Exact sequences in cohomology and an application

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is\nthe first step of Voevodsky's proof of this conjecture for arbitrary prime l)\nin a rather clear and elementary way. Assuming this conjecture, we construct a\n6-term exact sequence of Galois cohomology with cyclotomic coefficients for any\nfinite extension of fields whose Galois group has an exact quadruple of\npermutational representations over it. Examples include cyclic groups, dihedral\ngroups, the biquadratic group Z/2\\times Z/2, and the symmetric group S_4.\nSeveral exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn\nare proven in this way. In addition, we introduce a more sophisticated version\nof the classical argument known as "Bass-Tate lemma". Some results about\nannihilator ideals in Milnor rings are deduced as corollaries.\n

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free equivariant cohomology over the cohomology of BT. Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.

We apply the G-theory and anomaly of ghost and antighost fields in the theory of supersymmetry to study a superspace over time series data for the detection of hidden general supply and demand equilibrium in the financial market. We provide proof of the existence of a general equilibrium point over 14 extradimensions of the new G-theory compared with the M-theory of the 11 dimensions model of Edward Witten. We found that the process of coupling between nonequilibrium and equilibrium spinor fields of expectation ghost fields in the superspace of time series data induces an infinitely long exact sequence of cohomology from a short exact sequence of moduli state space model. If we assume that the financial market is separated into two topological spaces of supply and demand as the D-brane and anti-D-brane model, then we can use a cohomology group to compute the stability of the market as a stable point of the general equilibrium of the interaction between D-branes of the market. We obtain the result that the general equilibrium will exist if and only if the 14thBatalin–Vilkovisky cohomology group with the negative dimensions underlying 14 major hidden factors influencing the market is zero.

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

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.

In a semiabelian category, a strictly exact sequence 0→A→B→C→0 of cochain complexes gives rise to the cohomology sequence ...→Hn(A) →Hn(B)→ Hn(C)→ Hn+1(A) →.... We study conditions for exactness of the homology sequence at a given term.

The second degree cohomology of finite orthogonal groups, II

When the coecients are crossed bimodules, Guin's non- abelian cohomology (2), (3) is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

We begin this note by pointing out that a few modifications in some of the notations and arguments of C131 will make these fit in more closely with results in the literature. We also complete the results of C131 in several points. In particular we point out that the spectral sequence used in C131 is not quite a genuine generalization of the Hochschild-Serre spectral sequence in Galois cohomology. However with a slightly different spectral sequence the results of C131 can also be obtained and we shall show in section 2 that this is indeed a genuine generalization of the Hochschild-Serre sequence for Galois cohomology. In section 3 we shall use some of the results of [13] to derive an exact sequence complementary to that of Proposition 7.8 of [13] from which we deduce the following result first pointed out to us by S. Shatz: Let C be a field, C, its separable algebraic closure and its algebraic closure. Then if X is the lift map [2, Def. 2. 3.1, we have that X : Hr(C,/C)- . ~ ' ( 6 1is~ ) an isomorphism for r = 1,2, ...