In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Sturmfels and Sullivant 2006. Apart from their inherent appeal to combinatorial commutative algebra, these ideals also generalize graph statistical models for binary data and are related to some statistical models for phylogenetic trees. Specifically, we consider minimal free resolutions for the cut ideals of trees. We propose a method to combinatorially estimate the Betti numbers of the ideals in this class. Using this method, we derive upper bounds for some of the Betti numbers, given by formulas exponential in the number of vertices of the tree. Our method is based on a common technique in commutative algebra whereby arbitrary homogeneous ideals are deformed to initial monomial ideals, which are easier to analyze while conserving some of the information of the original ideals. The cut ideal of a tree on n vertices turns out to be isomorphic to the Segre product of the cut ideals of its n-1 edges (in particular, its algebraic properties do not depend on its shape). We exploit this product structure to deform the cut ideal of a tree to an initial monomial ideal with a simple combinatorial description: it coincides with the edge ideal of the incomparability graph of the power set of the edges of the tree. The vertices of the incomparability graph are subsets of the edges of the tree, and two subsets form an edge whenever they are incomparable. In order to obtain algebraic information about these edge ideals, we apply an idea introduced by Dochtermann and Engström in 2009 that consists in regarding the edge ideal of a graph as the (monomial) Stanley-Reisner ideal of the independence complex of the graph. Using Hochster\''s formula for computting Betti numbers of Stanley-Reisner ideals by means of simplicial homology, the computation of the Betti numbers of these monomial ideals is turned to the enumeration of induced subgraphs of the incomparability graph. That the resulting values give upper bounds for the Betti numbers of the cut ideals of trees is an important well-known result in commutative algebra. In the second chapter, we focus on some combinatorial features of triangulations of the point configuration obtained as the cartesian product of two standard simplices. These were explored in collaboration with César Ceballos and Arnau Padrol, and had a two-fold motivation. On the one hand, we intended to understand the influence of the product structure on the set of triangulations of the cartesian product of two point configurations; on the other hand, the set of all triangulations of the product of two simplices is an intricate and interesting object that has attracted attention both in discrete geometry and in other fields of mathematics such as commutative algebra, algebraic geometry, enumerative geometry or tropical geometry. Our approach to both objectives is to examine the circumstances under which a triangulation of the polyhedral complex given by the the product of an (n-1)-simplex times the (k-1)-skeleton of a (d-1)-simplex extends to a triangulation of an (n-1)-simplex times a (d-1)-simplex. We refer to the former as a partial triangulation of the product of two simplices. Our main result says that if d >= k > n, a partial triangulation always extends to a uniquely determined triangulation of the product of two simplices. A somewhat unexpected interpretation of this result is as a finiteness statement: it asserts that if d is sufficiently larger than n, then all partial triangulations are uniquely determined by the (compatible) triangulations of its faces of the form “(n-1)-simplex times n-simplex”. Consequently, one can say that in this situation ‘\''triangulations of an (n-1)-simplex times a (d-1)-simplex are not much more complicated than triangulations of an (n-1)-simplex times an n-simplex\''\''. The uniqueness assertion of our main result holds already when d>=k>=n. However, the same is not true for the existence assertion; namely, there are non extendable triangulations of an (n-1)-simplex times the boundary of an n-simplex that we explicitly construct. A key ingredient towards this construction is a triangulation of the product of two (n-1)-simplices that can be seen as its ``second simplest triangulation\''\'' (the simplest being its staircase triangulation). It seems to be knew, and we call it the Dyck path triangulation. This triangulation displays symmetry under the cyclic group of order n that acts by simultaneously cycling the indices of the points in both factors of the product. Next, we exhibit a natural extension of the Dyck path triangulation to a triangulation of an (n-1)-simplex times an n-simplex that, in a sense, enjoys some sort of ‘\''rigidity\''\'' (it also seems new). Performing a ‘\''local modification\''\'' on the restriction of this extended triangulation to the polyhedral complex given by (n-1)-simplex times the boundary of an n-simplex yields the non-extendable partial triangulation. The thesis includes two appendices on basic commutative algebra and triangulations of point configuration, included to make it slightly self-contained.

