Higher Algebraic K-Theory: An Overview (Lecture Notes in Mathematics) [Englisch] [Taschenbuch]

This book goes considerably further then these relatively elementary considerations, in that it treats the higher K-groups and the connection with topological K-theory. Readers will need an extensive background in algebra and topology to appreciate the constructions in this book, which are mostly formal and thus there is the canonical inverse relationship between rigor and understanding. There are many places in the book though where readers can gain useful insights into a mature and highly developed branch of mathematics.

As was hinted above, for a ring R, K0 gives a measure of the failure of finitely generated projective R-modules from having a dimension theory like that of vector spaces. The first algebraic K-group K1 of a ring R is the quotient group of the infinite general linear group GL(R) modulo the infinite elementary group E(R) (the infinite elementary group comes from considering those matrices which differ from the identity only by an off diagonal element). Whitehead's Lemma shows that E(R) is a normal subgroup of GL(R). One can show that K1 of the integers is just {-1, 1}, and, for more general commutative rings R, that the determinant on GL(R) to the units R* of R induces a universal homomorphism and K1(R) is equal to these units. Thus the determinant gives in this case a universal invariant as was noted above. The second algebraic K-group of a ring R is then defined by generalizing the elementary group to the `Steinberg group' and then taking limits. There is an epimorphism from the Steinberg group to the elementary group and after passing to the infinite limit, the kernel of this epimorphism is defined as the second algebraic K-group of the ring R. K2(R) measures to what extent the Steinberg relations do not define the relations for the elementary group. One can show that K2 of the integers Z is Z/2, and that K2 of the direct product of two rings is the direct sum of their K2 groups.

Topological K-theory, also discussed in detail throughout the book, has its origins in the theory of vector bundles. Two vector bundles are called `stably-equivalent' if they are isomorphic after taking their direct sum with trivial bundles. Stable equivalence forms an equivalence relation and the stable classes form a ring under direct sum and tensor product. This ring is called the K-ring K(X) of the space X on which the vector bundles are defined. If X is compact and E is a vector bundle over X, then the sheaf of sections of this vector bundle is a finitely generated projective module over the ring C(X) of continuous functions on X. This result is known as the Serre-Swan theorem and allows one to discuss the K-theory of the space X in terms of the K-theory of C(X). The properties of this K-theory satisfy those needed to make it a cohomology theory, except for the dimension axiom. Topological K-theory also has the property of Bott periodicity, wherein the K-groups at one dimension are isomorphic to those of two dimensions less.

The higher topological K-theory groups have a counterpart in algebraic K-theory. This can be shown in several different ways, but this book discusses the Quillen or `Q-construction' of higher algebraic K-theory. Dependent on the notion of a `nerve' of a category and its classifying space, the Q-construction involves starting with an exact category M and defining a new category QM with the same objects but with morphisms satisfying certain properties of admissibility and composition. For a small exact category M, the ith higher algebraic K-group of this category is defined as the (i+1)-th homotopy group of the classifying space of M. The book also discusses, for a ring R, the `+-construction' of Quillen, which was the first definition of higher algebraic K-theory, and is considerably less esoteric than the Q-construction since it involves the well-known result that GL(R) is the first homotopy group of the classifying space of GL(R) and the intuitive geometric construction of adding cells to the classifying space to form a new space that has certain useful properties. The ith-higher algebraic K-group is then defined as the ith-homotopy group of this space. Although it is not done in this book, this definition coincides with the Q-construction when the latter is applied to the category of finitely generated projective R-modules.