Suppose we're given a filtered algebra A over a field k with filtration F &#x2

Dayanara Terry 2022-06-30 Answered
Suppose we're given a filtered algebra A over a field k with filtration F A over the subspaces of A:
{ 0 } F 0 A F i A A ,
and suppose that
g r F A := i N 0 g r i F A
is the associated graded algebra of A, where g r i F A := F i A / F i 1 A and g r 0 F A = F 0 A.
If g r F A is commutative, does it follow that F i + j A F i A F j A for all i , j N 0 ?
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Answers (1)

Zackery Harvey
Answered 2022-07-01 Author has 21 answers
Nope. To keep things really simple let's take A itself to be commutative and graded (with the filtration where F n ( A ) is sums of elements of degree at most n, so the associated graded is just A again): specifically, take
A = k [ x 1 , x 2 ] / ( x 1 2 = x 2 2 = 0 )
where deg x i = i. (It doesn't really matter whether we impose x 2 2 = 0 or not, the point is just to have an element of degree 2 that isn't a sum of products of elements of degree 1.)
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