Group cohomology


In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients.

These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of can be thought of as the singular cohomology of the circle S1, and similarly for and

A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.


A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. We will write G multiplicatively and M additively.

Given such a G-module M, it is natural to consider the submodule of G-invariant elements:


Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in   are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology   is to precisely measure this difference.

The group cohomology functors   in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.


The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property   for all g in G and x in M). Sending each module M to the group of invariants   yields a functor from the category of G-modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors.[a] Their values are abelian groups and they are denoted by  , "the n-th cohomology group of G with coefficients in M". Furthermore, the group   may be identified with  .

Cochain complexesEdit

The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful.[1] For  , let   be the group of all functions from   to M (here   means  ). This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms


One may check that   so this defines a cochain complex whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex


Here the groups of n-cocycles, and n-coboundaries, respectively, are defined as


The functors Extn and formal definition of group cohomologyEdit

Interpreting G-modules as modules over the group ring   one can note that


i.e., the subgroup of G-invariant elements in M is identified with the group of homomorphisms from  , which is treated as the trivial G-module (every element of G acts as the identity) to M.

Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism


These Ext groups can also be computed via a projective resolution of  , the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective  -resolution (e.g. a free  -resolution) of the trivial  -module  :


e.g., one may always take the resolution of group rings,   with morphisms


Recall that for  -modules N and M, HomG(N, M) is an abelian group consisting of  -homomorphisms from N to M. Since   is a contravariant functor and reverses the arrows, applying   to F termwise and dropping   produces a cochain complex  :


The cohomology groups   of G with coefficients in the module M are defined as the cohomology of the above cochain complex:


This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of  , that is, functions   that obey


The coboundary operator   is now naturally defined by, for example,


The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains  , is given by reparameterizing so that


and so on. Thus


as in the preceding section.

Group homologyEdit

Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m − m, g ∈ G, m ∈ M. Assigning to M its so-called coinvariants, the quotient


is a right exact functor. Its left derived functors are by definition the group homology


The covariant functor which assigns MG to M is isomorphic to the functor which sends M to   where   is endowed with the trivial G-action.[b] Hence one also gets an expression for group homology in terms of the Tor functors,


Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches:

  • superscripts correspond to cohomology H* and invariants XG while
  • subscripts correspond to homology H and coinvariants XG := X/G.

Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolution F of the trivial  -module   as in the previous section. Apply the covariant functor   to F termwise to get a chain complex  :


Then Hn(G, M) are the homology groups of this chain complex,   for n ≥ 0.

Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.

The group homology   of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra  .[c]

Low-dimensional cohomology groupsEdit

H 1Edit

The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : GM satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : GM given by f(g) = gmm for some fixed mM. This follows from the definition of cochains above.

If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms GM, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i.e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary.

On the other hand, consider the case of   where   denotes the non-trivial  -structure on the additive group of integers, which sends a to -a for every  ; and where we regard   as the group  . By considering all possible cases for the images of  , it may be seen that crossed homomorphisms constitute all maps   satisfying   and   for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy   for some integer m: hence every crossed homomorphism   sending -1 to an even integer   is principal, and therefore:


with the group operation being pointwise addition:  , noting that   is the identity element.

H 2Edit

If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H2(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H2(G,M) classifies the isomorphism classes of all extensions   of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with an isomorphic G-module structure.

In the example from the section on   immediately above,   as the only extension of   by   with the given nontrivial action is the infinite dihedral group, which is a split extension and so trivial inside the   group. This is in fact the significance in group-theoretical terms of the unique non-trivial element of  .

An example of a second group cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:


See also [[1]].

Basic examplesEdit

Group cohomology of a finite cyclic groupEdit

For the finite cyclic group   of order   with generator  , the element   in the associated group ring is a divisor of zero because its product with  , given by




This property can be used to construct the resolution[2][3] of the trivial  -module   via the complex


giving the group cohomology computation for any  -module  . Note the augmentation map gives the trivial module   its  -structure by


This resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groups


showing that applying the functor   to the complex above (with   removed since this resolution is a quasi-isomorphism), gives the computation




For example, if  , the trivial module, then  ,  , and  , hence


Explicit cocyclesEdit

Explicit cocycles for the group cohomology of a cyclic group using the Bar resolution can be given explicitly[4]prop 2.3. We get a complete set of generators of  -cocycles for   odd as the maps


given by


for   odd,  ,   a primitive  -th root of unity,   a field containing  -th roots of unity, and


for a rational number   denoting the largest integer less than  . Also, we are using the notation


where   is a generator for  . Note that for   non-zero even indices the cohomology groups are trivial.

