Let A be an unital, associative, graded algebra over a base ring k. I’m happy to assume that k is a field if need be, and will insist that A free and of finite rank in each degree (locally finite). Further, A is connected: it vanishes in negative degrees and is of rank 1 (generated by the unit) in degree 0; then the projection onto k=A/A>0 makes k a graded A-module.

Write Hq,r(A)=Extq,rA(k,k) for the cohomology of the ring A. This is bigraded: if we compute this as the cohomology of the cobar complex for A, then classes in Hq,r(A) arise as the dual of tensors of the form [a1|…|aq] with the sum of the degrees of the ai being r. Notice that since non-unit ai are in positive degrees, r≥q.

The whole of the cohomology H∗,∗(A) is a bigraded algebra, and so for any constant j≥0, the summand

Hj:=∞⨁q=0Hq,q+j(A)

is a module of the subring H0≤H∗,∗(A).

My question is:Are there known conditions on A which ensure that Hj is finitely generated over H0?Of course H0 is always finitely generated over itself, so I’m really interested in j>0. In that vein, I know one condition which ensures finite generation: if A is generated in degree 1 and is Koszul, then by definition Hj=0 for j>0. Is there anything less stringent than this?

**Answer**

**Attribution***Source : Link , Question Author : Craig Westerland , Answer Author : Community*