Relative cohomology of groups

In [1] the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence.Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three meaningful relative groups, and we analyze the corresponding definition of relative Cheeger-Simons characters in each case.We then extract another definition, corresponding to the one fitting into a long exact sequence in [1].Finally, we show how the explicit formulas for the holonomy, the transgression maps and the integration can be extended to the relative case.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

Cohomology and extensions of inverse semigroups

Let p be a prime, R a commutative ring. There are two familiar group scheme sequences yielding computations of the flat (or etale) cohomology group Hi(R, Z/pZ): the Kummer exact sequence I~7Z/pZ~G,,~G,,~I, applicable where p is invertible and a primitive p-th root of unity ( is in R, and the ArtinSchreier sequence 1 ~71/pZ~ G , ~ Ga ~ 1 that is valid when p is zero in R (see e.g. [6, pp. 126-127]). In this paper I shall show that these are both special cases of a single sequence of the same kind defined over Z [(]. The two groups involved (besides Z/pZ) have H 1-cohomology computable by familiar K-theory, and hence we have a way of computing H 1 (R, 7Z/pZ) for all Z [(]-algebras R. In particular, the computation works for all rings when p = 2, and we recover the description of etale quadratic algebras derived previously by ad hoc methods (see e.g. [7]). The group schemes occurring in the exact sequence are smoothed versions of congruence subgroups, objects which (I believe) were first described in general in my joint work with Boris Weisfeiler classifying smooth connected group schemes of dimension one [8].

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the Jones polynomial of knots.

Negligible degree two cohomology of finite groups

Let G be a finite group and ℱ be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ℱ-projective resolution for ℤ when ℱ is the family of all subgroups H ≤ G with rk H ≤ rkG − 1. We answer this question negatively by calculating the relative group cohomology ℱH*(G, 𝔽2) where G = ℤ/2 × ℤ/2 and ℱ is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ℱH*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ℱ. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E 2 page is isomorphic to the relative group cohomology of G.

We establish two exact sequences for the lattice cohomology associated with non-degenerate plumbing graphs. The first is the analogue of the surgery exact triangle proved by Ozsvath and Szabo for the Heegaard-Floer invariant HF^+; for the lattice cohomology over Z_2-coefficients it was proved by J. Greene. Here we prove it over the integers, and we supplement it by some additional properties valid for negative definite graphs. The second exact sequence is an adapted version which does not mix the classes of the characteristic elements (spin^c-structures); it was partially motivated by the surgery formula for the Seiberg-Witten invariant obtained by Braun and the author. For this we define the `relative lattice cohomology' and we also determine its Euler characteristic in terms of Seiberg-Witten invariants.

We provide a suitable axiomatic framework for differential cohomology in the relative case and we deduce the corresponding long exact sequences. We also construct the relative version of the generalized Cheeger-Simons characters and we define the integration map when the fibre has a boundary.

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

In the preceeding paper we constructed an infinite exact sequence a la Villamayor–Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the present paper we interpret the middle cohomology group in the second level of the sequence. We introduce the notion of coring categories and we obtain that the mentioned middle cohomology group is isomorphic to the relative group of Azumaya quasi coring categories. This result is a categorical generalization of the classical Crossed Product Theorem, which relates the relative Brauer group and the second Galois cohomology group with respect to a Galois field extension. We construct the colimit over symmetric finite tensor categories of the relative groups of Azumaya quasi coring categories and the full group of Azumaya quasi coring categories over vec. We prove that the latter two groups are isomorphic.

