Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

by Lok Tsan-ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-75). / Introduction --- p.2 / Chapter 1 --- "Rings, Modules and Categories" --- p.4 / Chapter 1.1 --- "Rings, Subrings and Ideals" --- p.5 / Chapter 1.2 --- Modules and Categories --- p.8 / Chapter 1.3 --- Module Theory --- p.13 / Chapter 2 --- Isomorphisms between Endomorphism rings of Quasiprogener- ators --- p.24 / Chapter 2.1 --- Preliminaries --- p.24 / Chapter 2.2 --- The Fundamental Theorem --- p.31 / Chapter 2.3 --- Isomorphisms Induced by Semilinear Maps --- p.41 / Chapter 2.4 --- Isomorphisms of General linear groups --- p.46 / Chapter 3 --- Endomorphism ring of projective module --- p.54 / Chapter 3.1 --- Preliminaries --- p.54 / Chapter 3.2 --- Main Theorem --- p.60 / Bibliography --- p.74

Η διπλωματική εργασία κινείται γύρω από το θεώρημα Quillen-Suslin (1976): “Κάθε πεπερασμένα γενόμενο προβολικό module επί του δακτυλίου των πολυωνύμων k[x_1,…,x_m ] (όπου k σώμα) είναι ελεύθερο”. Το πρόβλημα ξεκίνησε το 1955, όταν ο J. P. Serre, σε υποσημείωση της ένδοξης εργασίας του “Faisceaux Algebriques Coherents” (σελίδα 243), σημειώνει: “ On ignore s’il existe des A-modules projectifs de type fini qui ne soient pas libres” (A=k[x_1,…x_m ], k σώμα).* Το πρόβλημα λύθηκε από τους Quillen και Suslin (ανεξάρτητα) είκοσι χρόνια μετά. Για την απόδειξη του θεωρήματος είναι απαραίτητο το αποτέλεσμα που οφείλεται στον ίδιο τον Serre (1958): “ Κάθε πεπερασμένα γενόμενο προβολικό k[x_1,…,x_m ]-module P είναι σταθερά ελεύθερο” (δηλαδή το P δέχεται πεπερασμένα γενόμενο ελεύθερο συμπλήρωμα F, ώστε το P⊕F να είναι ελεύθερο). Στo Κεφάλαιο 2 αυτής της εργασίας, θα παρουσιάσουμε την απόδειξη του ανωτέρω θεωρήματος του Serre και τελικά, στο Κεφάλαιο 3, θα σκιαγραφήσουμε την απόδειξη του θεωρήματος Quillen-Suslin, με τη μέθοδο του Suslin. *Αγνοούμε αν υπάρχουν πεπερασμένα γενόμενα προβολικά A-modules που δεν είναι ελεύθερα. / This work is about the Quillen-Suslin Theorem (1976): “If k is a field , then every finitely generated projective k[x_1,…,x_m ]-module is free”. This problem started in 1955, when J.P. Serre, in his glorious paper “FaisceauxAlgebriquesCoherents” (page 243), noted: “On ignore s’ilexiste des A-modules projectifs de type fini qui ne soient pas libres ” (A=k[x_1,…x_m ],k is field).* This problem was solved from Quillen and Suslin (independently) twenty years after. For the proof of this theorem is necessary the result, due to Serre (1958): “Every finitely generated projective k[x_1,…,x_m ]-module P is stably free ” (ie. P admits a finitely generated free complement F, so that P⊕F is free). In Chapter 2 of this work, we will represent the proof of the above Serre’s Theorem and, finally, in Chapter 3, we will sketch the proof of Quillen-Suslin's Theorem, with Suslin’s method. *We ignore, if exist finitely generated projective A-modules, that they are not free.

In this paper we have developed a general theory of characteristic classes of modules. To a given invariant map defined on a Lie algebra, we associate a cohomology class by using the curvature form of a certain kind of connections. Here we present a very simple proof of the invariance theorem (Theorem 12), which states that equivalent connections give rise to the same characteristic class. We have used those invariant maps of {9} to define Chern classes of projective modules and we have derived their basic properties. It might be interesting to observe that this theory could be applied to define characteristic classes of bilinear maps. In particular, the Euler classes of {6} can be obtained in this way.

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.

Ce mémoire porte sur la théorie des représentations de l’algèbre quantique Uqsl2 en q une racine de l’unité. Il étudie plus précisément certains modules de l’algèbre LUqsl2, l’extension de Lusztig de Uqsl2, lorsque q² est une p-racine primitive de l’unité pour p un entier supérieur ou égal à 2. Quatre familles de LUqsl2-modules de dimension finie, qualifiés de modules remarquables, sont identifiées : les modules simples et projectifs ainsi que les modules et comodules de Weyl. L’algèbre Uqsl2 possède une structure d’algèbre de Hopf ; cette dernière peut être étendue sur LUqsl2. L’antipode découlant de cette structure permet de définir la notion de dualité de LUqsl2-modules, à partir de laquelle sont construits les comodules de Weyl, tandis que le coproduit permet de définir le produit tensoriel de LUqsl2-modules, aussi appelé la fusion de modules. Le mémoire détermine les règles de fusion des modules remarquables : le produit tensoriel de toute paire de modules remarquables est exprimé comme une somme directe de modules indécomposables. Quoique les règles de fusion entre modules simples et projectifs aient été obtenues par Bushlanov, Feigin, Gainutdinov et Tipunin (cf. [7]), celles impliquant au moins un module ou comodule de Weyl sont nouvelles. / This thesis is devoted to the representation theory of the quantum algebra Uqsl2 for q a root of unity. More precisely it studies some modules of the algebra LUqsl2, the Lusztig extension of Uqsl2, when q² is a primitive p-root of unity for p an integer greater than or equal to 2. Four families of finite dimensional LUqsl2-modules, called remarkable modules, are identified: simple and projective modules as well as Weyl modules and comodules. The algebra Uqsl2 has a Hopf algebra structure; the latter can be extended to LUqsl2. The antipode of this structure is used to define a duality of LUqsl2-modules, from which the Weyl comodules are built, while the coproduct is used to define a tensor product of LUqsl2-modules, also called fusion of modules. This thesis determines the fusion rules of remarkable modules: the tensor product of any pair of remarkable modules is expressed as a direct sum of indecomposable modules. Although the fusion rules between simple and projective modules were obtained by Bushlanov, Feigin, Gainutdinov and Tipunin (cf. [7]), those involving at least one Weyl module or comodule are new.