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Nov 23, 2022 at 21:07 comment added user43326 OP is asking about one specific degree $p+q$, which means that there is no ring structure. He also supposes that $H_{p+i}(B,H_{q-i}F)=0$ for $i\neq 0$.
Nov 23, 2022 at 12:02 comment added Eduardo de Lorenzo I'm actually working with rational coefficients. I edited the question to reflect that.
Nov 23, 2022 at 11:43 history answered Neil Strickland CC BY-SA 4.0