Cohomological Dimension Research Articles

Cohomological Properties of Multinorm-one Tori

Cohomological Properties of Multinorm-one Tori

On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

On cohomological dimension of homomorphisms

The (co)homological dimension of a homomorphism ϕ : G → H \phi :G\to H is the maximal number k k such that the induced homomorphism in k k -th (co)homology groups is nonzero for coefficients in some H H -module. It is known that for geometrically finite groups G , c d ( G ) = h d ( G ) G, cd(G)=hd(G) and c d ( G × G ) = 2 c d ( G ) cd(G\times G)=2cd(G) . We prove analogous theorems for homomorphisms of geometrically finite groups. The analogy stops working on the Eilenberg-Ganea equality c d ( G ) = g d ( G ) cd(G)=gd(G) where c d ( G ) > 2 cd(G)>2 and g d ( G ) gd(G) is the geometric dimension of G G . We show that for every k > 2 k>2 there is a group homomorphism ϕ k : π k → Z k \phi _k:\pi _k\to \mathbb {Z}^k with c d ( ϕ k ) > k cd(\phi _k)>k and g d ( ϕ k ) = k gd(\phi _k)=k where π k \pi _k is the fundamental group of a closed aspherical ( k + 1 ) (k+1) -dimensional manifold.

The Genesis of a Theorem in the Galois Theory of <i>p</i>-Extensions of ℚ with Restricted Tame Ramification

This article traces the genesis of a theorem that gives for the first time examples of the Galois group <I>G<SUB>S</SUB></I> of the maximal <i>p</i>-extension of ℚ, unramified outside a finite set of primes not containing an odd <i>p</i>, that are of cohomlogical dimension 2 if the primes in <I>S</I> satisfy a certain linking condition. Because the ramification is tame the pro-<i>p</i>-group <I>G<SUB>S</SUB></I> has all of its derived factors finite which is a strong finitenesss condition on <I>G<SUB>S</SUB></I>. The paper starts with a question of Serre on one relator pro-<i>p</i>-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-<i>p</i>-groups where the linking condtion on the primes of <I>S</I> is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that <I>G<SUB>S</SUB></I> was of cohomological dimension 2 for certain sets <I>S</I> had it been applied to investigate whether the linking condition was true for certain sets <I>S</I>. It is not known if the cohomological dimension of <I>G<SUB>S</SUB></I> is 2 if <I>S</I> does not satisfy this linking condition.

On the sequential topological complexity of group homomorphisms

We define and develop a homotopy invariant notion for the sequential topological complexity of a map f:X→Y, denoted TCr(f), that interacts with TCr(X) and TCr(Y) in the same way Jamie Scott's topological complexity map TC(f) interacts with TC(X) and TC(Y). Furthermore, we apply TCr(f) to studying group homomorphisms ϕ:Γ→Λ.In addition, we give the characterization of cohomological dimension of group homomorphisms.

Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties

This paper explores the possibility of constructing multivariate generating functions for all cohomology dimensions of all holomorphic line bundles on certain complex projective varieties of Fano, Calabi-Yau and general type in various dimensions and Picard numbers. Most of the results are conjectural and rely on explicit cohomology computations. We first propose a generating function for the Euler characteristic of all holomorphic line bundles on complete intersections in products of projective spaces and toric varieties. This generating function is constructed by expanding the Hilbert-Poincaré series associated with the coordinate ring of the variety around all possible combinations of zero and infinity and then summing up the resulting contributions with alternating signs. Similar generating functions are proposed for the individual cohomology dimensions of all holomorphic line bundles on certain complete intersections, including examples of Mori and non-Mori dream spaces. Surprisingly, the examples studied indicate that a single generating function encodes both the zeroth and all higher cohomologies upon expansion around different combinations of zero and infinity, raising the question whether such generating functions determine the variety uniquely.

