Let Fn be the free group on n generators, and PΣn be the group of automorphisms of Fn which send each generator to a conjugate of itself. Let Kn be the kernel of the homomorphism PΣn → PΣn−1 induced by mapping one of the free group generators to the identity. We show that Kn has cohomological dimension (n − 1), and that Hi(Kn, Z) is infinitely generated for 2 ≤ i ≤ (n − 1).

Let A be a central simple algebra over a field F . Let k1, . . . , kr be cyclic extensions of F such that k1 ⊗F · · · ⊗F kr is a field. We investigate conditions under which A is a tensor product of symbol algebras where each ki is in a symbol F -algebra factor of the same degree as ki. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of...

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and ...

We develop a theory of parafree augmented algebras similar to the groups and explore some questions related Parafree Conjecture. provide an example finitely generated algebra infinite cohomological dimension. Motivated by this example, we prove version Composition-Diamond lemma for complete give sufficient condition be residually nilpotent in terms its relations.

In the past two decades, gauge theoretic methods became indispensable when considering manifolds in dimension four. Initially, research centred around the moduli spaces of Yang-Mills instantons. Simon Donaldson had introduced the instanton equations into the field. Using cohomological data of the corresponding moduli spaces, he defined invariants which could effectively distinguish differentiab...

We investigate the Galois group GS(p) of the maximal pextension unramified outside a finite set S of primes of a number field in the (tame) case, when no prime dividing p is in S. We show that the cohomology of GS(p) is ‘often’ isomorphic to the étale cohomology of the scheme Spec(Ok rS), in particular, GS(p) is of cohomological dimension 2 then. 2000 Mathematics Subject Classification: 11R34, ...

Let p be an odd prime number and let K be an imaginary quadratic number field whose class number is not divisible by p. For a set S of primes of K whose norm is congruent to 1 modulo p, we introduce the notion of strict circularity. We show that if S is strictly circular, then the group G(KS(p)/K) is of cohomological dimension 2 and give some explicit examples.

We consider Voevodsky’s slice tower for a finite spectrum E in the motivic stable homotopy category over a perfect field k. In case k has finite cohomological dimension (in characteristic two, we also require that k is infinite), we show that the slice tower converges, in that the induced filtration on the bi-graded homotopy sheaves Πa,bfnE is finite, exhaustive and separated at each stalk. Thi...

The complex of cycles on a surface is a cell complex that encodes all the ways that an element of first homology can be written as an embedded cycle. It was introduced by Bestvina–Bux–Margalit and plays an important role in their calculation of the cohomological dimension of the Torelli group. We give a detailed proof that this complex is contractible, expanding upon one of the proofs given by ...

We prove a new upper bound for the dimension of space cohomological automorphic forms fixed level and growing parallel weight on $\mathrm{GL}_2$ over number field which is not totally real, improving one obtained by Marshall. The main tool proof mod $p$ representation theory $\mathrm{GL}_2(\mathbb{Q}_p)$ as started Barthel-Livne Breuil, developed Paskunas.