We give a projective resolution of a finite dimensional 𝛫-algebra 𝛬 over its enveloping algebra 𝛬<SUP>𝑒</SUP> = 𝛬<SUP>𝑜𝑝</SUP> ⨂<SUB>𝛫</SUB>𝛬. The description of this resolution is related to decompositions of the first syzygy module of 𝛬 as an 𝛬<SUP>𝑒</SUP> module. Resolutions of right 𝛬 modules 𝑀<SUB>𝛬</SUB> may be obtained by tensoring 𝑀 over 𝛬 with this bimodule resoution. We describe how to obtain such a resolution when 𝑀 is simple or when 𝑀 is given in the form of a projective presentation. Computations of <I>𝐸𝑥𝑡</I><SUB>𝛬</SUB><SUP>𝑛</SUP>(𝑆<SUB>𝑣</SUB>,𝑆<SUB>𝑤</SUB>) for certain classes of algebras 𝛬 are made using these resolutions, and applied to obtain results on global dimension. / Ph. D.

L'implicitisation d'une surface algébrique rationnelle, c'est-à-dire le passage de la paramétrisation à une représentation implicite, est un<br />problème géométrique classique. Dans ce travail de thèse, nous utilisons la théorie des syzygies pour représenter implicitement une surface par une matrice dont les mineurs de taille maximale ont l'équation implicite comme plus grand diviseur commun. Dans les deux premiers chapitres, nous traitons deux classes de surfaces spéciales pour lesquelles il est toujours possible de construire une matrice carrée qui correspond au résultant d'une μ-base : les surfaces réglées et les surfaces canales. Dans les chapitres suivants, le cas général de surfaces rationnelles paramétrées sur une variété torique de dimension 2 est étudié. Nous montrons qu'une telle matrice peut être construite en n'utilisant que des syzygies linéaires et nous décrivons un algorithme simple et efficace pour son calcul.

