An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author's attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and ?; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.

An Introduction to Homological Algebra

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Group-graded rings, smash products, and group actions

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Continuous Étale Cohomology.

(1996). Hopf bigalois extensions. Communications in Algebra: Vol. 24, No. 12, pp. 3797-3825.

Hopf bigalois extensions

Introduction. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Noncommutative separable algebras have been studied in [2]. Commutative separable algebras have been studied in [1] and in [2], [6] where the main ideas are based on the classical Galois theory of fields. This paper depends heavily on these three papers and the reader should consult them for relevant definitions and basic properties of separable algebras. We shall be concerned with commutative separable algebras in two situations. Let R be an arbitrary commutative ring with no idempotents except 0 and 1. We first consider separable P-algebras that are finitely generated and projective as an P-module. We later drop the assumption that the algebras are projective but place restrictions on P — e.g., P a local ring or a Noetherian integrally closed domain. In §1 we give a proof due to D. K. Harrison that any finitely generated, projective, separable P-algebra without proper idempotents can be imbedded in a Galois extension of P also without proper indempotents. We also give a number of preliminary results to be used in later sections. In §2 we generalize some of the results about polynomials over fields to the case of aground ring P with no proper idempotents. We show that certain polynomials (called \"separable\") admit \"splitting rings\" which are Galois extensions of the ground ring. We apply this to show that any finitely generated, projective, separable homomorphic image of R[X] has a kernel generated by a separable polynomial. In §3 we restrict our attention to separable algebras over a local ring. (The term \"local\" will not imply any finiteness conditions.) In certain cases every finitely generated, separable algebra is a homomorphic image of a finitely generated, projective, separable algebra. This gives an external characterization of the separable algebras. In §4 we consider the internal structure of seperable algebras over a Noetherian integrally closed domain. We show that a finitely generated separable algebra is the direct sum of projective, separable domains containing the ground ring and an algebra that is separable but not faithful over the ground ring. For the case of a Dedekind domain we can obtain specific infor-

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Separable algebras over commutative rings

This paper has two sections concerned with the characterization of the algebraic structures of certain types of infinite abelian groups such as compact groups, or tensor products of torsion groups. It also has a section which gives an approach to the problems of mixed abelian groups (those which are neither torsion nor torsion-free) by establishing a duality between torsion groups and those abelian groups which are as " mixed as possible" (that is, they are as far away as possible from being just a direct sum of a torsion group and a torsion-free group). There is a final section containing some miscellaneous results on torsion-free abelian groups. For the reader unfamiliar with homological algebra, we give in Section 1 a brief outline of some results and methods of that theory. Almost all our proofs will depend heavily on homological methods. Section 2 is devoted to the duality of which we have spoken. A reduced group is called co-torsion if it is always a direct summand whenever it appears as a subgroup with a torsion-free quotient group. In the same way that Pontrjagin duality reduces the problems of compact groups to those of discrete groups, the problems of co-torsion groups are reduced to those of torsion groups by a one-to-one duality between the groups of these two classes. A method is then given by which all the mixed groups with a given torsion group can be constructed from the co-torsion group dual to that torsion group. The main result of Section 3 is that an abelian group can be a compact topological group if and only if it is isomorphic to a direct product (unrestricted direct sum) of copies of finite cyclic groups, p-adic integers, the reals, and the groups Z(p-) where, for each prime p, the number of copies of Z(p-) does not exceed the number of copies of the reals. It is also proved that if G is an abelian group with the multiples n! G considered as neighborhoods of the identity, then essentially G is a direct summand of a direct product of finite cyclic groups if and only if it is complete in the metric defined by these neighborhoods, while G is a direct sum of cyclic groups if and only if it is as far from complete as possible in this metric. At the very end of Section 3 it is shown that a torsion-free group is a direct summand of a direct product of finite 366

D. K. Harrison

Papers

Infinite Abelian Groups and Homological Methods

Galois Theory and Cohomology of Commutative Rings

Finite and infinite primes for rings and fields

Abelian extensions of commutative rings

A grothendieck ring of higher degree forms