Self-injective algebras and the second Hochschild cohomology group

In this paper we study the second Hochschild cohomology group ${\mathrm{HH}}^2(\Lambda )$ of a finite dimensional algebra Λ. In particular, we determine ${\mathrm{HH}}^2(\Lambda )$ where Λ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Λ; we give a basis for ${\mathrm{HH}}^2(\Lambda )$ in the few cases where it is not zero.     

Some bounds for the annihilators of local cohomology and Ext modules

Czechoslovak Mathematical Journal - Let $$\mathfrak{a}$$ be an ideal of a commutative Noetherian ring R and t be a nonnegative integer. Let M and N be two finitely generated R-modules. In certain...     

The second bounded cohomology of hypo-Abelian groups

Fujiwara [K. Fujiwara, The second bounded cohomology of a group with infinitely many ends, math.GR/9505208] conjectured that the second bounded cohomology of a group is zero or infinite-dimensional as a vector space over R. However, it is known that there are some linear groups for which the second bounded cohomology is not zero but finite-dimensional. In this paper, by using the transfinitely extended derived series, we prove that Fujiwara's conjecture is true for the hypo-Abelian groups, that is, groups with no non-trivial perfect subgroups.     

Entropy bounds for perfect matchings and Hamiltonian cycles

For a graph G = (V,E) and x: E → ℜ⁺ satisfying Σ _e∋υ x _e = 1 for each υ ∈ V, set h(x) = Σ _e x _e log(1/x _e ) (with log = log₂). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and x _e =Pr(e∈f),

On lower bounds for cohomology growth in p-adic analytic towers

Let \(p\) and \(\ell \) be two distinct prime numbers and let \(\Gamma \) be a group. We study the asymptotic behaviour of the mod- \(\ell \) Betti numbers in \(p\) -adic analytic towers of finite index subgroups. If \(\Theta \) is a finite \(\ell \) -group of automorphisms of \(\Gamma \) , our main theorem allows to lift lower bounds for the mod- \(\ell \) cohomology growth in the fixed point group \(\Gamma ^\Theta \) to lower bounds for the growth in \(\Gamma \) . We give applications to \(S\) -arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.     

Entropy bounds for endomorphisms commuting withK actions

Shereshevsky has shown that a shift-commuting homeomorphism from the two-dimensional full shift to itself cannot be expansive, and asked if such a homeomorphism can have finite positive entropy. We formulate an algebraic analogue of this problem, and answer it in a special case by proving the following: ifT : X → X is a mixing endomorphism of a compact metrizable abelian groupX, andT commutes with a completely positive entropyZ ²-actionS onX by continuous automorphisms, thenT has infinite entropy. Dedicated to the memory of Dr. Elizabeth Mary Hartley (1923–1998) The authors gratefully acknowledge support from EPSRC award no. 9570016X, N.S.F. grant No. DMS-94-01093, and the hospitality of the Warwick Mathematics Research Institute.     

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.     

Order bounds for second derivative approximations

Rational approximations to the exponential function with denominator (1−λz−μz ²) ^s arise as stability functions of second derivative generalizations of Runge–Kutta methods. The purpose of this paper is to derive order barriers for approximations of this and related forms. Although some of these barriers are already known, we will analyse them in a new way. Order arrows were originally proposed as a complement to order stars for establishing barriers for A-stable methods but they are shown also to be a powerful tool for analysing the type of order barrier considered in this paper.     

Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Generalized Witt algebras, over a field of characteristic , were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras over , where the ingredients are an abelian group , a vector space over , and a map which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra
, with the right kernel of being , are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.

     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

On the second cohomology group of a pinched Riemannian manifold

Summary Let M be a compact orientable Riemannian manifold of dimension even. We assume that the manifold can carry a metric, which is positively k-pinked. In the present paper some properties of the second cohomology group of such a manifold are obtained, when the number k is greater than a number which depends on the dimension of the manifold. These properties have some applications on the topological product of some special manifolds. This research was supported by S. F. B. grant. Entrata in Redazione il 6 dicembre 1969.     

Entropy in the category of perfect complexes with cohomology of finite length

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic.Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is “additive”. Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.