Showing papers on "Ring (mathematics) published in 2015"

FI-modules and stability for representations of symmetric groups

Abstract: In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: • the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold; • the diagonal coinvariant algebra on r sets of n variables; • the cohomology and tautological ring of the moduli space of n -pointed curves; • the space of polynomials on rank varieties of n × n matrices; • the subalgebra of the cohomology of the genus n Torelli group generated by H 1 ; and more. The symmetric group S n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n . In particular, the dimension is eventually a polynomial in n . In this framework, representation stability (in the sense of Church–Farb) for a sequence of S n -representations is converted to a finite generation property for a single FI-module.

GL-equivariant modules over polynomial rings in infinitely many variables

Abstract: Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellen- berg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

Baxter's relations and spectra of quantum integrable models

Abstract: Generalized Baxter's relations on the transfer-matrices (also known as Bax- ter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the cate- gory O introduced by Jimbo and the second author in (HJ) involving infinite-dimensional representations constructed in (HJ), which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Com- bining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (FR1). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.

Chiral algebras for trinion theories

Abstract: It was recently understood that one can identify a chiral algebra in any four-dimensional $$ \mathcal{N}=2 $$ superconformal theory. In this note, we conjecture the full set of generators of the chiral algebras associated with the T n theories. The conjecture is motivated by making manifest the critical affine module structure in the graded partition function of the chiral algebras, which is computed by the Schur limit of the superconformal index for T n theories. We also explicitly construct the chiral algebra arising from the T 4 theory. Its null relations give rise to new T 4 Higgs branch chiral ring relations.

Short Accountable Ring Signatures Based on DDH

Abstract: Ring signatures and group signatures are prominent cryptographic primitives offering a combination of privacy and authentication. They enable individual users to anonymously sign messages on behalf of a group of users. In ring signatures, the group, i.e. the ring, is chosen in an ad hoc manner by the signer. In group signatures, group membership is controlled by a group manager. Group signatures additionally enforce accountability by providing the group manager with a secret tracing key that can be used to identify the otherwise anonymous signer when needed. Accountable ring signatures, introduced by Xu and Yung CARDIS 2004, bridge the gap between the two notions. They provide maximal flexibility in choosing the ring, and at the same time maintain accountability by supporting a designated opener that can identify signers when needed. We revisit accountable ring signatures and offer a formal security model for the primitive. Our model offers strong security definitions incorporating protection against maliciously chosen keys and at the same time flexibility both in the choice of the ring and the opener. We give a generic construction using standard tools. We give a highly efficient instantiation of our generic construction in the random oracle model by meticulously combining Camenisch's group signature scheme CRYPTO 1997 with a generalization of the one-out-of-many proofs of knowledge by Groth and Kohlweiss EUROCRYPT 2015. Our instantiation yields signatures of logarithmic size in the size of the ring while relying solely on the well-studied decisional Diffie-Hellman assumption. In the process, we offer a number of optimizations for the recent Groth and Kohlweiss one-out-of-many proofs, which may be useful for other applications. Accountable ring signatures imply traditional ring and group signatures. We therefore also obtain highly efficient instantiations of those primitives with signatures shorter than all existing ring signatures as well as existing group signatures relying on standard assumptions.

Coupling of chiralities in spin and physical spaces: the Möbius ring as a case study.

Abstract: We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic M\"obius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the M\"obius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the M\"obius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the M\"obius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the M\"obius ring on its geometrical parameters and on the value of the easy-normal anisotropy is explored numerically.

The big de Rham–Witt complex

Abstract: This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a λ-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a λ-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the λ-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kahler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to etale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

Provably Weak Instances of Ring-LWE

Abstract: The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of the Decision Ring-LWE problem. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus.

A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula

Abstract: We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the “Euler characteristic integral” of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are etale, we compute this integral in terms of Morel’s identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck‐Witt ring. In particular, we show that the Euler characteristic of an etale algebra corresponds to the class of its trace form in the Grothendieck‐Witt ring. 14F42; 47H10, 11E81

The Beltrami equations and prime ends

Abstract: We first study the boundary behavior of ring Q-homeomorphisms in terms of Caratheodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation \( \overline{\partial} \)f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane ℂ.

Neutral Black Rings in Five Dimensions are Unstable.

Abstract: We study nonaxisymmetric linearized gravitational perturbations of the Emparan-Reall black ring using numerical methods. We find an unstable mode whose onset lies within the ``fat'' branch of the black ring and continues into the ``thin'' branch. Together with previous results using Penrose inequalities that fat black rings are unstable, this provides numerical evidence that the entire black ring family is unstable.

