Relative Cohomology Groups Research Articles

On the complete relative cohomology of Frobenius extensions

Let A be an algebra over a commutative ring K and F a subalgebra. Suppose that the extension A/F is a Frobenius extension. Then in [3, section 3], the complete relative cohomology group Hl^niM, ―) is introduced for an arbitrary left J-module M and reZ. We denote the opposite rings of A and F by A0 and F° respectively. Put P=A KA° and let 5 denote the natural image of F(Z)KF0 in P. Then the extension P/S is also a Frobenius extension. Since A is a left P-module with the natural way, we have HlP,s->(A,―). We will denote this Hu>.s>(A, ―) by Hr(A, F, -) for [6, section 3]. In this paper, we will study this complete relative cohomology H{A, F, ―). In section 1, we will study relative complete resolutions of A and in section 2, we willintroduce the dual of the fundamental exact sequence of [4, Proposition 1 and Theorem 1] for complete relative cohomology groups. In section 3, we will study an internal product like as in [9, section 2] which we will call the cup product. If the basic ring of the Frobenius extension is commutative, the cup product in this paper coincides with the product V in [2, Exercise 2 of Chapter XI]

Relative Lie algebra cohomology revisited

In this paper it is shown that relative Lie algebra cohomology is related to relative cohomology for restricted Lie algebras by a spectral sequence. Also an interpretation of the relative cohomology groups for Lie algebras in terms of the relative right derived functors of a certain relative derivative functor is given.

On the twistor description of sourced fields

Massless fields with source on an analytic world-line are double-valued, and it was shown by Bailey (1985) that a large family of such fields have a twistor description in terms of relative cohomology groups. In this paper it is proved that all right-handed massless fields are obtained in this way, and that if the sheaves O ( n - 2) are quotiented by the polynomials, then the relative cohomology of the resulting sheaves describes all left-handed sourced massless fields. The proof for right-handed fields uses techniques developed by Singer (1987, 1988) for applying the Penrose transform to situations in which the ‘pull-back mechanism’ is non-trivial. For the left-handed fields it is necessary to use some additional arguments involving the conserved quantities (e. g. momentum and angular momentum for spin 2) of these fields; it is shown that the conserved quantities are the obstructions to a twistor description of left-handed fields in terms of the cohomology of O ( n - 2).

Erratum: Erratum to "Relative Barr-Rinehart and Cotriple Cohomology Groups are Isomorphic"

Erratum: Erratum to "Relative Barr-Rinehart and Cotriple Cohomology Groups are Isomorphic"

Relative Barr-Rinehart and Cotriple Cohomology Groups are Isomorphic

Relative Barr-Rinehart and Cotriple Cohomology Groups are Isomorphic

Spaces with finitely many non-trivial homotopy groups

It is well known that the homotopy category of connected CW-complexes X whose homotopy groups n;(X) are trivial for i> 1 is equivalent to the category of groups. One of the objects of this paper is to prove a similar equivalence for the connected CW-complexes X whose homotopy groups are trivial for i > n + 1 (where n is a fixed non-negative integer). For n = 1 the notion of crossed module invented by J.H.C. Whitehead [13], replaces that of group and gives a satisfactory answer. We reformulate the notion of crossed module so that it can be generalized to any n. This generalization is called an ‘n-cat-group’, which is a group together with 2n endomorphisms satisfying some nice conditions (see 1.2 for a precise definition). With this definition we prove that the homotopy category of connected CW-complexes X such that Xi(X) = 0 for i > n + 1 is equivalent to a certain category of fractions (i.e. a localization) of the category of n-cat-groups. The main application concerns a group-theoretic interpretation of some cohomology groups. It is well known [lo, p. 1121 that the cohomology group H2(G; A) of the group G with coefficients in the G-module A is in one-to-one correspondence with the set of extensions of G by A inducing the prescribed G-module structure on A. Use of n-cat-groups gives a similar group-theoretic interpretation for the higher cohomology groups H”(G; A) and H”(K(C, k); A) where K(C, k) is an Eilenberg-MacLane space with k 2 1. In [8] we proved that crossed modules could be used to interpret a relative cohomology group. Here we show that the notion of n-cat-group is particularly suitable to interpret some ‘hyper-relative’ cohomology groups. The usefulness of this last result appears in its application to algebraic Ktheory where it leads to explicit computations [4]. This was in fact our primary motivation for a generalization of crossed modules. Section 1 contains the definitions of n-cat-groups and of n-cubes of fibrations. There are two functors: 5 : (n-cubes of fibrations) + (n-cat-groups)