We study partially ordered sets of prime ideals as found in commutative Noetherian rings. These structures, commonly known as prime spectra, have long been a popular topic in the field of commutative algebra. As a consequence, there are many related questions that remain unanswered. Among them is the question of what partially ordered sets appear as Spec(A) of some Noetherian ring A, asked by Kaplansky during the 1950's. As a partial case of Kaplansky's question, we consider finite posets that are ring spectra of commutative Noetherian rings. Specifically, we show that finite spectra of such rings are always order-isomorphic to a bipartite graph. However, the most significant undertaking of this study is that of devising a constructive methodology for finding a ring with prime spectrum that is order-isomorphic to an arbitrary bipartite graph. As a result, we prove that any complete bipartite graph is order-isomorphic to the prime spectrum of some ring of essentially finite type over the field of rational numbers. Moreover, a series of potential generalizations and extensions are proposed to further enhance the constructive methodology. Ultimately, the results of this study constitute an original contribution and perspective on questions related to commutative ring spectra.

In his study of spectra of f-rings via pointfree topology, Banaschewski [6] considers lattices of l-ideals, radical l-ideals, and saturated l-ideals of a given f-ring A. In each case he shows that the lattice of each of these kinds of ideals is a coherent frame. This means that it is compact, generated by its compact elements, and the meet of any two compact elements is compact. This will form the basis of our main goal to show that the lattice-ordered rings studied in [6] are coherent frames. We conclude the dissertation by revisiting the d-elements of Mart nez and Zenk [30], and characterise them analogously to d-ideals in commutative rings. We extend these characterisa-tions to algebraic frames with FIP. Of necessity, this will require that we reappraise a great deal of Banaschewski's work on pointfree spectra, and that of Mart nez and Zenk on algebraic frames. / Mathematical Sciences / M. Sc. (Mathematics)

Dans cette thèse, nous interprétons géométriquement la torsion de l'algèbre symétrique d'un faisceau d'idéaux I_Z d'un schéma Z défini par n+1 équations dans une variété n-dimensionnelle. Ceci revient à étudier la géométrie de la projectivisation de I_Z. Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l'espace projectif de dimension 3 au sujet duquel nous construisons des transformations birationnelles explicites qui ont le même degré algébrique que leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous généralisons une caractérisation des courbes libres en étendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersurfaces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degré 5 en caractéristique 3. La dernière application concerne le calcul de l'inverse d'une transformation birationnelle. / In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf I_Z of a scheme Z defined by n+1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of I_Z. The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map.

Ce mémoire est une présentation des travaux que l'auteur a réalisé en théorie des aneaux et des modules. Plus précisément l'auteur s'est consacré à l'étude des anneaux commutatifs arithmétiques et plus particulièrement aux anneaux de valuation (non nécessairement intègres). Le résultat le plus remaquable est la démonstration du théorème qui dit que tout annean local à type de module borné est un anneau de valuation presque maximal. Sont aussi présentés des résultats sur la localisation des modules injectifs et sur les enveloppes pure-injectives de certains modules.

Dans cette thèse, nous définissons algébriquement le résultant déterminantiel d'un morphisme f de modules libres de type dont la matrice a en entrée des polynômes homogènes f_i,j. A l'aide des complexes d'Eagon-Northcott et de Buchsbaum-Rim associés au morphisme P nous proposons des méthodes effectives pour calculer ce résultant déterminantiel ainsi que son degré. Dans le cas où les polynômes f_i,j sont à deux variables, nous montrons que ce résultant déterminantiel est donné par le déterminant d'une matrice en les coefficients des f_i,j , qui est une généralisation de la matrice de Sylvester de deux polynômes. Dans la deuxième partie de la thèse, nous étudions des problèmes d'intersection de courbes et surfaces de Bézier en évitant la fameuse conversion instable entre la base de Bernstein et la base monomiale. Ces problèmes jouissent d'une structure particulière qui est dégénérée pour le résultant de Macaulay. Nous prouvons l'existence d'un résultant anisotrope adapté à ces systèmes dégénérés et proposons un algorithme pour le calculer.

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Die Arbeit basiert auf zwei Veröffentlichungen zur graduierten kommutativen Algebra: Thema des ersten Artikels ist die Übertragung eines klassischen Ergebnisses zur Höhe von Primidealen in Polynomringen auf allgemeine multigraduierte Ringe; einige Anwendungen für die multigraduierte Dimensionstheorie werden vorgestellt. Der zweite Artikel behandelt Hilbertreihen von Moduln über einem standard-graduierten Polynomring über einem Körper. Ausgehend von einem grundlegenden Ergebnis über gewisse formale Laurentreihen werden unter anderem die möglichen Hilbertreihen und h-Vektoren solcher Moduln charakterisiert.

Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This thesis is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals. Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. First, we focus on the binomial edge ideals of block graphs. We give a lower bound for their Castelnuovo-Mumford regularity by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs. Secondly, we consider some classes of Cohen-Macaulay binomial edge ideals. We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs. Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. It is known that the polyomino ideal of simple polyominoes is prime. We consider multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.

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