Cohomology of free groupsEdit

Using a resolutionEdit

Given a set   the associated free group   has an explicit resolution[5] of the trivial module   which can be easily computed. Notice the augmentation map


has kernel given by the free submodule   generated by the set  , so


Because this object is free, this gives a resolution


hence the group cohomology of   with coefficients in   can be computed by applying the functor   to the complex  , giving


this is because the dual map


sends any  -module morphism


to the induced morphism on   by composing the inclusion. The only maps which are sent to   are  -multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other maps


can be generated by the  -basis of maps sending   for a fixed  , and sending   for any  .

Using topologyEdit

The group cohomology of free groups   generated by   letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group   there is a topological space  , called the classifying space of the group, which has the property


In addition, it has the property that its topological cohomology is isomorphic to group cohomology


giving a way to compute some group cohomology groups. Note   could be replaced by any local system   which is determined by a map


for some abelian group  . In the case of   for   letters, this is represented by a wedge sum of   circles  [6] which can be showed using the Van-Kampen theorem, giving the group cohomology[7]


Group cohomology of an integral latticeEdit

For an integral lattice   of rank   (hence isomorphic to  ), its group cohomology can be computed with relative ease. First, because  , and   has  , which as abelian groups are isomorphic to  , the group cohomology has the isomorphism


with the integral cohomology of a torus of rank  .


In the following, let M be a G-module.

Long exact sequence of cohomologyEdit

In practice, one often computes the cohomology groups using the following fact: if


is a short exact sequence of G-modules, then a long exact sequence is induced:


The so-called connecting homomorphisms,


can be described in terms of inhomogeneous cochains as follows.[8] If   is represented by an n-cocycle   then   is represented by   where   is an n-cochain   "lifting"   (i.e.   is the composition of   with the surjective map MN).


Group cohomology depends contravariantly on the group G, in the following sense: if f : HG is a group homomorphism, then we have a naturally induced morphism Hn(G, M) → Hn(H, M) (where in the latter, M is treated as an H-module via f). This map is called the restriction map. If the index of H in G is finite, there is also a map in the opposite direction, called transfer map,[9]


In degree 0, it is given by the map


Given a morphism of G-modules MN, one gets a morphism of cohomology groups in the Hn(G, M) → Hn(G, N).


Similarly to other cohomology theories in topology and geometry, such as singular cohomology or de Rham cohomology, group cohomology enjoys a product structure: there is a natural map called cup product:


for any two G-modules M and N. This yields a graded anti-commutative ring structure on   where R is a ring such as   or   For a finite group G, the even part of this cohomology ring in characteristic p,   carries a lot of information about the group the structure of G, for example the Krull dimension of this ring equals the maximal rank of an abelian subgroup  .[10]

For example, let G be the group with two elements, under the discrete topology. The real projective space   is a classifying space for G. Let   the field of two elements. Then


a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of  

Künneth formulaEdit

If, M = k is a field, then H*(G; k) is a graded k-algebra and the cohomology of a product of groups is related to the ones of the individual groups by a Künneth formula:


For example, if G is an elementary abelian 2-group of rank r, and   then the Künneth formula shows that the cohomology of G is a polynomial k-algebra generated by r classes in H1(G; k).,


Homology vs. cohomologyEdit

As for other cohomology theories, such as singular cohomology, group cohomology and homology are related to one another by means of a short exact sequence[11]


where A is endowed with the trivial G-action and the term at the left is the first Ext group.

Amalgamated productsEdit

Given a group A which is the subgroup of two groups G1 and G2, the homology of the amalgamated product   (with integer coefficients) lies in a long exact sequence


The homology of   can be computed using this:


This exact sequence can also be applied to show that the homology of the   and the special linear group   agree for an infinite field k.[12]

Change of groupEdit

The Hochschild–Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G). From it, one gets the inflation-restriction exact sequence.

Cohomology of the classifying spaceEdit

Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism[13]


The expression   at the left is a classifying space for  . It is an Eilenberg–MacLane space  , i.e., a space whose fundamental group is   and whose higher homotopy groups vanish).[d] Classifying spaces for   and   are the 1-sphere S1, infinite real projective space   and lens spaces, respectively. In general,   can be constructed as the quotient  , where   is a contractible space on which   acts freely. However,   does not usually have an easily amenable geometric description.