This elementary paper originated in the process of reading Katok and Spatzier's paper [3] concerning the first cohomology group of Z kand Rk-Anosov actions. It was proved in [3] that for the so-called standard Anosov actions any 1-cocycle (in C ~ and HSlder categories) with coefficients in R was cohomologous to a constant one. We will consider the .simplest class of actions, namely, the action of the discrete commutative group C SL (n, Z) by automorphisms of the n-toms T ~. We say that E is cohomologicalIy rigid if any ~ -cocyc l e : E ---* R is C~-cohomologous to a homomorphism ~ e Hom (E, R), i.e., Hi(Z, R) = H o m (~, R). It is proved in [3], in particular, that Z is cohomologically rigid provided that it is Anosov (i.e., contains an element a that has no eigenvalue of absolute value 1) and standard, i.e., contains a subgroup E' "-~ Z 2 that consists of ergodic elements (i.e., any nontrivial b E E' is ergodic on T~). Note that this result implies cohomological rigidity in the HSlder category [3], which is a special case of Katok and Schmidt's result [4] for mixing and expanding actions of Z k by automorphisms on a compact abetian group (see also [5]). On the other hand, higher cohomology groups for the Anosov ~ C SL (n, Z) were calculated in [2]. We will study more thoroughly the first cohomology group Hi(Z, R) of C~-cocycles with coefficients in R over the ergodic action of the commutative group Z C SL (n, Z) on T n by automorphisms. Our purpose here is to find a criterion for cohomological rigidity of ~. In fact, as Katok mentioned to the author, the result of [3] holds without the assumption that Z is Anosov. This requires involving the Lipschitz-type theorem of Veech [6] for ergodic automorphisms of a toms; we discuss this in Sec. 1. On the other hand, the cohomological rigidity easily implies that T" has no ~..-invariant proper subtorus T C T ~ such that the quotient action of ~ on T ~ / T is of rank I, i.e., contains a subgroup of finite index isomorphic to Z (see Sec. 1 for more details). Henceforth by a subtorus we mean a connected closed subgroup T C T~; thus T " / T ~_ T ~ .for some integer m and the quotient action of ~ lies in SL (m, Z). We will prove that the absence of rank-1 quotients is, in fact, sufficient for cohomological rigidity. The main result of the paper is as follows.

The second degree cohomology of finite orthogonal groups, II

The purpose of this paper is to generalize the seven terms exact sequence given by Chase, Harrison and Rosenberg [8]. Our work was motivated by Kanzaki [16] and, of course, [8], [9]. The main theorem holds for any generalized crossed product, which is a more general one than that in Kanzaki [16]. In §1, we define a group P(A/B) for any ring extension A/B, and prove some preliminary exact sequences. In §2, we fix a group homomorphism J from a group G to the group of all invertible two-sided B-submodules of A.

A fibration, in the Zariski topology, of algebraic varieties leads to an exact sequence of cohomology of the relative units functor. Under suitable hypotheses, the exact sequence can be interpreted as a sequence including the Picard groups of the base variety, the total space, and the fibre. Let X Y be a fibration with fibre F of algebraic varieties (over an algebraically closed field) which is locally trivial in the Zariski topology on Y. If Y and F are smooth and F is rational, Fossum and Iverson [2, Proposition 5, p. 81 constructed an exact sequence relating Pic(Y), Pic(X), and Pic(F). The purpose of the present note is to interpret that sequence in terms of the cohomology, in the Zariski topology, of the relative units sheaf. Besides explaining the Fossum-Iverson sequence, this interpretation helps isolate the hypotheses for the existence of the sequence, which leads to some mild generalizations of the original result. We fix throughout the following conventions: k denotes an algebraically closed field and all varieties considered are over k and irreducible. Unless otherwise specified, all sheaves are for the Zariski topology, and all sheaf cohomology is in that topology. If V is a variety, U an open subset of V and Y a sheaf on V, then (ff)IU denotes Yf restricted to U. If A is an abelian group then (A)v denotes the constant sheaf on V associated to A. We recall that the higher cohomology groups of constant sheaves on V are trivial, and, hence by descent, that all locally constant sheaves on V are constant. Gm denotes the units sheaf on V. Definition 1. Let V be a variety. Uk V denotes the presheaf cokernel of ((k*)v G v) We refer to U k, V as the sheaf of relative units on V. Lemma 2. Let V be a variety. (i) Uk V is a sheaf on V. (ii) H(V, Uk V) = H(V, Gm ) for all i > 1. (iii) Pic(V) = H1(V, UkV)Received by the editors October 25, 1974. AMS (MOS) subject classifications (1970). Primary 14D99, 14F10, 14F20.