Homological algebra. (English) Zbl 0075.24305

Princeton Mathematical Series. 19. Princeton, New Jersey: Princeton University Press. xv, 390 p. (1956).
The title “Homological Algebra” is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. The particular formal aspect of this theory stemming from algebraic topology is that of a preoccupation with endomorphisms of square 0 in graded modules. The conceptual flavor of homological algebra derives less specifically from topology than from the general “naturalistic” trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behavior under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones.
In the recent requisite mathematical terminology, this means that the principal objects of study in homological algebra are functors from categories of modules to other categories of modules. If \(R\) and \(S\) are two rings (with identity elements that act as the identity on all modules considered) a covariant functor (of one variable) from the category of \(R\)-modules to the category of \(S\)-modules is a function \(T\) attaching to each \(R\)-module \(A\) an \(S\)-module \(T(A)\), and to each \(R\)-homomorphism \(h: A'\to A\) an \(S\)-homomorphism \(T(h): T(A') \to T(A)\) such that, if \(k\) is a second homomorphism \(A\to A''\), \(T(k \circ h) = T(k) \circ T(h)\), and \(T(h)\) is the identity map \(T(A)\to T(A)\) whenever \(h\) is the identity map \(A\to A\). For a contravariant functor, one has instead \(T(h): T(A)\to T(A')\), and \(T(h)\circ T(k)\). The definition of a functor in several variables in different categories, and some covariant, some contravariant, is the evident extension of the above, with an additional commutativity requirement for maps acting on arguments in different positions in the functor.
The functor \(T\) is said to be additive if, for any two homomorphisms \(h_1\) and \(h_2\) between the same modules, one has \(T(h_1+h_2) = T(h_1) + T(h_2)\). As a linear theory, homological algebra deals only with additive functors.
A functor is called exact if (for each variable separately) it maps every exact sequence into an exact sequence. It is the essence of homological algebra to channel any deviation of a functor from exactness into the construction of new functors. For these constructions, the following notions are basic.
A module \(M\) is called projective if every homomorphism of \(M\) into a factor module \(A/B\) can be lifted to \(A\); \(M\) is called injective if every homomorphism of \(B\) into \(M\) can be extended to \(A\). Every module is a homomorphic image of a projective module and can be imbedded in an injective module. Hence every module \(A\) has a “projective resolution”, i.e., an exact sequence
\[ \cdots \to X_n\to \cdots \to X_0\to A\to 0\]
in which the \(X_i\) are projective, and an “injective resolution”, i.e., an exact sequence
\[ 0\to A\to X^0\to \cdots \to X^n\to \cdots \]
in which the \(X^i\) are injective. If projective resolutions are substituted for all the contravariant variables in a functor and injective resolutions are substituted for all the covariant variables a complex is obtained whose homology groups are, up to natural isomorphisms, independent of the particular choice of the resolutions. The \(n\)-th homology group of this complex is the module value of the \(n\)-th right derived functor \(R^nT\) of the given functor \(T\). Similarly, one obtains the left derived functors \(L_nT\) by substituting injective resolutions for the contravariant variables and projective resolutions for the covariant variables. These derived functors have the same variance as \(T\), and vanish for \(n<0\).
The derived functors are related by the so-called connecting homomorphisms which increase the degree by 1 in the case of the right derived functors and lower the degree by 1 in the case of the left derived functors. These result from any exact sequence when \(0\to A'\to A\to A''\to 0\) when \(A', A, A''\)are substituted in turn for any one of the variables. If the functor is covariant in this variable the connecting homomorphism sends \(R^nT(A'')\)) into \(R^{n+1}T(A')\) and, with the usual functor homomorphisms, forms an exact sequence
\[ \cdots\to R^nT(A')\to R^nT(A)\to R^nT(A'') \to R^{n+1}T(A')\to \cdots. \]
The case of contravariance, and the case of the left derived functors, are obtained by making the evident appropriate changes.
