Affine Scheme Research Articles

A generalization of formal multiple zeta values related to multiple Eisenstein series and multiple q-zeta values

We present the τ-invariant balanced quasi-shuffle algebra Gf, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular, Gf has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra Zf of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra Gf. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra Gf onto Zf. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit q→1 of multiple q-zeta values.

Local dual spaces and primary decomposition

Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine scheme defined by the ideal.

K-spaces of non-domain-valued geometric points

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a k-algebra and this new “k-space” becomes a generalization of the corresponding Zariski space. We prove that a k-space is T0, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such k-spaces. We conclude with a question about the construction of a sheaf of k-spaces similar to affine schemes.

A scheme associated to modules over commutative rings

For any module M over a commutative ring R with identity, we consider Sch ( M ) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch ( M ) and introduce a sheaf O Sch ( M ) of commutative rings on it, which makes ( Sch ( M ) , O Sch ( M ) ) into a scheme. In particular, if M is a module over a Noetherian ring R, then ( Sch ( M ) , O Sch ( M ) ) is a locally Noetherian scheme. Among others, we give sufficient conditions for Sch ( M ) to be an affine scheme.

Gluing compactly generated t-structures over stalks of affine schemes

Gluing compactly generated t-structures over stalks of affine schemes

Noncommutative Poisson vertex algebras and Courant–Dorfman algebras

We introduce the notion of double Courant–Dorfman algebra and prove that it satisfies the so-called Kontsevich–Rosenberg principle, that is, a double Courant–Dorfman algebra induces Roytenberg's Courant–Dorfman algebras on the affine schemes parametrizing finite-dimensional representations of a noncommutative algebra. The main example is given by the direct sum of double derivations and noncommutative differential 1-forms, possibly twisted by a closed Karoubi–de Rham 3-form. To show that this basic example satisfies the required axioms, we first prove a variant of the Cartan identity [LX,LY]=L[X,Y] for double derivations and Van den Bergh's double Schouten–Nijenhuis bracket. This new identity, together with noncommutative versions of the other Cartan identities already proved by Crawley-Boevey–Etingof–Ginzburg and Van den Bergh, establishes the differential calculus on noncommutative differential forms and double derivations and should be of independent interest. Motivated by applications in the theory of noncommutative Hamiltonian PDEs, we also prove a one-to-one correspondence between double Courant–Dorfman algebras and double Poisson vertex algebras, introduced by De Sole–Kac–Valeri, that are freely generated in degrees 0 and 1.

Unified variable regularization factor based static compensator for power quality enhancement in distribution grid

A unified variable regularization factor based strategy for harmonic filtering at the supply side of the distribution system is presented in this paper. For this purpose, a three-phase distribution grid along with a distribution static compensator and a rectifier based nonlinear load is considered. A combined regularization affine projection sign algorithm-based control technique is used to calculate the weight values that denote the active power and reactive power segments of the load current. The proposed algorithm contains a variable mixing factor of two adaptive filters with different regularization parameters. The variable mixing factor enhances the tracking performance of the algorithm and hence it offers quick convergence and less steady-state error when compared with other affine projection-based filtering schemes. The control algorithm is evaluated under varying load patterns to derive its robustness and stability. Empirical assessment is carried out through real time simulation analysis aided by dSPACE-1202 controller. The performance of the applied control technique is found good-enough in mitigating the supply side harmonics and the total harmonic distortion levels rendered by the experimentation are in-line with the IEEE-519 standard.

On the direct product of fields with an application

In this paper, the (infinite) direct product of fields is investigated. In particular, the finiteness of a given set is characterized in terms of some ring-theoretic observations. Next, a certain localization (whose multiplicative set formed by cofinite sets) of the direct product of fields is studied. Finally, it is shown that every set [Formula: see text] can be made into a separated scheme, and this scheme is an affine scheme if and only if [Formula: see text] is a finite set.

The Balmer spectrum of certain Deligne–Mumford stacks

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.

