Subject: Homology of Algebras
Date: Sun, 17 Mar 2002 02:49:46
From: "Avraham Goldstein"
To: dmd1@lehigh.edu
Dear Prof. Davis,
Can I, please, post the following to the list ?
Let $A$ be a connected algebra over a field $F$. $I$ be an augmentation
ideal of $A$, $G \subset I$ be some minimal generating set of $A$, $A'$ be a
free algebra, generated by $G$, $I'$ - the augmentation ideal of $A'$ and
$Rel$ - the kernel ideal of the projection of $A'$ onto $A$.
It is a well known fact that $Tor_1(A)$ is isomorphic, as $F$-vector space,
to $I/(I\cdot I) = I'/(I'\cdot I') .
Futhermore, there is a result of C.T.C. Wall, which can be restated as:
$Tor_2(A)$ is isomorphic to $Rel/(Rel\cdot I'+I'\cdot Rel)$
In what I hope to become my thesis I prove that $Tor_3(A)$ is isomorphic
to $(I'\cdot Rel \cap Rel\cdot I')/(Rel\cdot Rel+I'\cdot Rel\cdot I')$.
Futhermore I outline how to continue such a description for all higher
$Tor_n(A)$. At present I am working to prove the "isomorphic" part for
$Tor_4(A)$.
One "usefullness" of this aproach is, that in a free algebra $A'$ the
homogeneous elements (products of generators) can be filtered by the number
of generators in them (called their weight). In the same manner one can
filter the homogenous elements in A'\otimes A'. This filtration can, then be
used to filter any elemtns in $A'$ and $A'\otimes A'$.
If $A$ is a Hopf algebra, one can lift the co-product to $A'$ and the
filtration by weight gives a nice way of calculating the cup-co-product
struture in the Homology of $A$.
Futhermore, in that case of the mod 2 Steenrod Algebra I was able to show
many relations on the products of $h$'s this way.
In the case of Steenrod Algebra cup_1-co-product is also nicely describle
via the "weight" filtration.
Is there any information about the description of Homology as quotients of
ideals in free algebra and usage of the "weight" filtration for describing
cup-co-products and cup_n-co-products ?
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