Complete intersections in tensor algebras II

In this paper we continue the study of’ the vanishing of the lower relative Poochschild cohomology groups. This paper is ;a continuation of [3]. Recall that if A is a k-algebra, where k is a commutative ring, a two-sided ideal I in A is a camplete intersection in ~4 if H*((@‘)//I,~ = 0 for all two-sided A-modules M. The use of the terminology “complete intersection” comes from the analogy between these relative cohomology groups and the cohomology groups studied in commutative deformation theory [see 2,3] . Suppose that k is a field, R and R 03+ R”P are semi-simple k-algebras and X is a two-sided R-module. Let T be the tensor k-algebra R + X + (6b~ X) + (6~ x) + l . In [3] 9 a homological criterion was found for an ideal I in T to be a complete intersection; namely, I/I* must be a left projective (T/f) Ok (T/Z)‘P-module. Section 1 of this paper gives a homological criterion for an ideal in a factor ring of T to be a complete intersection. For the rest of the paper we restrict our attention further, to special Artin tensor k-algebras, defined in section 2, and their factor rings. After defining an appropriate concept of “minimal generating set” of an ideal, we study the question of when an ideal with one generator, in a factor ring of a special tensor k-a!gebra, is a complete intersection. Section 3 deals with the question of when a subideal of a complete intersection in a special tensor k-algebra is again a complete intersection. We show that if I is a complete intersection in T and if I has generators (more precisely, J-generators, see section 2) x1, . . . . x, then, after reordering the xi in a natural way, the ideals generated by x1, . . . . xi are also complete intersections in T, for i = 1, . . . . rn. Once this is shown, we show that the generators of complete intersections have properties similar to those of R-sequences, where R is a commutative ring (see Theorem 4.1). Unfortunately, this analogy does not go too far, as shown by example in section 4.

Smoothness and rigidity of tensor algebras and their factors

In this paper, we begin a study of the lower dimensional relative cohomology groups of algebras defined by Hochschild [7]. Section 1 gives a few known but basic results. The results show how the relative cohomology groups for noncommutative rings of dimensions 1,2, and 3 correspond to the cohomology groups studied in commutative deformation theory. We show, for example, that they are the cohomology groups of a noncommutative cotangent complex. This correspondence motivates our definitions of smoothness and rigidity. From the results of Section 1, we see that if R is a semisimple ring and A is an augmented ring over R, i.e., there are ring morphisms R -+ A -R whose composition is the identity, then A is smooth over R if it is a tensor algebra over R. In Section 2, we show that the converse is also true under some additional hypothesis on A. For the remainder of the paper, we only consider tensor algebras that are associated to K-species, where K is a field (see Section 3 for definitions). Section 4 gives results on the rigidity of factors of these tensor algebras. It is shown that there are tensor algebras, each of whose factor rings is rigid. We call such tensor algebras factor rigid and classify them in Section 5. Note that the phenomenon of factor rigidity does not occur in the commutative theory, where a polynomial ring over a field K is factor rigid over k if and only if it is just the field k itself. I would like to thank D.S. Rim for introducing me to this subject and for his many helpful suggestions.

A cohomology for foliated manifolds

for a e T(v A V ) , X1? . . . , Xk + 1 e T(T). Since the curvature tensor of V restricted to T is identically zero we have that d o d = 0. Denote the homology of this complex by F*(T; V). This is the cohomology of the Lie algebra of vector fields tangent to the foliation with coefficients in sections of the normal bundle, the representation being given by the connection [GF]. In general the groups F\x ; v) are not finitely generated (the complex is not elliptic) but they satisfy the following. (i) F* is a functor from the category of foliated manifolds and transverse maps to the category of abelian groups and homomorphisms. (ii) If ƒ : N -» M is an embedded transverse submanifold, we can define relative cohomology groups F*(T; v, ƒ) and obtain the usual long exact sequence. (iii) F* is an invariant of the diffeomorphism type of the foliation. However, F* is not an invariant of the integrable homotopy type of the foliation when M is an open manifold.

A Note on Exponents of Relative Cohomology Groups

A Note on Exponents of Relative Cohomology Groups

A cohomology for foliated manifolds

for a e T(v A V ) , X1? . . . , Xk + 1 e T(T). Since the curvature tensor of V restricted to T is identically zero we have that d o d = 0. Denote the homology of this complex by F*(T; V). This is the cohomology of the Lie algebra of vector fields tangent to the foliation with coefficients in sections of the normal bundle, the representation being given by the connection [GF]. In general the groups F\x ; v) are not finitely generated (the complex is not elliptic) but they satisfy the following. (i) F* is a functor from the category of foliated manifolds and transverse maps to the category of abelian groups and homomorphisms. (ii) If ƒ : N -» M is an embedded transverse submanifold, we can define relative cohomology groups F*(T; v, ƒ) and obtain the usual long exact sequence. (iii) F* is an invariant of the diffeomorphism type of the foliation. However, F* is not an invariant of the integrable homotopy type of the foliation when M is an open manifold.

