integers 0 = a(0) < a(1) < & centerdot; & centerdot; & centerdot; < a(k) =: d. A gap of this sequence is a(i+1)-a(i)-1. Gruson-Lazarsfeld-Peskine bound (1983) says that reg(C) <= d-k +2, which is equal to the sum of all gaps plus 2. Lvovsky (1996) showed that it is enough to take the sum of two largest gaps plus 2. In this paper, under some specific conditions, we give several new bounds which are better than Lvovsky's bound. Our method relies on the study of Apery sets and Frobenius numbers. From this we can give new criteria to check the (arithmetically) Cohen-Macaulay and Buchsbaum properties of C. Algorithms are provided to check these properties as well as to compute reg(C) and other invariants. We also give an application to studying the structure of sumsets. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The conjugation representation of a finite group G is the complex permutation module defined by the action of G on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for G = SL2(Z/p (R)), where r >= 1 and p >= 5 is a prime, any irreducible representation of G that is trivial on the centre of G is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to p = 2 or 3. We answer the Hain-Tiep question in the affirmative and also prove analogous statements for SL2 and GL(2) over any finite local principal ideal ring with residue field of odd characteristic. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Let Fqn be a finite field with qnelements. In this paper, we employ standard results on character sums, combined with a novel approach involving linearized polynomials, to derive an explicit formula for the number of Fqn-rational points on the affine superelliptic curve yd = F(x), where F(x) is a q-polynomial over Fqn satisfying some particular algebraic conditions. In particular, we provide an explicit formula for the number of Fqn-rational points on the generalized Artin-Schreier curve yd = beta(xqm-alpha x), when d is suitable. (c) 2026 Published by Elsevier B.V.
We study the syzygies of canonical curves of genus g >= 3 over an algebraically closed field F of characteristic p > 0. We provide a new proof of generic Green's Conjecture for p >= ( g +4) /2 . Using the techniques from the even-genus case, we establish a significant case of the Geometric Syzygy Conjecture for the last syzygy space of a general even-genus canonical curve (assuming p > g). In characteristic 0, it was shown in prior work that this case implies the full conjecture. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
