Asymptotic dimension of finitely presented groups

We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.

     

Representation dimension and finitely generated cohomology

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

On Serre dimension of monoid algebras and Segre extensions

Let R be a commutative noetherian ring of dimension d and M be a commutative, cancellative, torsion-free monoid of rank r. Then S-$dim(R[M])\le max\{1,dim(R[M])?1\}$. Further, we define a class of monoids ${\{{\mathfrak{M}}_n\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal, then S-$dim(R[M])\le dim(R[M])?n=d+r?n$, where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over R, S-$dim(S_{mn}(R))\le dim(S_{mn}(R))?\left[\frac{m+n?1}{min\{m,n\}}\right]=d+m+n?1?\left[\frac{m+n?1}{min\{m,n\}}\right]$.     

When Essential Extensions of Finitely Generated Modules are Finitely Generated

For certain classes 𝒞 of R-modules, including singular modules or modules with locally Krull dimensions, it is investigated when every module in 𝒞 with a finitely generated essential submodule is finitely generated. In case 𝒞 = Mod-R, this means E(M)/M is Noetherian for any finitely generated module M_R. Rings R with latter property are studied and shown that they form a class 𝒬 properly between the class of pure semisimple rings and the class of certain max rings. Duo rings in 𝒬 are precisely Artinian rings. If R is a quasi continuous ring in 𝒬 then R ? A ⊕ T where A is a semisimple Artinian ring and T ∈ 𝒬 with Z(T_T) ≤_ess T_T.     

Betti numbers of finitely presented groups and very rapidly growing functions

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the kth Betti number b_k(G)= rank H_k(G) be providing that G has length ≤N and b_k(G) is finite? We prove that for every k≥3 the maximum b_k(N) of the kth Betti numbers of all such groups is an extremely rapidly growing function of N. It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, b_k grows as the third busy beaver function that measures the maximal productivity of Turing machines with ≤N states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines.We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function.Also, we outline a construction of a finitely presented group all of whose homology groups are either or trivial such that its Betti numbers form a random binary sequence.     

Left (N,U)-coherent Dimensions and Excellent Extensions of Rings

Let S be an excellent extension of a ring R and U a flat right 5-module. We show in this paper that the left (N, U)-coherent dimension of S is equal to that of R.     

On transfinite extension of asymptotic dimension

We prove that a transfinite extension of the asymptotic dimension asind is trivial. We introduce a transfinite extension of the asymptotic dimension asdim and give an example of a metric proper space which has transfinite infinite dimension.     

Excellent扩张和弱维数

在这篇文章里，我们证明了，当环Ｓ是Ｒ的ｅｘｃｅｌｌｅｎｔ扩张，Ｍ是Ｓ－模时，Ｍ做为Ｓ－模的弱维数与Ｍ做为Ｒ－模的弱维数相等。     

On variations of m,n-simply presented abelian p-groups

We define a few new classes of p-primary abelian groups,and study their crucial properties,one of which is the so-called Nunke’s-esque property.They are also closely related to n-simply presented groups from Keef and Danchev(2012)and m,n-simply presented groups from Keef and Danchev(2013).     

Loeb measure,finitely additive measures,and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of -saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.      

Relative Flatness and Coherence

For a fixed given right R-module M, so-called M-flat left R-modules are investigated, which are flat relative to short exact sequences in the subcategory σ[M] subgenerated by M. The properties of M-flat modules lead naturally to a concept of an M-coherent module and M-coherent ring R. Various properties and characterizations of M-coherent modules and rings are proven.

Communicated by R. Wisbauer     

Applications and adaptations of the low index subgroups procedure

The low-index subgroups procedure is an algorithm for finding all subgroups of up to a given index in a finitely presented group and hence for determining all transitive permutation representations of of small degree. A number of significant applications of this algorithm are discussed, in particular to the construction of graphs and surfaces with large automorphism groups. Furthermore, three useful adaptations of the procedure are described, along with parallelisation of the algorithm. In particular, one adaptation finds all complements of a given finite subgroup (in certain contexts), and another finds all normal subgroups of small index in the group . Significant recent applications of these are also described in some detail.

     

Essential domains and two conjectures in dimension theory

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.

     

Finitely Presented Functors and Degenerations#

Given two d-dimensional Λ-modules M and N, then M degenerates to N if and only if there exists an exact sequence of the form 0→ U→ U ⊕ M→ N→ 0 for some U ? mod Λ (Zwara, 1998 Zwara , G. ( 1998 ). A degeneration-like order for modules . Arch. Math. 71 : 437 – 444 . [CSA] [CROSSREF]  [Google Scholar]). Having this as a starting point, in this article we give a characterization of degenerations by the existence of a certain finitely presented functor. This gives new information about U in the sequence above. We show how this new information can be used to prove that M   _deg N even when M ⊕ X ≤  _deg N ⊕ X for some X for modules over Λ _q  = k〈 x,y〉/〈 x ²,y ²,xy + qyx〉, q ≠ 0 in k, where k is an algebraically closed field.