Geometric Models for Algebraic Suspensions

Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $X$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the ${\mathbb P}^{1}$ suspension of $X$; we also analyze a host of variations on this observation. Our approach yields many examples of ${\mathbb A}^{1}$-$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine ${\mathbb A}^{1}$-contractible smooth schemes.

Correlated structure viscoplastic self-consistent polycrystal plasticity: Application to modeling strain rate sensitive deformation of Ti-6Al-4 V

This paper presents a multi-level viscoplastic self-consistent (VPSC) model for correlated structures (CS-VPSC). A polycrystalline aggregate response is homogenized over inclusions, which are linearized using the generalized affine scheme. Response of the inclusions is that of either a single grain or a correlated structure of multiple single crystals. The single crystal response is a strain rate adjusted power-law viscoplastic formulation, which considers not only the driving force in the glide direction but also the two orthogonal shear and three normal components of the driving force. The resistance to slip is grain size sensitive and based on the evolution of dislocation density. Furthermore, the resistances of prismatic and basal slip systems are adjusted based on the geometry of slip transfer between adjacent correlated structures. The overall model linking the single crystal micro-scale to a correlated structure meso‑scale to an aggregate macro-scale response is also coupled with the implicit finite elements (FE-CS-VPSC). The model is used for modeling strain rate sensitive mechanical response and microstructural evolution of additively manufactured Ti-6Al-4 V (Ti64) alloy in two conditions. As-built and stress-relived (SR) microstructure consisted of correlated α-lath/lamellar structures, while thermo-hydrogen refined microstructure (THRM) consisted primarily of α- and β-globular structures and some correlated α-lath/lamellar structures. Electron backscatter diffraction data, neutron diffraction data, and mechanical data were used to initialize, calibrate, and validate the model. To this end, the flow stress and texture evolution of the materials in tension and compression under quasi-static and high strain rate deformations were predicted using CS-VPSC, while the texture evolution during rolling and impact was simulated using FE-CS-VPSC. Good predictions of the texture evolution and mechanical response including the tension/compression asymmetry allowed us to discuss the roles of correlated structures, slip system activation stresses, size effects, strain partitioning between phases, and slip activities in such predictions.

The Arens-Michael envelopes of the Jordan plane and $U_q(\mathfrak{sl}(2))$

The Arens-Michael functor in noncommutative geometry is an analogue of the analytification functor in algebraic geometry: out of the ring of “algebraic functions” on a noncommutative affine scheme, it constructs the ring of “holomorphic functions” on it when viewed as a noncommutative complex analytic space. In this paper, we explicitly compute the Arens-Michael envelopes of the Jordan plane and the quantum enveloping algebra U_q(\mathfrak{sl}(2)) of \mathfrak{sl}(2) for |q|=1 .

Noetherian operators in Macaulay2

A primary ideal in a polynomial ring can be described by the variety it defines and a finite set of Noetherian operators, which are differential operators with polynomial coefficients. We implement both symbolic and numerical algorithms to produce such a description in various scenarios as well as routines for studying affine schemes through the prism of Noetherian operators and Macaulay dual spaces.

Reformulation of expressions for thermoelastic properties of crystals using harmonic mapping

We introduce alternative statistical mechanics expressions for thermoelastic properties of monatomic crystals, in the classical canonical ensemble, using the mapped-averaging framework. Two reference models are examined as a basis for the coordinate mapping: ideal gas (IG) and quasiharmonic (QH) crystal. The former prescribes affine mapping and yields the conventional expressions for the properties, while the latter has, in addition, a nonaffine component and yields alternative formulas, denoted as ``harmonically mapped averaging'' (HMA). While the (strain-dependent) affine mapping is universal and does not require system-specific information, the HMA method uses a system- (and strain-) dependent matrix to specify the mapping; this mapping matrix is derived from the force-constant matrix of the QH reference and its variation with strains. In practice, implementing HMA (and affine) schemes depends on the availability of analytical (Lagrangian) derivatives of potential energy with respect to strains (e.g., Born term) and coordinates (e.g., Hessian matrix); however, when these derivatives are unavailable, numerical derivatives can be computed via finite difference by moving atoms according the mapping matrices, while straining the crystal. In no circumstances does mapped averaging affect sampling of configurations; it involves only the calculation of averages. Results show that HMA provides CPU speedup ranging from a factor of 4, to over two orders of magnitude, compared to the conventional formulas (especially at low temperature), with the best performance obtained with isotropic-deformation properties. In addition to its application to molecular simulation, HMA can open avenues for developing new theories for thermoelastic properties.

