Exact cohomology sequences with integral coefficients for torus actions

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

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.

The purpose of this note is to prove that under certain hypotheses, the graded ring of integral automorphic forms, with respect to an arithmetic group operating On a tube domain, is generated as a graded algebra over the complex numbers by a finite number of automorphic forms having rational integral Fourier coefficients. The result, which seems to have some potential numbertheoretic interest, was inspired by notes of M. Eichler [8, esp. Satz 49] and by questions raised in correspondence with that author and with I. I. Pyatetskii-Shapiro. One should also note the similarity of ideas here with those used in the proof of Satz D of [9]. We wish to remark, furthermore, that in most cases lengthy computations will be needed to verify the applicability of our theorem here, which should therefore be regarded more as a technical lemma than as a substantial contribution in itself. As in [2], let ~ denote a (hermitian) symmetric tube domain, let F be a discrete, arithmetically defined subgroup of the group of all holomorphic automorphisms of ~ , and assume that ~ has a zerodimensional rational boundary component Fo, which we take to be the zero-dimensional rational boundary component of ~ at infinity as in [2]. Let

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.

I ° Introduction This talk is to present a few results on cohomology of groups arising from an investigation into the cohomological behaviour of inverse semigroups ([6], [7], [8]). In this context, relative cohomology in the sense of Auslander ([i]) is just a special case of a more general situation, and various long exact sequences, some of which are well-known ([9], [ii]) and have been established by various authors individually, can be obtained by one and the same method. The starting point for our consideration is the concept of a semilattice of groups, that is a functor S from a semilattice E regarded as a category to the category Gv of groups. Thus for e, f ( E , e ~_ f , there is exactly one group homomorphism ~e,f : S(e) ~ S(f) such that ~e,e is the identity on S(e) ,

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 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.

We study problem of global classification of ordinary differential equations with the linear-fractional right-hand side with rational coefficients with respect to a symmetry group. We find the field of differential invariants and obtain the equivalence criterion for two such equations. We adduce certain examples for applying of this criterion. These examples were obtained by means of computer.

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.

In 1946 Friedman [2] found all schlicht functions, with certain restrictions, that have rational integral coefficients. It is the purpose of this article to find all schlicht functions similarly restricted, whose nth powers, n a positive integer, have coefficients belonging to an integral domain I, of characteristic zero, and that has no integer except zero of absolute value less than 1. 1. Notations and definitions. A function is said to be schlicht in a domain D if for any two points z1 and Z2, belonging to D, we have f(Z1) =f(Z2) only if z1 = z2. We shall seek all functions