More generally, one can attach to any  -module   a local coefficient system on   and the above isomorphism generalizes to an isomorphism[14]


Further examplesEdit

Semi-direct products of groupsEdit

There is a way to compute the semi-direct product of groups using the topology of fibrations and properties of Eilenberg-Maclane spaces. Recall that for a semi-direct product of groups   there is an associated short exact sequence of groups


Using the associated Eilenberg-Maclane spaces there is a Serre fibration


which can be put through a Serre spectral sequence. This gives an  -page


which gives information about the group cohomology of   from the group cohomology groups of  . Note this formalism can be applied in a purely group-theoretic manner using the Lyndon–Hochschild–Serre spectral sequence.

Cohomology of finite groupsEdit

Higher cohomology groups are torsionEdit

The cohomology groups Hn(G, M) of finite groups G are all torsion for all n≥1. Indeed, by Maschke's theorem the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology.

If the order of G is invertible in a G-module M (for example, if M is a  -vector space), the transfer map can be used to show that   for   A typical application of this fact is as follows: the long exact cohomology sequence of the short exact sequence (where all three groups have a trivial G-action)


yields an isomorphism


Tate cohomologyEdit

Tate cohomology groups combine both homology and cohomology of a finite group G:


where   is induced by the norm map:


Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory, see class formation.

Tate cohomology of finite cyclic groups,   is 2-periodic in the sense that there are isomorphisms


A necessary and sufficient criterion for a d-periodic cohomology is that the only abelian subgroups of G are cyclic.[15] For example, any semi-direct product   has this property for coprime integers n and m.


Algebraic K-theory and homology of linear groupsEdit

Algebraic K-theory is closely related to group cohomology: in Quillen's +-construction of K-theory, K-theory of a ring R is defined as the homotopy groups of a space   Here   is the infinite general linear group. The space   has the same homology as   i.e., the group homology of GL(R). In some cases, stability results assert that the sequence of cohomology groups


becomes stationary for large enough n, hence reducing the computation of the cohomology of the infinite general linear group to the one of some  . Such results have been established when R is a field[16] or for rings of integers in a number field.[17]

The phenomenon that group homology of a series of groups   stabilizes is referred to as homological stability. In addition to the case   just mentioned, this applies to various other groups such as symmetric groups or mapping class groups.

Projective representations and group extensionsEdit

In quantum mechanics we often have systems with a symmetry group   We expect an action of   on the Hilbert space   by unitary matrices   We might expect   but the rules of quantum mechanics only require


where   is a phase. This projective representation of   can also be thought of as a conventional representation of a group extension   of   by   as described by the exact sequence


Requiring associativity


leads to


which we recognise as the statement that   i.e. that   is a cocycle taking values in   We can ask whether we can eliminate the phases by redefining


which changes


This we recognise as shifting   by a coboundary   The distinct projective representations are therefore classified by   Note that if we allow the phases themselves to be acted on by the group (for example, time reversal would complex-conjugate the phase), then the first term in each of the coboundary operations will have a   acting on it as in the general definitions of coboundary in the previous sections. For example,  


Cohomology of topological groupsEdit

Given a topological group G, i.e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G-modules, i.e., requiring that the action


is a continuous map. For such modules, one can again consider the derived functor of  . A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology is called Galois cohomology.

Non-abelian group cohomologyEdit

Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.

The zeroth cohomology of G with coefficients in A is defined to be the subgroup


of elements of A fixed by G.

The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map   to be a 1-cocycle is that   and   if there is an a in A such that  . In general,   is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group,   which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.

Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let


be a short exact sequence of G-groups, then there is an exact sequence of pointed sets


History and relation to other fieldsEdit

The low-dimensional cohomology of a group was classically studied in other guises, well before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert's Theorem 90 in 1897; this was recast into Emmy Noether's equations in Galois theory (an appearance of cocycles for  ). The idea of factor sets for the extension problem for groups (connected with  ) arose in the work of Otto Hölder (1893), in Issai Schur's 1904 study of projective representations, in Otto Schreier's 1926 treatment, and in Richard Brauer's 1928 study of simple algebras and the Brauer group. A fuller discussion of this history may be found in (Weibel 1999, pp. 806–811).

In 1941, while studying   (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula (Hopf 1942), which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:


where   and F is a free group.

Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States (Rotman 1995, p. 358); Hopf and Beno Eckmann in Switzerland; Hans Freudenthal in the Netherlands (Weibel 1999, p. 807); and Dmitry Faddeev in the Soviet Union (Arslanov 2011, p. 29, Faddeev 1947). The situation was chaotic because communication between these countries was difficult during World War II.

From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying space BG as discussed above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the CartanEilenberg theory of homological algebra in the early 1950s.