Tese(doutorado) - Universidade Federal do Rio Grande, Programa de Pós–Graduação em Oceanografia Biológica, Instituto de Oceanografia, 2012. / Submitted by Cristiane Gomides (cristiane_gomides@hotmail.com) on 2013-11-18T14:07:49Z No. of bitstreams: 1 anders.pdf: 5772507 bytes, checksum: e763efddc5c877c6fc7d449f1cc01c6d (MD5) / Approved for entry into archive by Angelica Miranda (angelicacdm@gmail.com) on 2013-11-18T19:18:53Z (GMT) No. of bitstreams: 1 anders.pdf: 5772507 bytes, checksum: e763efddc5c877c6fc7d449f1cc01c6d (MD5) / Made available in DSpace on 2013-11-18T19:18:53Z (GMT). No. of bitstreams: 1 anders.pdf: 5772507 bytes, checksum: e763efddc5c877c6fc7d449f1cc01c6d (MD5) Previous issue date: 2012 / O caranguejo-uçá, Ucides cordatus, é uma espécie chave para o manguezal, tem grande importância socioeconômica e seus estoques vêm sendo reduzidos por mortalidades em massa. Para subsidiar a gestão deste recurso pesqueiro, este estudo objetivou gerar conhecimento sobre sua reprodução e recrutamento. As ocorrências das chamadas “andadas” reprodutivas foram registradas de 2006 a 2011, sendo que, em 2008 e 2010, a abundância e comportamento de U. cordatus foram monitorados. Um total de 1441 caranguejos foi observado, dos quais 80% exploravam tocas. Em torno de 90% dos caranguejos que estavam explorando tocas eram machos a procura de parceiras. Foram registrados 78 caranguejos escalando árvores, sendo 78% deles, fêmeas realizando extrusão de ovos. Estes movimentos foram utilizados para análise crítica de classificações unificadoras de movimento animal e concluiu-se que é ainda necessária uma nomenclatura que permita uma classificação sem ambiguidades e um agrupamento coerente de movimentos de diferentes taxa, de acordo com suas causas evolutivas. Foi constatado que as andadas podem ocorrer de dia e de noite e que o ritmo de procura de parceiros é ligado ao “ciclo de desigualdade de maré de sizígia”. Em sincronia com este ciclo, as andadas variaram entre lua cheia ou nova, dependendo de qual fase apresentava a maior amplitude de maré. A causa evolutiva mais provável da andada é permitir a maior sobrevivência das larvas através da liberação sincronizada com a maré de maior amplitude, um mês depois. Este conhecimento facilitará a determinação de períodos de defeso, reduzindo conflitos entre os coletores e o órgão gestor. Para estudar o recrutamento, juvenis foram amostrados mensalmente de Abril/08 a Maio/09. Dos 474 recrutas encontrados, 96% estavam associados a tocas de coespecíficos, indicando facilitação ecológica. O recrutamento foi maior em zonas intermediárias do gradiente ambiental, com altura de inundação estimada de 18,5 cm, evitando a alta predação das zonas mais inundadas e o estresse físico-químico das menos inundadas. Amostragens em outras 6 áreas, com diferentes padrões de zonação, também revelaram maior recrutamento em zonas de manguezal arbóreo menos inundadas, próximas à transição para terra-firme. Ressalta-se, portanto, a importância da conservação e da nomenclatura adequada desta zona de transição, sugerindo-se que esta seja definida tendo como critério a estrutura herbácea de sua vegetação e não medições de salinidade e inundação, que podem variar espaço-temporalmente e, portanto, serem subjetivas. O conhecimento sobre recrutamento habitat específico poderá indicar se ações de repovoamento são realmente necessárias para a recuperação de estoques de U. cordatus afetados por mortalidades em massa. / The land crab, Ucides cordatus, is a keystone species for the mangrove, has an important socioeconomic role and their stocks have been reduced by mass mortalities. To support the management of this fisheries resource, the aim of the study was produce knowledge about its reproduction and recruitment. The occurrences of the so-called reproductive andadas (= “walk” in Portuguese), were recorded from 2006 to 2011. In 2008 and 2010, the abundance and behaviour of U. cordatus were also monitored. A total of 1441 crabs were observed, 80% exploring burrows. About 90% of exploring crabs were males searching for mates. We recorded 78 crabs climbing on trees and 78% of them were females extruding eggs. We used these movements for the critical assessment of unifying animal movement classifications. We conclude that a nomenclature allowing an unambiguous classification and a coherent grouping of movements across different taxa, according to their ultimate causes is still lacking. We found that andada occurred during the day and night and that the rhythm of mate searching was linked to the “syzygy tide inequality cycle” (STIC). In sync with this cycle, andada shifted between new and full moon, depending upon which moon phase had the higher amplitude tides. The likely ultimate cause of andada is increased larval survival after synchronous release at highest amplitude spring tides one month later. This knowledge will support the establishment of capture bans and reduce conflicts between fishers and regulatory agencies. To study the recruitment, juveniles were sampled monthly from April 2008 to May 2009. Of the 474 recruits found, 96% were associated with burrows of conspecifics, indicating ecological facilitation. Recruitment was higher in intermediate zones of the environmental gradient, with 18.5 cm of estimated flood height, avoiding high predation at the most inundated zone and the physical-chemical stress at the less flooded zone. Sampling in other 6 areas, with different zonation patterns, also showed higher recruitment in intermediate zones of arboreous mangrove near the transition to hinterland. Therefore it was emphasized the importance of conservation and of a proper nomenclature of this transition zone. It is suggested that the transition zone must be defined based on the herbaceous structure of its vegetation rather than measurements of salinity and inundation, which can vary in space and time and therefore be subjective. Knowledge about habitat-specific recruitment could indicate whether stock enhancement actions are really necessary for the recovery of U. cordatus stocks affected by mass mortalities.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / David Mumford introduced the concept of regularity of a coherent beam into the projective space in terms of local cohomology, generalizing a classic argument of Castelnuovo. In this dissertation under view of commutative algebra, we will introduce the concept of regularity of finitely generated graduated modules on the ring of polynomials. First, we perform a preliminary study on dimension theory and especially on Hilbert’s function. We also studied the basics of Cohen- Macaulay modules, properties of Betti’s graduated numbers, and the local cohomology functors. In the main chapter, we define the regularity of Castelnuovo-Mumford using the free resolution shifts. Soon after, we show that the definition of regularity can be given in terms of local cohomology, with emphasis on the cases of Artinian and Cohen-Macaulay modules. / David Mumford introduziu o conceito de regularidade de um feixe coerente no espac¸o projetivo em termos de cohomologia local, generalizando um argumento cl´assico de Castelnuovo. Nessa dissertac¸ ˜ao sob a vis˜ao da ´algebra comutativa, introduziremos o conceito de regularidade de m´odulos graduados finitamente gerados sobre o anel de polinˆomios. Primeiramente realizamos um estudo preliminar sobre teoria da dimens˜ao e em especial sobre a func¸ ˜ao de Hilbert. Tamb´em estudamos noc¸ ˜oes b´asicas em m´odulos Cohen-Macaulay, propriedades dos n´umeros graduados de Betti e dos funtores de cohomologia local. No cap´ıtulo principal, definimos a regularidade de Castelnuovo-Mumford utilizando os shifts de resoluc¸ ˜oes livres. Logo ap´os, mostramos que a definic¸ ˜ao de regularidade pode ser dada em termos de cohomologia local, dando ˆenfase aos casos de m´odulos Artinianos e Cohen-Macaulay. / São Cristóvão, SE

If <i>E</i> is an exterior algebra on a finite dimensional vector space and <i>M</i> is a graded <i>E</i>-module, the relationship between the resonance varieties of <i>M</i> and the minimal free resolution of <i>M </i>is studied. In the context of Orlik–Solomon algebras, we give a condition under which elements of the second resonance variety can be obtained. We show that the resonance varieties of a general <i>M</i> are invariant under taking syzygy modules up to a shift. As corollary, it is shown that certain points in the resonance varieties of <i>M</i> can be determined from minimal syzygies of a special form, and we define syzygetic resonance varieties to be the subvarieties consisting of such points. The (depth one) syzygetic resonance varieties of a square-free module <i>M</i> over <i>E</i> are shown to be subspace arrangements whose components correspond to graded shifts in the minimal free resolution of <i>_SM</i>, the square-free module over a commutative polynomial ring <i>S </i>corresponding to <i>M</i>. Using this, a lower bound for the graded Betti numbers of the square-free module<i> M</i> is given. As another application, it is shown that the minimality of certain syzygies of Orlik–Solomon algebras yield linear subspaces of their (syzygetic) resonance varieties and lower bounds for their graded Betti numbers.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.