Let $\bk$ be a field of characteristic zero. A Pham-Brieskorn ring is a $\bk$-algebra of the form $B_{a_0,\dots,a_n} = \bk[X_0,\dots,X_n] / \lb X_0^{a_0} + \cdots + X_n^{a_n} \rb$, where $n \geq 2$ and $a_0, \dots, a_n$ are positive integers. A ring $B$ is rigid if the only locally nilpotent derivation $D : B \to B$ is the zero derivation. Consider the following conjecture. \begin{conjnonumber}\label{PBConjectureAbstract} Let $n \geq 2$, and let $B_{a_0, \dots, a_n} = \bk[X_0, \dots, X_n] / \langle X_0^{a_0} + \cdots + X_n^{a_n} \rangle$ be a Pham-Brieskorn ring. If $\min\{a_0, \dots,a_n \} \geq 2$ and at most one element $i$ of $\{0,\dots ,n\}$ satisfies $a_i = 2$, then $B_{a_0, \dots, a_n}$ is rigid. \end{conjnonumber} The $n = 2$ case of the Conjecture is known to be true. In this thesis, we make progress towards solving the above conjecture. Our main results are: \begin{enumerate}[\rm(1)] \item For any $n \geq 3$, in order to prove the above conjecture, it suffices to prove rigidity of $B_{a_0, \dots, a_n}$ in the cases where $\bk = \Comp$ and $\cotype(a_0, \dots, a_n) = 0$. \item For any $n \geq 2$, $X = \Proj B_{a_0, \dots, a_n}$ is a well-formed quasismooth weighted complete intersection if and only if $\cotype(a_0, \dots, a_n) = 0$. \item When $n = 3$ and $\cotype(a_0, a_1, a_2, a_3) = 0$, $B_{a_0, a_1, a_2, a_3}$ is rigid, except possibly in the cases where, up to a permutation of the $a_i$, $(a_0, a_1, a_2, a_3) \in \{(2,3,4,12), (2,3,5,30)\}$. \item We summarize the list of 3-dimensional Pham-Brieskorn rings $B_{a_0, a_1, a_2, a_3}$ for which rigidity is known. It follows in particular that if $B_{2,3,4,12}$ and $B_{2,3,5,30}$ are rigid then the $n = 3$ case of the above conjecture is true. \end{enumerate} In addition to the above, we develop techniques for proving rigidity of rings in general; prove rigidity of many Pham-Brieskorn rings whose dimension is greater than 3; give simple examples of rational projective surfaces with quotient singularities that have an ample canonical divisor and prove that the members of a certain family of singular hypersurfaces are not rational.

<p>The goal of this dissertation is to broaden the classes of ideals for which the generators of residual intersections are known. This is split into two main parts.</p> <p>The first part is Chapter 5, where we prove that, for an ideal I in a local Cohen-Macaulay ring R, under suitable technical assumptions, we are able to express s-residual intersections, for s ≥ μ(I) − 2, in terms of (μ(I) − 2)-residual intersections. This result implies that s- residual intersections can be expressed in terms of links, if μ(I) ≤ ht(I) + 3 and some other hypotheses are satisfied. In Chapter 5, we prove our result using two different methods and two different sets of technical assumptions on the depth conditions satisfied by the ideal I. For Section 5.2 and Section 5.3 we use the properties of Fitting ideals and methods developed in [33] to prove our main result. In these sections, we require I to satisfy the Gs condition and be weakly (s − 2)-residually S2. In Section 5.4, we prove analogous results to those in Section 5.2 and Section 5.3 using disguised residual intersections, a notion developed by Bouca and Hassansadeh in [5].</p> <p>The second part is Chapter 6 where we prove that the n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. To prove this, we require a series of technical results. We begin by proving the main theorem for this chapter in a special case, using the results of Section 6.1 to compute the generators of the relevant links in a our special case, and then using these generators to compute the Gro ̈bner Basis for the sum of links in Section 6.2. The computation of the Gro ̈bner basis, as well as an application of graph theoretic results about binomial edge ideals [17], allow us to show that our main theorem holds in this special case. Lastly, we conclude our proof in Section 6.3, where we show that n-residual intersections of ideals generated by maximal minors of 2 × n generic matrices commute with specialization maps, and use this to show that the generic n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. This allows us to prove the main theorem of Chapter 6.</p>

Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext_R/I<i>(M,N)</i> -> Ext_R<i>(M,N)</i> and Tor^R_N<i>(M, N)</i> -> Tor^R/I_n<i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞_n=o Tor^R/I_n <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X_r] with deg X_i = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ωⁿ_R/IR/I(M),I) is a Golod pair over <i>R</i> where Ωⁿ_R/IR/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹_R/IR/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>. Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms. / Ph. D.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25090号 / 理博第4997号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 COLLINSBenoit Vincent Pierre, 教授 泉 正己, 准教授 窪田 陽介 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Pour l'algèbre de Banach complexe A ayant l'élément unité, on dénote par G(A) l'ensemble des éléments inversibles de A, et par G1(A) on dénote la composante qui contient l'unité. Le spectre de a ∈ A est l'ensemble de tous les nombres complexes λ tels que λ1 - a ∉ G(A), et le spectre exponentiel de a est l'ensemble de tous les nombres complexes tels que λ1 - a ∉ G1(A). Évidemment, pour chaque élément de l'algèbre, son spectre exponentiel contient le spectre habituel. Il est bien connu que le spectre habituel a une propriété que l'on nommera "propriété de commutativité". Cela signifie que, pour chaque choix des deux éléments a; b ∈ A, nous avons Sp(ab) \ {0} = Sp(ba) \ {0}, où Sp est le spectre. Avons-nous la même propriété pour les spectres exponentiels? Cette question n'est toujours pas résolue. L'objectif de ce mémoire est d'étudier le spectre exponentiel, et plus particulièrement sa propriété de commutativité. Dans le premier chapitre, nous donnerons les définitions d'algèbre de Banach complexe, spectre et spectre exponentiel de ses éléments, et leurs propriétés de base. Aussi nous établirons des relations topologiques entre les spectres exponentiel et habituel. Dans le deuxième chapitre, nous définirons les fonctions holomorphes sur une algèbre de Banach, et discuterons du problème de la propriété de commutativité de spectre exponentiel, en établissant des résultats positifs connus. Dans le troisième et dernier chapitre, nous examinerons quelques exemples d'algèbres de Banach, décrivant les ensembles G(A) et G1(A), et discuterons de la propriété de commutativité pour ces algèbres. / For a complex Banach algebra A with unit element, we denote by G(A) the set of invertible elements of A, and by G1(A) we denote the component of G(A) which contains the unit. The spectrum of a ∈ A is the set of all complex numbers λ such that λ1 - a ∉ G(A), and the exponential spectrum of a is the set of all complex numbers λ such that λ1 - a ∉ G1(A). Of course for each element of the algebra its exponential spectrum contains the usual spectrum. It is well known that the usual spectrum has the so-called commutativity property. This means that, for any two elements a and b of A, we have Sp(ab) \ {0} = Sp(ba) \ {0}, where Sp denotes the spectrum. Does this property hold for exponential spectra? This is still an open question. The purpose of this memoir is to study the exponential spectrum, and particularly its commutativity property. In chapter one, we will give definitions of a complex Banach algebra, the spectrum and exponential spectrum of its elements, and their basic properties. Also we will establish topological relations between exponential and usual spectra. In chapter two, we will define holomorphic functions on a Banach algebra, and also discuss the commutativity property problem for the exponential spectrum, establishing some known positive results. In the last chapter, we will consider some examples of Banach algebras, describing the sets G(A) and G1(A), and discuss the commutativity property for these algebras.