The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) (Weibel 1999, p. 822). Some refinements in the theory post-1960 have been made, such as continuous cocycles and John Tate's redefinition, but the basic outlines remain the same. This is a large field, and now basic in the theories of algebraic groups.

The analogous theory for Lie algebras, called Lie algebra cohomology, was first developed in the late 1940s, by Claude Chevalley and Eilenberg, and Jean-Louis Koszul (Weibel 1999, p. 810). It is formally similar, using the corresponding definition of invariant for the action of a Lie algebra. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics.

Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as symmetry-protected topological states.[18][19]

See alsoEdit


  1. ^ This uses that the category of G-modules has enough injectives, since it is isomorphic to the category of all modules over the group ring  
  2. ^ Recall that the tensor product   is defined whenever N is a right  -module and M is a left  -module. If N is a left  -module, we turn it into a right  -module by setting ag = g−1a for every gG and every aN. This convention allows to define the tensor product   in the case where both M and N are left  -modules.
  3. ^ For example, the two are isomorphic if all primes p such that G has p-torsion are invertible in k. See (Knudson 2001), Theorem A.1.19 for the precise statement.
  4. ^ For this, G is assumed to be discrete. For general topological groups,  .


  1. ^ Page 62 of Milne 2008 or section VII.3 of Serre 1979
  2. ^ Dummit, David Steven; Foote, Richard M. Abstract algebra (Third ed.). Hoboken, NJ. p. 801. ISBN 0-471-43334-9. OCLC 52559229.
  3. ^ Brown, Kenneth S. Cohomology of groups. New York, New York. p. 35. ISBN 978-1-4684-9327-6. OCLC 853269200.
  4. ^ Huang, Hua-Lin; Liu, Gongxiang; Ye, Yu (2012-06-23). "The braided monoidal structures on a class of linear Gr-categories". {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ Evens, Leonard. (1991). The cohomology of groups. Oxford: Clarendon Press. ISBN 0-19-853580-5. OCLC 23732584.
  6. ^ Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. p. 43. ISBN 0-521-79160-X. OCLC 45420394.
  7. ^ Webb, Peter. "An Introduction to the Cohomology of Groups" (PDF). Archived (PDF) from the original on 6 May 2020.
  8. ^ Remark II.1.21 of Milne 2008
  9. ^ (Brown 1972), §III.9
  10. ^ Quillen, Daniel. The spectrum of an equivariant cohomology ring. I. II. Ann. Math. (2) 94, 549-572, 573-602 (1971).
  11. ^ (Brown 1972), Exercise III.1.3
  12. ^ (Knudson 2001), Chapter 4
  13. ^ Stasheff, James D. (1978-07-01). "Continuous cohomology of groups and classifying spaces". Bulletin of the American Mathematical Society. 84 (4): 513–531. doi:10.1090/s0002-9904-1978-14488-7. ISSN 0002-9904.
  14. ^ (Adem & Milgram 2004), Chapter II.
  15. ^ (Brown 1972), §VI.9
  16. ^ Suslin, Andrei A. (1984), "Homology of  , characteristic classes and Milnor K-theory", Algebraic K-theory, number theory, geometry and analysis, Lecture Notes in Mathematics, vol. 1046, Springer, pp. 357–375
  17. ^ In this case, the coefficients are rational. Borel, Armand (1974). "Stable real cohomology of arithmetic groups". Annales Scientifiques de l'École Normale Supérieure. Série 4. 7 (2): 235–272. doi:10.24033/asens.1269. Archived from the original on 2016-04-15. Retrieved 2016-04-02.
  18. ^ Wang, Juven C.; Gu, Zheng-Cheng; Wen, Xiao-Gang (22 January 2015). "Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond". Physical Review Letters. American Physical Society (APS). 114 (3): 031601. arXiv:1405.7689. doi:10.1103/physrevlett.114.031601. ISSN 0031-9007.
  19. ^ Wen, Xiao-Gang (4 May 2015). "Construction of bosonic symmetry-protected-trivial states and their topological invariants via G×SO(∞) nonlinear σ models". Physical Review B. American Physical Society (APS). 91 (20): 205101. arXiv:1410.8477. doi:10.1103/physrevb.91.205101. ISSN 1098-0121.

Works citedEdit

Further readingEdit

  • Serre, Jean-Pierre (1994), Cohomologie galoisienne, Lecture Notes in Mathematics, vol. 5 (Fifth ed.), Berlin, New York: Springer-Verlag, doi:10.1007/BFb0108758, ISBN 978-3-540-58002-7, MR 1324577
  • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08017-8, MR 0347778
  • Chapter 6 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.