A unified treatment of some basic problems in homotopy theory

0. Introduction. This note announces some new methods in algebraic topology, based on the results of [1], [2] and [3]. The relative lifting problem [5, p. 415] is fundamental to that subject. It includes the extension problem, the retraction problem, the lifting problem, the section problem, the relative section problem and the computation of [X, Y] problem, among its particular cases. These problems (excluding the last one) are usually taken to concern just the existence of extensions, retractions, etc. ; we use the terms in the wider sense, to include both the existence question and the homotopy classification question, for the respective extensions, retractions, etc. We will : (i) prove that many cases of the above problems (including the computation of the absolute and relative cohomology groups of the total space of a fibration, and the computation of certain homotopy groups) are equivalent to problems, involving the restricted fibered mapping projection (pq;a); (ii) give some examples of solutions of parallel problems. Our solutions include one lifting problem and two extension problems. The extension results have some immediate consequences; they give new derivations for the exact cohomology sequences of Serre and Wang. Gysin's sequence is derived separately. Future papers will discuss these and other applications in detail. Our argument is valid in several convenient categories, including the category of i-spaces [4], [2], [7] and the category of quasi-topological spaces [6], [1], [2]. This category of I-spaces, for a definition see [2, p. 276], contains the usual category of Hausdorff fc-spaces (= compactly generated spaces = Kelley spaces) as a subcategory.

On the relative group cohomology ring

The product structure on the Hochschild-Serre spectral sequence generalizes to the spectral sequences of Butler-Horrocks. It is shown that Evens’ proof of the finite generation of the integral cohomology ring of a finite group does not generalize to the relative cohomology groups of Adamson.

Relative Cohomology of Algebraic Linear Groups, II

Let G be an algebraic linear group over a field F of characteristic 0, and let H be an algebraic subgroup of G. Let A, M be rational G-modules. In [4], we defined (A, M), and, in particular, relative cohomology groups Hn(G, H, M) were defined as (F, M).

A Remark on Relative Homology and Cohomology Groups of a Group

Let G be a group and H a subgroup of G. With a left G-module M, relative cohomology groups Hn(G, H, M) of G on M, relative to H, have been defined by Adamson [1] and may be expressed as (Z, M) in the notation of relative homological algebra of Hochschild [2], where Z denotes the G-module of rational integers (acted by G trivially). Regarding M as a right G-module, (M, Z) are similarly relative homology groups Hn(G, H, M). In case H is of finite index in G, Hochschild [2] defines further negative-dimensional relative homology and cohomology groups.

On Amitsur’s complex

and Fk* the group of units of Fk. Define ring homomorphisms ei (i = 1, , k+ 1) of Fk to Fk+l by(2'3) ei(fl? * ?fk) =fl? * fi?1 fi ... ?fk and define a multiplicative homomorphism Ak: Fk* -> F(k+l)* by Ak(X) = el(x) (E2(X)) * ... (ek+l(x))+1. Then the groups Fk* and mappings Ak form the Amitsur complex; the kth homology group Ker Ak+l/Im Ak we denote by Hk(F). In case F is a finite dimensional extension field, Amitsur showed that H2(F) is isomorphic to the Brauer group of central simple C-algebras split by F. He also showed that in case F is a normal separable extension, Hn(F) is isomorphic to Hn(G, F*) the nth cohomology group of the Galois group G of F over C with coefficients in the group of nonzero elements of F. In this paper, we extend and simplify Amitsur's results. We begin by showing (?2) that in case F is a separable field extension of C, Hn(F) --'Hn([G: H], K*), where K is a normal closure of F with Galois group G, H is the subgroup corresponding to F, and the cohomology group on the right side is the relative cohomology group as introduced in [I]. Next, we study Hn(F) when C is not necessarily a field but when n = 2. In ?3, under weak hypotheses on F, we exhibit a homomorphism of H2(F) to the (generalized) Brauer group of central separable algebra classes split by F [5]. This homomorphism is an isomorphism under stronger hypotheses on F and C. These hypotheses are slightly weaker than assuming all projective C, F, and F2 modules are free and include the cases (10) C is semilocal (not necessarily Noetherian) and F is a C-algebra which is a-finitely generated projective C-module containing C 1 as a direct summand and (20) C=K[x], F=L[x], with K a field and L a finite dimensional commutative K-algebra. Hochschild in [13] has given a description of the Brauer group in case F is a purely inseparable extension field of C of exponent 1. In [3, ?7], Amitsur

The reduction theorem of the relative cohomology group in algebras, and its application

The reduction theorem of the relative cohomology group in algebras, and its application

A Note on the Relative 2-Dimensional Cohomology Group in Complete Fields with Respect to a Discrete Valuation

A Note on the Relative 2-Dimensional Cohomology Group in Complete Fields with Respect to a Discrete Valuation

Effectiveness at the origin of the sum set of basic sets of polynomials

an equivalence relation only at one stage, namely, when cocycles are identified modulo coboundaries. Actually, we arrived at this definition of HP(X, A) while trying to set up the relative cohomology groups subject to the following two conditions. (i) HP(X, A) should be the cohomology group of a cochain complex associated with the pair (X, A). (ii) The use of the process of identification should be minimized. Condition (i) is motivated by the desire to take advantage of the technical convenience of the general theory of cochain complexes, while condition (ii) is concerned with conceptual simplicity.