A covariant functor \(T\) of one variable is said to be left exact if, for every exact sequence \(0\to A'\to A\to A''\to 0\) the sequence \(0\to T(A')\to T(A)\to T(A'')\) is exact. In case \(T\) is contravariant, interchange \(A''\) and \(A'\) in the last sequence. For functors of several variables, one demands that this property hold for each variable. One says that \(T\) is right exact if, instead of the above, the sequence \(T(A')\to T(A)\to T(A'')\to 0\) is exact, etc. If \(T\) is left exact then the left derived functors \(L_nT\) are 0 (except for \(n = 0)\) and \(R^0T\) may be identified with \(T\). If \(T\) is right exact the right derived functors \(R^nT\) are 0 (except for \(n = 0)\) and \(L_0T\) may be identified with \(T\).
A functor \(T\) is said to be right balanced if substitution of any projective module for a contravariant variable, or substitution of any injective module for a covariant variable, yields an exact functor in the remaining variables. The definition of a left balanced functor is obtained by interchanging the words projective and injective. Much of the power of the mechanism of derived functors results from the following facts: if \(T\) is right balanced then its full right derived functors \(R^nT\) may be identified with the partial right derived functors obtained by using resolutions for any non-empty subset of variables and treating all the other variables as constants in the construction; if \(T\) is left balanced then the same result holds for the left derived functors \(L_nT\).
The above basic principles and results are established in chapters I–V. Chapter VI concentrates on the two basic functors of homological algebra and their derived functors. The first basic functor is the twice covariant functor \(\otimes_R\) that attaches to each pair \((A,B)\), where \(A\) is a right \(R\)-module and \(B\) a left \(R\)-module, the tensor product \(A\otimes_RB\), regarded as a module over the ring \(Z\) of the integers, in general. This functor is right exact and left balanced. The second basic functor is the contravariant-covariant functor \(\operatorname{Hom}_R\) that attaches to each pair \((A,B)\) of left \(R\)-modules the \(Z\)-module \(\operatorname{Hom}_(A,B)\) of all \(R\)-homomorphisms of \(A\) into \(B\). This functor is left exact and right balanced. The left derived functors of \(\otimes_R\) are denoted \(\text{Tor}_n^R\), and the right derived functors of \(\operatorname{Hom}_R\) are denoted \(\text{Ext}_R^n\).
Some of the simpler interpretations and applications of these functors are given in Chapters VI and VII. An \(R\)-module \(A\) is said to be of projective dimension \(\le n\) if it has a projective resolution \(\cdots\to X_i\to\cdots\to X_0\to A\to 0\) such that \(X_i = 0\), for all \(i > n\). There is a similar, but less frequently used, notion of injective dimension. One says that a ring \(R\) has left global dimension \(\le n\) if every left \(R\)-module has projective dimension \(\le n\) or – equivalently – if \(\text{Ext}_R^{n+1}=0\). Another equivalent condition is that every left \(R\)-module be of injective dimension \(\le n\).
The significance of this notion for ring theory is already indicated by the following facts: the rings of global dimension 0 are precisely the semisimple rings, i.e., the rings \(R\) such that every \(R\)-module is a direct sum of simple submodules; a ring has left global dimension \(\le 1\) if and only if every left ideal is a projective module; an integral domain has this property if and only if it is a Dedekind ring, i.e., if and only if the classical ideal theory holds for the ring.
The functor \(\text{Tor}_1^R\) plays an important role in the Künneth relations between the homology groups of a tensor product of complexes and those of the factors. In fact, it arose for the first time exactly in this context. Here, the Künneth relations are obtained in full group-theoretic form and under far more general conditions than those obtaining in the original topological situation of the integral homology groups of product spaces.
Chapter VIII introduces the homology theory of augmented rings from which the specific homology (and cohomology) theories of associative algebras, groups, and Lie algebras are obtained by specialization. The structure of an augmented ring consists of a ring \(R\) and an \(R\)-epimorphism of \(R\) onto an \(R\)-module \(Q\). The \(n\)-th homology group of the augmented ring \(R\) in a right \(R\)-module \(A\) is defined as \(\text{Tor}_n^R(A,Q)\). The \(n\)-th cohomology group of \(R\) in a left \(R\)-module \(A\) is defined as \(\text{Ext}_R^n(Q,A)\). These notions apply rather directly to a dimension theory for local rings and graded rings yielding, for instance, various generalizations and extensions of Hilbert’s theorem on chains of syzygies of homogeneous polynomial ideals.