Surface counterexamples to the Eisenbud-Goto conjecture

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in P 5 \mathbb {P}^5 , and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P 4 \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

Correction: Injective dimension of cofinite modules and local cohomology

Correction: Injective dimension of cofinite modules and local cohomology

Fourier calculus from intersection theory

Building on recent advances in studying the cohomological properties of Feynman integrals, we apply intersection theory to the computation of Fourier integrals. We discuss applications pertinent to gravitational bremsstrahlung and deep inelastic scattering in the saturation regime. After identifying the bases of master integrals, the latter are evaluated by means of the differential equation method. Finally, new results with exact dependence on the spacetime dimension D are presented. Published by the American Physical Society 2024

Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion

Abstract Let p be a prime number. One of the most difficult and important questions in Galois theory these days is determining which pro-p groups can occur as maximal pro-p Galois groups. Some restrictions of the structure of maximal pro-p Galois groups are already known, and therefore, the families of all pro-p groups satisfying these restrictions are in the center of the research. In the current paper we deal with three of these families: The family of all pro-p groups which satisfy hereditarily the n-vanishing Massey product property, the family of 1-cyclotomic oriented pro-p groups, and the family of pro-p groups that are hereditarily of p-absolute Galois type. We show that all these families are closed under taking profinite completion of each order.

Strongly locally homogeneous generalized continua of finite cohomological dimension

The notion of a Vn-continuum was introduced by Alexandroff [1] as a generalization of the concept of n-manifold. In this note we consider the cohomological analogue of Vn-continuum and prove that any strongly locally homogeneous generalized continuum X with cohomological dimension dimG⁡X=n is a generalized Vn-space with respect to the cohomological dimension dimG. In particular, every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in [12].An analog of the Mazurkiewicz theorem that no subset of covering dimension ≤n−2 cuts any region of the Euclidean n-space is also obtained for strongly locally homogeneous generalized continua X of cohomological dimension dimG⁡X=n.

Derivations, extensions, and rigidity of subalgebras of the Witt algebra

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W=Der(k[t,t−1]) and the one-sided Witt algebra W≥−1=Der(k[t]). In the first part of the paper, we consider finite codimension subalgebras of W≥−1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to ExtU(L)1(M,L), where L is a subalgebra of W≥−1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W≥−1 extends to an automorphism of W≥−1.The second part of the paper is devoted to explaining the observed rigidity. We define a notion of “completely non-split extension” and prove that W≥−1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W≥−1 as abstract Lie algebras, they remember that they are contained in W≥−1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations.Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper.

Topological and dynamical properties of Torelli groups of partitioned surfaces

Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.

On metric and cohomological properties of Oeljeklaus-Toma manifolds

On metric and cohomological properties of Oeljeklaus-Toma manifolds

Failure of the well-rounded retract for outer space and Teichmüller space

The well-rounded retract for S L n ( Z ) SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X n X_n of metric graphs of rank n n and the Teichmüller space T g \mathcal {T}_g of closed hyperbolic surfaces of genus g g , we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for T g \mathcal {T}_g by Thurston) are spines for X n X_n but that their dimension is larger than the virtual cohomological dimension of O u t ( F n ) Out(F_n) in general.

On low dimensional cohomology of crossed modules with nontrivial coefficients

On low dimensional cohomology of crossed modules with nontrivial coefficients

Second cohomology group and quadratic extensions of metric Hom-Jacobi–Jordan algebras

In this paper, we introduce and study the low dimensional cohomology of metric Hom-Jacobi–Jordan algebras. We establish one-to-one correspondence between the equivalence classes of abelian quadraticextensions of a Hom-Jacobi–Jordan algebra and its second cohomology group.

Projectively coresolved Gorenstein flat dimension of groups

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group G over a commutative ring R and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of RG in terms of this dimension and the Gorenstein global dimension of R. Moreover, we study special cases of groups, such as ▪-groups, and show that for such a group every Gorenstein projective RG-module is Gorenstein flat when the global dimension of R is finite.

Properties of gradient maps associated with action of real reductive group

Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].

Arithmetical rank and cohomological dimension of generalized binomial edge ideals

Let [Formula: see text] be a connected and simple graph on the vertex set [Formula: see text]. To the graph [Formula: see text] one can associate the generalized binomial edge ideal [Formula: see text] in the polynomial ring [Formula: see text]. We provide a lower bound for the cohomological dimension of [Formula: see text]. We also study when [Formula: see text] is a cohomologically complete intersection. Finally, we show that the arithmetical rank of [Formula: see text] equals the projective dimension of [Formula: see text] in several cases.