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The author begins the lectures with stating the main goal of the book, namely for proving that for the polynomial ring A in N variables over the integers or integers modulo p, the general linear group of n by n matrices GL(n,A) over this ring is finitely generated for n greater than or equal to N + 3. To meet this goal he reviews the properties of elementary matrices in lecture 1. For a ring A, by considering the elementary subgroups E(n,A) of GL(n,A), these subgroups consisting of matrices satisfying certain relations, the author shows that for a surjective ring homomorphism between rings A and A', the homomorphism from E(n,A) to E(n,A') is surjective, even though it is not for GL(n,A) to GL(n,A'). E(n,A) is shown to be stable under transposition and shown to be commutator subgroup of GL(n,A) for large n. This is the origin of the stability issues in K-theory, and these are discussed in lecture 2. The author shows just why it is advantageous to consider taking the union GL(A) of GL(n,A) (and E(n,A)) for all n and why stability is important in the proof of the above result.
The "Whitehead group" K(1,A) is defined as GL(A)/E(A), and its use in the proof of the theorem results from the map GL(n,A)/E(n,A) to K(1,A) being a bijection for large n and that K(1,A) is finitely generated. Following this matrix characterization of K(1,A), the author reduces the proof of the theorem to showing that for a "regular" ring A, every unipotent element in GL(A) represents 0 in K(1,A), and that the rings in the theorem are indeed regular. Noting the analogy between determinants of matrices and determinants of endomorphisms of vector spaces, the author begins the proof of these assertions with a different description of K(1,A). This description involves the consideration of Grothendieck and Whitehead groups of categories with exact sequences.
The Whitehead group is now defined as the quotient of the Grothendieck group, the latter being the abelian group whose generators are essentially isomorphism classes of objects from an admissible Abelian category. The Whitehead group K(1,A) for a ring A is then related to the Whitehead group K(1,M) for an admissible category M. This definition is due to Grothendieck and involves showing that their is an isomorphism between K(1,A) and K(1, P(A)) where P(A) is the category of finitely generated projective A-modules. P(A) is not abelian, and therefore must be enlarged, without changing K(1,A), to one that is. The author shows that P(A) must be abelian in order to kill unipotents K(1,A). The enlarged P(A) is abelian as long as A is regular, the latter meaning that A is right Noetherian and that any finitely generated A-module has finite homological dimension. As the name implies, homological dimensions involves some discussion of homology theory, and is defined to the least n for which there is a projective resolution of the A-module of length n. The proof of the above theorem then follows, as the author shows, from Hilbert's syzygy theorem.
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in detail....!
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An American Jew and an Arab-American write fact-filled, partisan opening statements, rebuttals and then enter into a face-to-face debate (transcribed and moderated by journalist and writer David Shipler).
Although written in the late 1980s, this book is exactly the right prescription for befuddled Americans who, more than they need a purportedly objective history of the region, need to get a pro-Arab and a pro-Israeli in a locked room to duke it out.
There are no easy answers, but readers get a look at the facts "through different eyes."
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To act both as a market bargaining agent and as a class actor has contradictions. It is widely held that most workers desire a union to be a body that focuses on and stabilizes their particular employment and wages - not those of an entire class. And resulting bargaining agreements tacitly assume that unions will exert control over workforces. It is a position that essentially leaves unchallenged the rights of capital to control workplaces and the economy, but it has proven to be quite vulnerable, as economic pressures have increased in the last of the 20th century. Early 20th century Italian factory councils or the more recent British shop steward system gained some degree of worker control on a location-specific basis though largely outside the purview of bureaucratic unions. But no trade union body has ever consolidated working class interests sufficiently to be able to cause a transformation of capitalism in favor of working class interests.
Political power or standing permeates the entire discussion. Repeatedly it is the ability of working class interests to be a part of the political process that is key to their relative standing in both the larger society and in workplaces. The social integration or corporatism that is found throughout continental Europe includes trade union central bodies in peak-level macroeconomic discussions as well as other working class considerations, such as the establishment of works councils in workplaces. Trade unions are, in essence, social partners in this climate. They are also class actors in their ability to aggregate working class political power. A purely collective bargaining approach seems to be indicative of minimal political power and, in this era, of minimal effectiveness.
There is no doubt that the book is thought provoking. Do the poles of market, class, and society provide an adequate framework for understanding trade unionism? For this reviewer, class and society would be subsumed under political power with that power countering and influencing economic power. The discussions of the three particular trade union movements are quite tedious and sometimes provide confusion as well as clarity as to how those movements fit in the author's general framework. The more theoretical parts of the book are also a little difficult to follow with the qualifiers and exceptions sometimes diluting the central argument. But I think the book should be read by anyone wanting to take a new look at understanding trade union orientation.