Comments on: Blackbox computation of A-infinity structures http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/ Blogging ... up to homotopy. Wed, 07 Jan 2009 03:41:24 +0000 http://wordpress.org/?v=2.5 By: Mikael Vejdemo Johansson http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-214 Mikael Vejdemo Johansson Thu, 27 Mar 2008 17:35:27 +0000 http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-214 $H^{*}(C_{5},Z/5)$ would indeed have arity up to 5, but it is a much more subtle interplay than just group order. For instance, my and Ainhoas results show that $H^{*}(C_{n}\times C_{n},Z/p)$ has unbounded structures with respect to the arity. So group order is far from the only factor in determining these structures. H *(C 5 ,Z/5 ) would indeed have arity up to 5, but it is a much more subtle interplay than just group order. For instance, my and Ainhoas results show that H *(C n×C n,Z/p) has unbounded structures with respect to the arity. So group order is far from the only factor in determining these structures.

]]> By: Jim Stasheff http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-213 Jim Stasheff Thu, 27 Mar 2008 16:00:10 +0000 http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-213 Mikael wrote: This example shows us that the only higher products and quasi-isomorphism components that do not vanish in H *(C 3,Z/3) have arity 2 and 3. Is it that arity greater than 3 is prohibited by the order of the group? e.g. would H *(C 5,Z/5) have arity up to 5? Mikael wrote:

This example shows us that the only higher products and quasi-isomorphism components that do not vanish in H *(C 3,Z/3) have arity 2 and 3.

Is it that arity greater than 3 is prohibited
by the order of the group?
e.g. would H *(C 5,Z/5) have arity up to 5?

]]> By: Mikael Vejdemo Johansson http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-196 Mikael Vejdemo Johansson Sat, 09 Feb 2008 11:50:22 +0000 http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-196 Actually, it occurred to me. When computing Ext(k,k) over a group ring, we have the obvious way to construct, step by step, a minimal resolution. From the minimal resolution, we can deduce generators and relations. And we know when we have it all, as it were, by virtue of the Carlson/Benson stopping criteria. For monomial rings, I know that there are good methods to get hold of the Poincare-Betti series given a presentation of the ring. And certainly, with an initial portion of a minimal resolution computed, it is easy to start constructing a finite presentation. However, I haven't heard of anything resembling these stopping criteria. John: are you able to sit down with a ring presentation (local, monomial, Adams graded - throw any nicety properties you want at it ;) and work out a finite presentation of Ext(k,k) as a k-algebra from it? Or is the obsession with Poincare-Betti series I remember rooted in ring structures being too difficult to get hold of? Actually, it occurred to me. When computing Ext(k,k) over a group ring, we have the obvious way to construct, step by step, a minimal resolution. From the minimal resolution, we can deduce generators and relations. And we know when we have it all, as it were, by virtue of the Carlson/Benson stopping criteria.

For monomial rings, I know that there are good methods to get hold of the Poincare-Betti series given a presentation of the ring. And certainly, with an initial portion of a minimal resolution computed, it is easy to start constructing a finite presentation. However, I haven’t heard of anything resembling these stopping criteria.

John: are you able to sit down with a ring presentation (local, monomial, Adams graded - throw any nicety properties you want at it ;) and work out a finite presentation of Ext(k,k) as a k-algebra from it? Or is the obsession with Poincare-Betti series I remember rooted in ring structures being too difficult to get hold of?

]]> By: Mikael Vejdemo Johansson http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-195 Mikael Vejdemo Johansson Sat, 09 Feb 2008 00:08:14 +0000 http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-195 I'm using the Magma modules built by Jon F. Carlson for computing the cohomology data. Of course, once the homological algebra handling in Magma gets expanded to handle non-group-algebra rings, there's nothing that prevents my code from working with those - however, the homological algebra code currently in there is built for the express purpose of computing group cohomology rings, and functions such as computing projective resolution and computing cohomology ring generators and relations only work if the BasicAlgebra (the data type for path algebra quotients) is a group ring. I’m using the Magma modules built by Jon F. Carlson for computing the cohomology data. Of course, once the homological algebra handling in Magma gets expanded to handle non-group-algebra rings, there’s nothing that prevents my code from working with those - however, the homological algebra code currently in there is built for the express purpose of computing group cohomology rings, and functions such as computing projective resolution and computing cohomology ring generators and relations only work if the BasicAlgebra (the data type for path algebra quotients) is a group ring.

]]> By: John Palmieri http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-194 John Palmieri Fri, 08 Feb 2008 19:38:11 +0000 http://infinity.blogseminar.net/2008/02/06/blackbox-computation-of-a-infinity-structures/#comment-194 Looks fun. I suppose it requires a group as input. Would it be conceivable to put in other sorts of augmented algebras instead, to find A-infinity structures on their Ext algebras? (I've never used Magma, so I don't know how much you're using already existing Ext calculations, versus writing your own.) Looks fun. I suppose it requires a group as input. Would it be conceivable to put in other sorts of augmented algebras instead, to find A-infinity structures on their Ext algebras?

(I’ve never used Magma, so I don’t know how much you’re using already existing Ext calculations, versus writing your own.)

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