Extensions of Rings and Modules

Abstract: Preliminaries and Basic Results.- Injectivity and Some of Its Generalizations.- Baer, Rickart, and Quasi-Baer Rings.- Baer, Quasi-Baer Modules, and Their Applications.- Triangular Matrix Representations and Triangular Matrix Extensions.- Matrix, Polynomial, and Group Ring Extensions.- Essential Overring Extensions - Beyond the Maximal Ring of Quotients.- Ring and Module Hulls.- Hulls of Ring Extensions.- Applications to Rings of Quotients and C* Algebras.- Open Problems and Questions.- References.- Index.

Efficient Zero-Knowledge Proofs for Commitments from Learning with Errors over Rings

Abstract: We extend a commitment scheme based on the learning with errors over rings $$\mathsf{RLWE}$$ problem, and present efficient companion zero-knowledge proofs of knowledge. Our scheme maps elements from the ring or equivalently, n elements from $$\mathbb F_q$$ to a small constant number of ring elements. We then construct $$\varSigma $$-protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of $$\mathcal {O}Mn\log q$$ and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known $$\varSigma $$-protocols for LWE-related languages only achieved a noticeable or even constant knowledge error thus requiring many repetitions of the protocol, or relied on "smudging" out the error which necessitates working over large fields, resulting in poor efficiency.

Stark points and -adic iterated integrals attached to modular forms of weight one

Abstract: Let be odd two-dimensional Artin representations for which is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms of respective weights two, one, and one, giving rise to via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain -adic iterated integrals attached to the triple -adic avatars of the leading term of the Hasse–Weil–Artin when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on —referred to as Stark points—which are defined over the number field cut out by . This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when are binary theta series attached to a common imaginary quadratic field in which splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing -adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on with Galois group a central extension of the dihedral group .

Polynomial degree bounds for matrix semi-invariants

Abstract: We study the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$ on $m$-tuples of $n \times n$ matrices with entries in an infinite field $K$. We show that invariants of degree $n^2- n$ define the null cone. Consequently, invariants of degree $\leq n^6$ generate the ring of invariants if $\operatorname{char}(K)=0$. We also prove that for $m \gg 0$, invariants of degree at least $n\lfloor \sqrt{n+1}\rfloor$ are required to define the null cone. We generalize our results to matrix invariants of $m$-tuples of $p\times q$ matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.

𝔏-Rickart Modules

Abstract: It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M R is a finitely generated 𝔏-Rickart module, we prove that End R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End R (M) is a left extending ring iff M is a Baer module and End R (M) is left cononsingular.

Discriminants in the Grothendieck ring

Abstract: We consider the limiting behavior of discriminants, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X and linear systems on X. These are connected—we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we ask whether the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and we propose a number of new conjectures, both arithmetic and topological.

Artin Approximation

Abstract: In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P{\l}oski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem, aimed at non-experts.

Frobenius categories, Gorenstein algebras and rational surface singularities

Abstract: We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

Affineness and chromatic homotopy theory

Abstract: Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and vanishing results about Tate spectra.

A family of quotients of the Rees Algebra

Abstract: A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.

A thick subcategory theorem for modules over certain ring spectra

Abstract: We classify thick subcategories of the1‐categories of perfect modules over ring spectra which arise as functions on even periodic derived stacks satisfying affineness and regularity conditions. For example, we show that the thick subcategories of perfect modules over TMF are in natural bijection with the subsets of the underlying space of the moduli stack of elliptic curves which are closed under specialization. 55P43, 18E30

Quantum codes from cyclic codes over q+vq+v2q+v3q

Abstract: We give a construction of quantum codes over 𝔽q from cyclic codes over a finite non-chain ring 𝔽q+v𝔽q+v2𝔽q+v3𝔽q, where q=pr, p is a prime, 3|(p−1) and v4=v.

Finite-dimensionality and cycles on powers of $K3$ surfaces

Abstract: For a K3 surface S, consider the subring of CH(S^n) generated by divisor and diagonal classes (with Q-coefficients). Voisin conjectures that the restriction of the cycle class map to this ring is injective. We prove that Voisin's conjecture is equivalent to the finite-dimensionality of S in the sense of Kimura-O'Sullivan. As a consequence, we obtain examples of S whose Hilbert schemes satisfy the Beauville-Voisin conjecture.