La présente thèse porte sur le développement d'une théorie d'algèbre linéaire, de géométrie et d'analyse basée sur les algèbres (Z2)n-commutatives, c'est-à-dire des algèbres (Z2)n-graduées associatives unitaires satisfaisant ab = (-1)<deg(a),deg(b)>ba, pour tout couple d'éléments homogènes a, b de degrés deg(a), deg(b) où <.,.> est le produit scalaire usuel). Cette généralisation de la supergéométrie a de nombreuses applications : en mathématiques (l'algèbre de Deligne des superformes différentielles, l'algèbre des quaternions et les algèbres de Clifford en sont des exemples) et même en physique (paraparticules). Dans ce travail, les notions de trace et de (super)déterminant pour des matrices à coefficients dans une algèbre gradué-commutative sont définies et étudiés. Une attention particulière est portée au cas des algèbres de Clifford : ce point de vue gradué fournit une nouvelle approche au problème classique du « bon » déterminant pour des matrices à coefficient non-commutatifs (quaternioniques). En outre, nous entreprenons l'étude de la géométrie différentielle (Z2)n-graduée. Privilégiant l'approche par les espaces annelés, les (Z2)n-supervariétés sont définies en choisissant l'algèbre (Z2)n-commutative des séries formelles en variables graduées comme modèle pour le faisceau de fonctions. Les résultats les plus marquants ainsi obtenus sont : le Berezinien gradué et son interprétation cohomologique (essentielle pour établir une théorie de l'intégration) ; le théorème des morphismes, attestant qu'on peut rétablir un morphisme entre (Z2)n-supervariétés à partir de sa seule expression sur les coordonnées ; le théorème de Batchelor-Gawedzki pour les (Z2)n-supervariétés lisses / The present thesis deals with a development of linear algebra, geometry and analysis based on (Z2)n-superalgebras ; associative unital algebras which are (Z2)n-graded and graded-commutative, i.e. statisfying ab=(-1)<deg(a),deg(b)>ba, for all homogeneous elements a, b of respective degrees deg(a), deg(b) in (Z2)n (<.,.> denoting the usual scalar product). This generalization widens the range of applications of supergeometry to many mathematical structures (quaternions and more generally Clifford algebras, Deligne algebra of superdifferential forms, higher vector bundles) and appears also in physics (for describing paraparticles) proving its worth and relevance. In this dissertation, we first focus on (Z2)n-superalgebra theory ; we define and characterize the notions of trace and (super)determinant of matrices over graded-commutative algebras. Special attention is given to the case of Clifford algebras, where our study gives a new approach to treat the classical problem of finding a “good” determinant for matrices with noncommuting (quaternionic) entries. Further, we undertake the study of (Z2)n-graded differential geometry. Privileging the ringed space approach, we define (smooth) (Z2)n-supermanifolds modeling their algebras of functions on the (Z2)n-commutative algebra of formal power series in graded variables, and develop the theory along the lines of supergeometry. Notable results are : the graded Berezinian and its cohomological interpretation (essential to establish integration theory) ; the theorem of morphism, which states that a morphism of (Z2)n-supermanifolds can be recovered from its coordinate expression ; Batchelor-Gawedzki theorem for (Z2)n-supermanifolds

The time coordinate is a common obstacle in the theory of non-commutative (nc.) spacetimes. Despite that, this work shows how the interplay between quantum fields and an underlying nc. spacetime can still be analyzed, even for the case of nc. time. This is done for the example of a general Moyal-type external potential scattering of the Dirac field in Moyal-Minkowski spacetime. The spacetime is a rare example of a Lorentzian non-compact nc. geometry. Elements of the associated spectral function algebra are shown to be operationally involved at the level of quantum field operators by Bogoliubovs formula. Furthermore, a similar task is attacked in the case of locally nc. spacetimes. An explicit star-product is constructed by a method of Kontsevich. It implements a decay of non-commutativity with increasing distance. This behavior should benefit the technical side - diverse interesting formal attempts are discussed. It is striven for unification of several toy models of nc. spacetimes and a general strategy to define quantum field operators. Within the latter one has to implement the usual quantum behavior as well as a new kind of spacetime behavior. It is shown how this two-fold character causes key difficulties in understanding.

