Vanishing Of Cohomology Groups Research Articles

Phase Transition in Cohomology Groups of Non-Uniform Random Simplicial Complexes

We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408–417].We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group $R$. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.

A transitivity result for ad-nilpotent ideals in type [formula omitted

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal I meets a unique largest nilpotent orbit OI in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals I such that OI=O for a fixed nilpotent orbit O. We include two applications of the result, one to the higher vanishing of cohomology groups of vector bundles on the flag variety and another to the Kazhdan–Lusztig cells in the affine Weyl group of the symmetric group. Finally, some combinatorial results are discussed.

On complete Finslerian Yamabe solitons

Here it is shown that any Finslerian compact Yamabe soliton with bounded above scalar curvature is of constant scalar curvature. Furthermore, this extension of Yamabe solitons is developed for inequalities and among the others, it is proved that a forward complete non-compact shrinking Yamabe soliton has finite fundamental group and its first cohomology group vanishes, providing the scalar curvature is strictly bounded above.

On compact Ricci solitons in Finsler geometry

Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work, it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group, and hence the first de Rham cohomology group vanishes.

Algebras determined by their supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander–Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.

Galois Coverings of Weakly Shod Algebras

We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence we show that a weakly shod algebra which is not quasi-tilted of canonical type is simply connected if and only if its first Hochschild cohomology group vanishes.

Geometry of domains with the uniform squeezing property

We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmüller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kähler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.

On Galois coverings and tilting modules

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Let T be a basic tilting A-module with arbitrary finite projective dimension. For a fixed group G we compare the set of isoclasses of connected Galois coverings of A with group G and the set of isoclasses of connected Galois coverings of End A ( T ) with group G. Using the Hasse diagram K → A (see [D. Happel, L. Unger, On a partial order of tilting modules, Algebr. Represent. Theory 8 (2) (2005) 147–156] and [C. Riedtmann, A. Schofield, On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1) (1991) 70–78]) of basic tilting A-modules, we give sufficient conditions on T under which there is a bijection between these two sets (these conditions are always verified when A is of finite representation type). Then we apply these results to study when the simple connectedness of A implies the one of End A ( T ) (see [I. Assem, E.N. Marcos, J.A. de la Peña, The simple connectedness of a tame tilted algebra, J. Algebra 237 (2) (2001) 647–656]). Finally, using an argument due to W. Crawley-Boevey, we prove that the type of any simply connected tilted algebra is a tree and that its first Hochschild cohomology group vanishes.

REGULARITY AND RESOLUTIONS FOR MULTIGRADED MODULES

This paper is concerned with the relationships between two concepts, vanishing of cohomology groups and the structure of free resolutions. In particular, we study the connection between vanishing theorems for the local cohomology of multigraded modules and the structure of their free multigraded resolutions.

Vanishing of cohomology groups and large eigenvalues of the Laplacian on p-forms

For collapsing of closed Riemannian manifolds, the first positive eigenvalue of the Laplacian on p-forms may or may not tend to infinity. In special cases, we show that the existence of the first positive eigenvalue tending to infinity is related to vanishing of cohomology groups of generic fibers.

Homothéties et images de Galois dans les représentations associées aux motifs

We prove that the image of Galois in the p-adic Galois-representation associated with a pure motive of non-zero weight on a number field contains for p large non-trivial homotheties of order prime to p. The vanishing of cohomology groups of the image of Galois with values in these representations follows.

On the vanishing of cohomology groups with values in a product sheaf.

On the vanishing of cohomology groups with values in a product sheaf.

Approximationseigenschaft, Lifting und Kohomologie bei lokalkonvexen Produktgarben

Motivated by examples like spaces of solutions of hypoelliptic operators, we treat product sheaves, prove density theorems in connection with the approximation property and use them for results on liftings and the vanishing of cohomology groups. Theorems of this type (2.4,2.9,3.3) are derived on regular subsets (2.3) of a product for product sheaves, where one factor has essentially a partition of unity. In the case of the compact open topology, we obtain the approximation property on arbitrary open subsets by a localization principle (4.5,4.9). The nuclearity of a sheaf in the co-topology turns out to imply strong nuclearity (1.11); the same is shown for the sheaf of holomorphic functions on the dual of a strongly nuclear (F)-space (1.12).

Analytic Equivalence of a Singular Set

This paper originates from the work of H. Hironaka on the equivalence of singularities and the work of M. Gerstenhaber on the deformations of algebras. Hironaka ([6], [7]) shows, in short, that the formal completion along the singular locus of a point in an algebraic variety is entirely determined by a finite truncation, whereas M. Gerstenhaber ([3]) shows that there is no non-trivial deformation if an appropriate cohomology group vanishes. This makes one to suspect that their works are somewhat related. Namely if an appropriate cohomology group is small enough then the whole structure may be determined by a finite datum. Now let k be a fixed commutative noetherian ring and consider a pair (A,I) where A is a commutative algebra of essentially finite type over k and I is an ideal in A. For such pair (A, I) we set A (n) IvlIv+n+l v

On the First Cohomology of Discrete Subgroups of Semi-Simple Lie Groups

Introduction. Let G be a connected semi-simple Lie group and r a discrete subgroup such that the quotient G/r is compact. Let po be a finite dimensional representation of G. Our aim in this paper, is to show that for a large class of representations po, the first cohomology group of r with coefficients in the representation pr (the restriction of po to r) is zero (for a precise statement, see Theorem 1). Our results say in particular that if po does not contain the trivial representation of G and if no simple component of G is compact or locally isomorphic to SO, (n, 1) or SU (n, 1), then this first cohomology group vanishes. Even if G has components locally isomorphic to SOo (n, 1) or SU (n, 1) we give a sufficient condition for the vanishing of the cohomology in terms of the highest weights of the irreducible components of the complexification p of po. The importance of these cohomology groups arises from the role they play in deformation theory; for instance, when poO is the adjoint representation, these cohomology groups are intimately connected with the theory of deformations of discrete subgroups of Lie groups [6]. It has been proved essentially by A. Weil [8] (see also [5] and [6]) that when po is the adjoint representation, this cohomology group vanishes if G has no compact or three dimensional components. This result is a special case of our theorem (see Corollary 1 to Theorem 1). The case of trivial representations has been treated by Y. Matsushima in [4].