Motivic Rigidity for Smooth Affine Henselian Pairs over a Field

Abstract Let $Z\to X$ be a closed immersion of smooth affine schemes over a field $k$, and let $X^h_Z$ denote the henselisation of $X$ along $Z$. We prove that $E(X^h_Z)\simeq E(Z)$ for every additive presheaf $E\colon \textbf {SH}(k)^{\textrm {op}}\to \textrm {Ab}$ on the stable motivic homotopy category over $k$ that is $l_\varepsilon $-torsion or $l$-torsion, where $l\in \mathbb Z$ is coprime to $\operatorname {char} k$, and $l_\varepsilon =\sum _{i=1}^n \langle (-1)^i \rangle $. More generally, the isomorphism holds for any homotopy invariant $l_\varepsilon $-torsion or $l$-torsion linear $\sigma $-quasi-stable framed additive presheaf on $\textrm {Sm}_{k}$. This generalises the result known earlier for local schemes. We prove the above isomorphism by constructing (stable) ${\mathbb {A}}^1$-homotopies of motivic spaces via algebraic geometry. To achieve this, we replace Quillen’s trick with an alternative and more general construction that provides relative curves required in our setting.

Primary Decomposition with Differential Operators

Abstract We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis–Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal decomposition for an ideal in a polynomial ring.

Improved incrementally affine homogenization method for viscoelastic-viscoplastic composites based on an adaptive scheme

We proposed an adaptive incrementally affine method, a micromechanics-based mean-field homogenization scheme for viscoelastic-viscoplastic particle-reinforced composites, which is applicable for predicting the mechanical response under complex loading conditions. The formulation is based on an incrementally affine scheme using algorithmic tangent operators while adaptively adjusting the mean strain of each constituent at every step of the loading process to reflect the changes in tangent operators and the shape of the reinforcing particles. We found that its prediction beyond the viscoelastic regime can be further improved by enforcing consistency between the accumulated strain states of each phase and the concentration tensors, excluding accumulated affine strain and stress. It was inevitable that the plastic deformation of the composite was initiated earlier than what was predicted by the mean-field theory owing to the local stress concentration near the particles. Hence, we proposed a yield reduction method that enforces the earlier initiation of plastic deformation in the matrix phase when obtaining an effective mechanical response. We observed that the predictions of the adaptive incrementally affine scheme adjusted with the yield reduction agreed well with various numerical simulations of particle-reinforced composites considering viscoelastic, elastic-viscoplastic, and viscoelastic-viscoplastic matrices under uniaxial, cyclic, and biaxial loadings.

Experiments on the Brauer map in high codimension

Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a Chow cover which works in all dimensions. As a corollary, we prove that cohomological Brauer classes are geometric on arbitrary separated surfaces. In higher dimensions, we obtain the first examples of non-quasi-projective algebraic spaces with $\mathrm{Br}(X)=\mathrm{Br}'(X)$.

Representations of General Linear Groups in the Verlinde Category

In this article, we construct affine group schemes GL(X) where X is any object in the Verlinde category in characteristic p and classify their irreducible representations. We begin by showing that for a simple object X of categorical dimension i, this representation category is semisimple and is equivalent to the connected component of the Verlinde category for SLi. Subsequently, we use this along with a Verma module construction to classify irreducible representations of GL(nL) for any simple object L and any natural number n. Finally, parabolic induction allows us to classify irreducible representations of GL(X) where X is any object in Verp.

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a catalogue raisonne of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a weakly functorial aspect of this result. Splinters are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma.