Algebraic Resolution Research Articles

Resolutions in Local Algebra and Singularity Theory

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

On the semifree resolutions of DG algebras over the enveloping DG algebras

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra B over the enveloping DG algebra B ⊗ A B , where A ⊆ B is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by ( B , D ) , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of ( B , D ) , we provide a new characterization of naïve liftability of semifree DG B-modules.

Rigidity dimensions of Hochschild extensions of hereditary algebras of type D

Rigidity dimension of algebras is a new homological dimension which measures the quality of resolutions of algebras by algebras of finite global dimension and large dominant dimension. In this paper, we calculate the rigidity dimension of the Hochschild extension of a hereditary algebra H of Dynkin type D n ( n ≥ 4 ) by the standard duality H -bimodule. It turns out that this dimension is independent of n and the orientation of D n .

On regularity bounds and linear resolutions of toric algebras of graphs

Let G be a simple graph. We show that if G is connected and R(I (G)) is normal, reg ⁡(R(I (G)))≤α0 (G), where α0 (G) is the vertex cover number of G. As a consequence, for every normal König connected graph G, reg ⁡(R(I (G)))= mat ⁡ (G), the matching number of G. For a gap-free graph G, we give various combinatorial upper bounds for reg ⁡(R(I (G))). As a consequence we give various sufficient conditions for the equality of reg ⁡(R(I (G))) and mat ⁡ (G). Finally we show that if G is a chordal graph such that K[G] has q-linear resolution (q≥4), then K[G] is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.

Use DG-methods to build a matrix factorization

Let P be a commutative Noetherian ring, 𝔠 be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and P¯ be the hypersurface ring P∕(f). Assume that 𝔠:f is a grade four Gorenstein ideal of P. We give a resolution N of P¯∕𝔠P¯ by free P¯-modules. The resolution N is built from a differential graded algebra resolution of P∕(𝔠:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.

The resolution of (xN,yN,zN,wN)

Let k be a field of characteristic zero, n<N be positive integers, P be the polynomial ring k[x,y,z,w], F be the homogeneous polynomial xn+yn+zn+wn, K be the ideal (xN,yN,zN,wN), and P‾ be the hypersurface ring P‾=P/(F). We describe the minimal multi-homogeneous resolution of P‾/KP‾ by free P‾-modules, the socle degrees of P‾/KP‾, and the minimal multi-homogeneous resolution of the Gorenstein ring P/(K:F) by free P-modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P‾/KP‾ by free P‾-modules is obtained from a Differential Graded Algebra resolution of P/(K:F) by free P-modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal.

A Didactical Analysis of Teaching Practices Around Differentials Equations in Tunisian School Context

The teaching of differential equations is often viewed from an interdisciplinary perspective that justifies the rationale for this notion in the secondary cycle. The history of the teaching of this notion in the Tunisian educational system shows a diversity of approaches which seem to evolve in the direction of modeling extra-mathematical situations and dialectic between semiotics registers. This research is part of the anthropological theory of didactics and analyzes the didactic transposition process developed by the Tunisian institution around differential equations over successive reforms. Three dimensions of analysis were taken into account in this study: a historical-epistemological dimension which makes it possible to identify the dynamic nature of the notion of differential equation through the stages of its constitution and to determine its meaning through the problems addressed in teaching and their development, an institutional dimension addressed by ecology and praxeology analyses based on the programs and official textbooks. Current institutional practices allow us to glimpse a dynamic at scale relating to the teaching of this notion. This aspect is nourished by sets of frameworks, flexibility between registers and interdisciplinary praxeology mobilized upstream and downstream of the work of algebraic resolution. Professional entry allows us to question the constraints for their effective implementation in classes. This dimension is analyzed from ten student notebooks considered as a first revealer of teaching practices in terms of resistance or change. This analysis of the notebooks make it possible to discover the main characteristics of the institutional relationship. The result of the analysis shows that the personal relationship of teachers perceived through the analysis of their didactic preparation with the object of knowledge differential equations, is not suitable to institutional relationship with this same object of knowledge.

Crossed and quadratic resolutions of algebras

In this work, using crossed resolutions, we will give a construction of a free reduced quadratic resolutions of a commutative k-algebra and explain its 2-skeleton.

On conversions from CNF to ANF

In this paper we discuss conversion methods from the conjunctive normal form (CNF) to the algebraic normal form (ANF) of a Boolean function. Whereas the reverse conversion has been studied before, the CNF to ANF conversion has been achieved predominantly via a standard method which tends to produce many polynomials of high degree. Based on a block-building mechanism, we design a new blockwise algorithm for the CNF to ANF conversion which is geared towards producing fewer and lower degree polynomials. In particular, we look for as many linear polynomials as possible in the converted system and check that our algorithm finds them. As an application, we suggest a new algebraic extension of the resolution calculus which is tailored to the output of the standard CNF to ANF conversion algorithm. Experiments show that the ANF produced by our algorithm outperforms the standard conversion in “real life” examples originating from cryptographic attacks, and that our algebraic resolution algorithm solves some SAT instances which are notoriously hard for SAT solvers.

Convergent presentations and polygraphic resolutions of associative algebras

Several constructive homological methods based on noncommutative Gr\"obner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the existence of a quadratic Gr\"obner basis of its ideal of relations. In this article, using a higher-dimensional rewriting theory approach, we give several improvements of these methods. We define polygraphs for associative algebras as higher-dimensional linear rewriting systems that generalise the notion of noncommutative Gr\"obner bases, and allow more possibilities of termination orders than those associated to monomial orders. We introduce polygraphic resolutions of associative algebras, giving a categorical description of higher-dimensional syzygies for presentations of algebras. We show how to compute polygraphic resolutions starting from a convergent presentation, and how these resolutions can be linked with the Koszul property.

La Motivación en la Resolución de Problemas Aritmético-algebraicos. Un Estudio con Alumnado de Educación Secundaria

Introducción. La motivación constituye un factor importante en el aprendizaje de las matemáticas. Dentro de este ámbito educativo, la resolución de problemas ocupa un lugar central en la mayoría de currículos de matemáticas en Educación Secundaria. El objetivo de esta investigación es detectar las diferencias en motivación en función de las estrategias empleadas para la resolución de problemas aritmético-algebraicos.Método. Se han analizado los procedimientos de resolución de tres problemas verbales aritmético-algebraicos y se han recopilado datos sobre la motivación mediante un cuestionario basado en el modelo de expectativa-valor de Eccles y sus colaboradores adaptado para la aplicación en el aprendizaje de las matemáticas. La muestra está compuesta por 598 estudiantes de 2º, 3º y 4º de Educación Secundaria Obligatoria (ESO) de la Comunidad Autónoma Vasca.Resultados. Los resultados obtenidos indican que el grupo de alumnos con perfil de resolución algebraico obtiene puntuaciones superiores tanto en valor de la tarea como en la autoeficacia percibida. Además, el grupo de resolución sin perfil definido declara que el estudio de las matemáticas supone una mayor pérdida de oportunidades para realizar otras actividades.Discusión. Por una parte, el alumnado que resuelve los problemas algebraicamente muestra una motivación superior al grupo que alterna el álgebra y la aritmética como estrategias de resolución; por tanto, parece ser que la supuesta flexibilidad en la resolución implica menor motivación. Por otra parte, se discute la baja motivación del alumnado sin perfil de resolución definido en relación a su bajo rendimiento.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Comparison morphisms between two projective resolutions of monomial algebras

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Algebraic resolution of the Burgers equation with a forcing term

We introduce an inhomogeneous term, f(t,x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f(t,x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra s l(2,R). When f is a function of only t, there are five symmetries with the algebra s l(2,R) ⊕ s 2A 1. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.

Box graphs and resolutions I

Box graphs succinctly and comprehensively characterize singular fibers of elliptic fibrations in codimension two and three, as well as flop transitions connecting these, in terms of representation theoretic data. We develop a framework that provides a systematic map between a box graph and a crepant algebraic resolution of the singular elliptic fibration, thus allowing an explicit construction of the fibers from a singular Weierstrass or Tate model. The key tool is what we call a fiber face diagram, which shows the relevant information of a (partial) toric triangulation and allows the inclusion of more general algebraic blowups. We shown that each such diagram defines a sequence of weighted algebraic blowups, thus providing a realization of the fiber defined by the box graph in terms of an explicit resolution. We show this correspondence explicitly for the case of SU(5) by providing a map between box graphs and fiber faces, and thereby a sequence of algebraic resolutions of the Tate model, which realizes each of the box graphs.

The structure of Gorenstein-linear resolutions of Artinian algebras

Let k be a field, A a standard-graded Artinian Gorenstein k-algebra, S the standard-graded polynomial ring Sym•kA1, I the kernel of the natural map ▪, d the vector space dimension dimk⁡A1, and n the least index with In≠0. Assume that 3≤d and 2≤n. In this paper, we give the structure of the minimal homogeneous resolution B of A by free S-modules, provided B is Gorenstein-linear. (Keep in mind that if A has even socle degree and is generic, then A has a Gorenstein-linear minimal resolution.)Our description of B depends on a fixed, but arbitrary, decomposition of A1 of the form kx1⊕V0, for some non-zero element x1 and some (d−1) dimensional subspace V0 of A1. Much information about B is already contained in the complex B‾=B/x1B, which we call the skeleton of B. One striking feature of B is the fact that the skeleton of B is completely determined by the data (d,n); no other information about A is used in the construction of B‾.The skeleton B‾ is the mapping cone of zero:K→L, where L is a well known resolution of Buchsbaum and Eisenbud; K is the dual of L; and L and K are comprised of Schur and Weyl modules associated to hooks, respectively. The decomposition of B‾ into Schur and Weyl modules lifts to a decomposition of B; furthermore, B inherits the natural self-duality of B‾.The differentials of B are explicitly given, in a polynomial manner, in terms of the coefficients of a Macaulay inverse system for A. In light of the properties of B‾, the description of the differentials of B amounts to giving a minimal generating set of I, and, for the interior differentials, giving the coefficients of x1. As an application we observe that every non-zero element of A1 is a weak Lefschetz element for A.

Projective resolutions of associative algebras and ambiguities

The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing thus a useful tool for computing invariants associated to the considered algebras. We illustrate how to use it by giving several examples in the last section of the article. In particular we give necessary and sufficient conditions for noetherian down–up algebras to be 3-Calabi–Yau.

Algebraic resolution of equations of the Black–Scholes type with arbitrary time-dependent parameters

We examine various forms of the Black–Scholes Equation using symmetry analysis and consider the standard problem of a terminal condition, namely u(T,x)=U. The models considered are the standard form of the equation to provide a benchmark, the nonhomogeneous form and two variations in which the parameters of the two basic models are explicitly time dependent. We find, subject to some constraints upon the parameters, a remarkable robustness in the results.

Different approaches to the complex of three Dirac operators

An attempt to study the compatibility conditions, and the general free resolution, for the system associated with the Dirac operator in \(k\) vector variables appeared already in Sabadini et al. (Math Z 239: 293–320, 2002), from the point of view of Clifford analysis, and in Sabadini et al. (Exp Math 12: 351–364, 2003) using the tool of megaforms. Other studies have been carried out in other papers, like Krump (Adv Appl Clifford Alg 19: 365–374, 2009), Krump and Soucek (17: 537–548, 2007), Salac (The generalized Dolbeault complexes in Clifford analysis, Praha 2012), using methods of representation theory. In this paper, we restrict our attention to the case of three variables and we describe the free resolution associate to the module from various different angles. The comparison has several noteworthy consequences. In particular, it gives the explicit description of all the maps contained in the algebraic resolution and shows that they are invariant with respect to the action of \(SL(3)\times SO(m)\). We also discuss how the methods used in this paper can be generalized to the case of \(k>3\) Dirac operators.

Intersection spaces, perverse sheaves and type IIB string theory

The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar\'e duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.