Chapter IX discusses the homology and cohomology theory of associative algebras \(L\) (with 1) over commutative rings \(K\) (with 1). Define \(L^e\) as the tensor product algebra \(L\otimes_K L'\), where \(L'\) denotes the usual anti-isomorph of \(L\). Together with the \(L^e\)-epimorphism \(L^e\to L\) that sends \(a\otimes b'\) onto \(ab\) this is an augmented ring structure. By using a suitable \(L^e\)-projective resolution of \(L\), it is shown that the usual cohomology groups \(H^n(L,A)\) of \(L\) in an \(L^e\)-module \(A\) are nothing but the groups \(\text{Ext}_L^n e(L,A)\). This makes the general theory available for the study of the cohomology of algebras and thus gives a high degree of control over it. Of particular interest is the cohomological dimension of \(L\) which is defined to be \(\le n\) if and only if \(H^{n+1}(L,A) = 0\) for all \(L^e\)-modules \(A\).
The homology theories for groups and Lie algebras have certain special features in common, and Chapter X introduces the requisite prespecialization of the augmented ring theory. The basic structure is that of a supplemented algebra. It consists of a \(K\)-algebra \(L\), together with a unitary algebra epimorphism of \(L\) onto \(K\). Close relations exist between the homology of \(L\) as a supplemented algebra and the homology of \(L^e\), augmented as above. In many situations, this leads to a strengthening of each theory. The homology and cohomology groups of a group \(G\) are obtained from the supplemented algebra structure consisting of the group algebra \(Z(G)\) and the coefficient sum epimorphism \(Z(G)\to Z\).
Chapter XI gives the general foundation for the multiplicative theory. Four “external products” are defined, involving two \(K\)-algebras \(L\) and \(M\), their tensor product \(L\otimes_K M\) and the functors \(\text{Tor}\) and \(\text{Ext}\) for these three algebras. “Internal products”; involving only one algebra, can be derived from these with the aid of algebra homomorphisms \(L\otimes_K L\to L\) or \(L\to L\otimes_K L\). In particular, in the case of groups or Lie algebras, one has natural homomorphisms \(L\to L\otimes_K L\) by means of which the cup and cap products are derived from two of the external products. The most complete and most powerful cohomology theory is that of a finite group \(G\).
Chapter XII develops the special formal machinery that governs this case. The main point (discovered by J. T. Tate in connection with the application to class field theory) is that the norm map in \(G\)-modules \((N(a) = \sum_{x\in G} x\cdot a)\) leads to a tie-up between homology and cohomology by which the homology groups play the role of cohomology groups of negative degree and link up with the usual cohomology groups to form the so-called complete derived sequence of \(G\). The connecting homomorphisms go through this complete sequence, and a product can be defined which extends simultaneously the cup and the cap product and applies without restriction to arbitrary members of the complete derived sequence.
Chapter XIII brings the homology and cohomology theory of Lie algebras into the framework of supplemented algebras. The critical device for doing this is to pass to the universal enveloping algebra; with every Lie algebra \(L\) one can uniquely associate an associative algebra \(L^e\) in such away that there is a natural \(1-1\) correspondence between the representations of \(L\) and those of \(L^e\). Assuming that \(L\) is free over the ground ring \(K\), one has \(L\subset L^e\), and the commutation in \(L\) is given by the multiplication in \(L^e\): \([x,y] = xy - yx\). There is a natural algebra epimorphism of \(L^e\) onto \(K\) whose kernel coincides with the subalgebra of \(L^e\) that is generated by \(L\). This is the supplemented algebra structure whose homology and cohomology is that of \(L\). It is, a priori, not at all evident that this actually yields the theory as originally formulated under the inspiration of the differential geometry on Lie groups. The identification requires the use of a certain \(L^e\)-projective resolution of \(K\) involving the exterior algebra built over \(L\).
