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In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.

In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.

Mali is currently the seventh largest country in Africa. It shares borders with Mauritania and Senegal in the west, Algeria in the north, Guinea and Ivory Coast in the south, and Burkina Faso and Niger in the east. After decades of dictatorship, in 1992, a new democratic constitution was adopted and today Mali is one of the most politically and socially stable countries in Africa. While Mali still has a long way to go with their economy_they are considered to be among the 10 poorest countries in the world_they continue to make progress and their increase in cereal and gold production are steps in the right direction. The fourth edition of the Historical Dictionary of Mali, through its chronology, bibliography, introductory essay, and over 500 cross-referenced dictionary entries on important persons, places, events, and institutions and significant political, economic, social, and cultural aspects, provides an important reference on this African country.

This book provides a radical new interpretation of the aims of the lesser German princes during the seventeenth and eighteenth centuries through the example of the duchy of Württemberg. Arguing that the princes' political ambitions were fundamental in shaping the internal development of their territories, the author sheds new light on the political importance of the notorious German 'soldier trade' and its role in international diplomacy. The wider social and political impact of these policies is also investigated in a comparative framework, while traditional interpretations of the dramatic struggle between duke and estates are challenged in a reassessment of the role of early modern representative institutions in German state development. The relationship of these internal political struggles to the different elements of the Holy Roman Empire is revealed, opening up new perspectives on the role of the German states within the imperial structure and revealing the empire as a flawed but functioning political system.

This book offers a balanced mixture of practice-oriented information and theoretical background as well as numerous references, clear illustrations, and useful data tables. Problems and solutions are accessible via a special website. This new edition has been completely revised and extended; it now includes three new chapters on tandem mass spectrometry, interfaces for sampling at atmospheric pressure, and inorganic mass spectrometry.