Mathematics, particularly research mathematics, or mathematics that is close to the frontiers of research would be considerably easier to learn if mathematicians would both explain what they are going to do and explain what historical context motivates the problems or issues they are interested in. This would grant the needed insights and "intuitions" into the subject, which are absolutely necessary if one is to understand a particular mathematical topic in depth. In their papers, monographs, and textbooks, mathematicians could perhaps include at various places in the text some `fundamental insights' into the concepts that are being discussed. In addition, they could describe in detail what their goals are and what strategies they are going to use to solve the problems as they develop.
Unfortunately not many works of mathematics are written in this manner, and so those who wish to learn a given topic must frequently undertake time-consuming literature searches and solve myriads of exercises and problem sets in order to gain the needed insight. This takes large blocks of time, and poses an even greater challenge for those, such as physicists, who not must not only master the physics but also the mathematical formalism used to frame physical theories. Physicists would welcome, and even in many cases grab at straws to learn mathematics in a way that they need to in order to push forward the frontiers of their own subject.
This book though is very different, and is one of the best works of mathematics put in print in the last three decades. The author has given the reader a work that is not only mathematically rigorous but also fits the learning framework discussed above. There is no doubt that K-theory is a highly esoteric subject, but it can be learned much more easily by the study of this book. Within its covers there are myriads of fundamental insights that the author shares with the reader that make the learning of K-theory much more palatable and actually downright fun. It should not be thought however that the reader does not have to exercise a fair amount of cognition when wrestling with the intricacies of K-theory. This is true for K-theory as with other branches of mathematics, but those hungry for a true understanding of K-theory will deeply appreciate the author's efforts in this book.
The goal of K-theory is to generalize linear algebra, the latter of which deals with linear transformations on vector spaces over fields. K-theory tries to find out, and make rigorous, what constructions in linear algebra carry over when the field is replaced by a ring R and the vector space is replaced by a module over this ring. The first issue that must be dealt with is that of the concept of dimension, which for the case of a vector space is well defined (it is the cardinality of its basis). But an R-module does not necessarily have a basis. So the strategy deployed by the author is find the R-modules that do have a dimension. To find out what it means for an R-module to have a basis, the author constructs R-modules that are generated by elements that are not "linearly related." These are called `free' R-modules and the generating set is called an `R-basis.' The author then characterizes free modules that have a finite basis. That this is a non-trivial exercise is proven by the fact that every nontrivial finite Abelian group is a Z-module, where Z is the ring of integers. In addition, every set is a basis of a free R-module so one must find `presentations' of R-modules. These allow the construction of R-modules satisfying certain needed properties. And then, as expected if one is to extend linear algebra, the author constructs `matrices' of R-linear maps between finitely generated free R-modules. However, a free R-module can have a basis with unequal cardinality, and therefore the author finds those rings R whose free modules have unique dimension and those over which every module is free. The needed property is called an `invariant basis number' and it turns out that most rings have an invariant basis number. But some finitely generated R-modules the author points out are too "small" to be free, and so he finds the appropriate generalization of free modules. These are the famous `projective modules' and are the objects on which algebraic K-theory is based.
The designation of "projective" refers to the familiar notion of a projection in ordinary vector space theory, i.e. a linear, idempotent operator. The author describes projective modules as being the `direct summands' of free R-modules, and so to obtain the needed generalization of dimension he constructs Abelian monoids of R-modules under the direct sum operation. This involves finding a universal construction of an Abelian group from a semigroup and this leads to the famous Grothendieck group K0(R) of finitely generated projective R-modules. The finitely generated projective R-modules are `stably isomorphic' if they become equal in K0(R); they are `stably equivalent' if they become congruent in K0(R) modulo finitely generated free modules. The strategy then becomes that of adding R to two finitely generated R-modules to make them stably isomorphic (or stably equivalent). In addition, one must find out to what extent K0(R) determines whether a finitely generated projective R-module is free. This brings up the notion of an R-module being `stably free', and the author finds those stably free R-modules which are free. This involves the notion of the `matrix completion' of a ring and of "shortening" unimodular rows. The author also studies the connection of K0 with number theory, eventually showing that the projective class group is isomorphic to the ideal class group when R is a Dedekind domain.
K0(R) is an element of a sequence of Abelian groups associated to each ring R. To find K1(R), the author finds an analog of the row operations in ordinary linear algebra. The elements of K1(R), the `Bass-Whitehead group' are row-equivalence classes of invertible matrices. A group homomorphism from K1(R) to a group G is the analog of the determinant in ordinary linear algebra, and is often called the `Whitehead-Bass' determinant. K1(R) can be thought of as the "abelianization" of the general linear group GL(R). The elements of K2(R) consist of the relations among the generators of the group of row operations on a matrix. The "standard" relations among these operations give the `Steinberg group', and K2(R) is the center of this group.