The first few chapters of the dissertation will catalogue what is known regarding z-ideals in commutative rings with identity. Some special attention will be paid to z-ideals in function rings to show how the presence of the topological description simplifies z-covers of arbitrary ideals. Conditions in an f-ring that ensure that the sum of z-ideals is a z-ideal will be given. In the latter part of the dissertation I will generalise a result in higher order z-ideals and introduce a notion of higher order d-ideals / Mathematical Sciences / M. Sc. (Mathematics)

Cette thèse porte sur la construction d'une famille de groupoïdes quantiques de transformations qui dans le cadre algébrique sont des algébroïdes de Hopf de multiplicateurs mesurés au sens de Timmermann et Van Daele et qui dans le cadre des algèbres d'opérateurs sont des C*-bimodules de Hopf sur une C*-base au sens de Timmermann.Dans le contexte purement algébrique, nous définissons d'abord une algèbre involutive de Yetter-Drinfeld tressée commutative sur un groupe quantique algébrique au sens de Van Daele et une intégrale de Yetter-Drinfeld sur elle. En utilisant ces objets nous construisons après un algébroide de Hopf de multiplicateurs involutif mesuré, ce nouvel objet nous l'appellons groupoïde quantique algébrique de transformations.Pour être capables de passer au cadre des algèbres d'opérateurs, nous donnons des conditions sur l'intégral de Yetter-Drinfeld qui vont nous permettre d'utiliser la construction Gelfand–Naimark–Segal pour étendre tous nos objets purement algébriques en des objets C*-algébriques. Dans ce contexte, notre construction se fait d'une manière similaire à celle présentée dans le travail de Enock et Timmermann, nous obtenons un nouvel objet mathématique que nous appellons un groupoïde quantique C*-algébrique de transformations, qui est définit en utilisant le langage des C*-bimodules de Hopf sur une C*-base. / This thesis is concerned with the construction of a family of quantum transformation groupoids in the algebraic framework in the form of the measured multiplier Hopf *-algebroids in the sense of Timmermann and Van Daele and also in the context of operator algebras in the form of Hopf C*-bimodules on a C*-base in the sense of Timmermann.In the purely algebraic context, we first give a definition of a braided commutative Yetter-Drinfeld *-algebra over an algebraic quantum group in the sense of Van Daele and a Yetter-Drinfeld integral on it. Then, using these objects we construct a measured multiplier Hopf *-algebroid, we call to this new object an algebraic quantum transformation groupoid.In order to pass to the operator algebra framework, we give some conditions on the Yetter-Drinfeld integral inspired by the properties of KMS-weights on C*-algebras which will allow us to use the Gelfand–Naimark–Segal construction to extend all the purely algebraic objects to the C*-algebraic level. At this level, we construct in a similar way to that used in the work of Enock and Timmermann, a new mathematical object that we call a C*-algebraic quantum transformation groupoid, which is defined using the language of Hopf C*-bimodules on C*-bases.

The time coordinate is a common obstacle in the theory of non-commutative (nc.) spacetimes. Despite that, this work shows how the interplay between quantum fields and an underlying nc. spacetime can still be analyzed, even for the case of nc. time. This is done for the example of a general Moyal-type external potential scattering of the Dirac field in Moyal-Minkowski spacetime. The spacetime is a rare example of a Lorentzian non-compact nc. geometry. Elements of the associated spectral function algebra are shown to be operationally involved at the level of quantum field operators by Bogoliubovs formula. Furthermore, a similar task is attacked in the case of locally nc. spacetimes. An explicit star-product is constructed by a method of Kontsevich. It implements a decay of non-commutativity with increasing distance. This behavior should benefit the technical side - diverse interesting formal attempts are discussed. It is striven for unification of several toy models of nc. spacetimes and a general strategy to define quantum field operators. Within the latter one has to implement the usual quantum behavior as well as a new kind of spacetime behavior. It is shown how this two-fold character causes key difficulties in understanding.