Interesting by-products are obtained concerning \(L^e\). The cohomological dimension of \(L^e\) coincides with the projective dimension of \(K\) as an \(L^e\)-module. If \(K\) is semisimple the cohomological dimension of \(L^e\) coincides with its right and left global dimensions. In particular, if \(L\) has rank \(n\) over \(K\), these dimensions are all equal to \(n\). This generalizes earlier results on polynomial algebras, because if \(L\) is abelian then \(L^e\) is simply \(K[x_1, \ldots, x_n]\).
Chapter XIV takes up the applications of cohomology theory to various extension theories. The simplest case is that of extensions of \(R\)-modules. To each such extension \(0 \to C\to X\to A\to 0\) one can associate an element of \(\text{Ext}_R^1(A, C)\) which is 0 if and only if the extension is split. This correspondence yields an isomorphism of the group of equivalence classes of the extensions of \(C\) by \(A\) (with the Baer composition) onto \(\text{Ext}_R^1(A, C)\). This is actually the origin of the notation \(\text{Ext}\).
The other three cases are extensions of \(K\)-projective \(K\)-algebras with kernels of square 0, group extensions with abelian kernels, and Lie algebra extensions with abelian kernels. In each of these cases one obtains an interpretation of the second cohomology group as a group of equivalence classes of extensions.
Chapter XV gives a complete development of the formalism of spectral sequences. In particular, two spectral sequences are associated with every double complex. The terms \(E_\infty\) of these spectral sequences are the graded modules associated with the homology module by the two natural filtrations of the double complex. The terms \(E_2\) of the spectral sequences are the compound homology modules obtained from the double complex by using the two differential operators in succession.
In Chapter XVI it is shown how these spectral sequences can be used for establishing connections between partial derived functors and full derived functors, and for investigating the behavior of Tor and Ext under homomorphisms of operator rings. It is also shown how the general method yields the spectral sequences that have been used for examining the relations between the homology and cohomology of a group or a Lie algebra and that of a normal subgroup or an ideal. The rest of Chapter XVI sketches some topological applications concerning spaces with groups of operators.
Chapter XVII reaches a very high degree of complexity. Resolutions of complexes take the place of resolutions of modules, and spectral sequences are obtained which connect the results of first applying a functor and then passing to homology with the results of first passing to homology and then applying the functor. In particular, this leads to a fuller and more general treatment of the Künneth relations.
The appendix by D. A. Buchsbaum proposes an abstract framework of “exact categories” that is capable of accommodating the functor theory of this book as well as additional structural elements that one may wish to introduce. The proposed theory includes an abstract notion of duality which makes it unnecessary, at least in principle, to give separate treatments for covariance and contravariance and for projectivity and injectivity.
The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level. It is probably with such expectations that the authors have put so much missionary zeal into the systematization of their approach and the cataloguing of the basic results.
A probably unavoidable effect of this is that the book cannot be consulted by the uninitiated in a local fashion. The reader is definitely forced to go through it starting at the beginning. Each chapter (with the exception of the last) is followed by a collection of exercises which are designed not so much to strengthen the reader by easy gymnastics (for they are generally not particularly easy) as to point out various ramifications and applications of the general theory.
A few misprints might lead to confusion: p. 80, in the diagram above Prop. 2.3, the top \(X\) should have a double prime and the \(X\)s in the bottom row should have single primes; p. 101, first line below Lemma 10.1, the middle term of the sequence should have a double prime; p. 152, lines 4 and 7, replace \(M/J_k\) by \(M/MJ_k\) and \(M/J_{k-1}\) by \(M/MJ_{k-1}\); p. 233, diagram (1), reverse the arrow labelled \(g\); p. 319, line 5 from bottom, replace \(d_r\) by \(d_k\); p. 322, third line above Theorem 3.2, replace \(dsx\) by \(sdx\).

MathOverflow Questions:

Cup product in Tate Cohomology Ring


18-02 Research exposition (monographs, survey articles) pertaining to category theory
18Gxx Homological algebra in category theory, derived categories and functors