Let $R$ be a graded ring (concentrated in nonnegative dimensions and maybe bounded from above). For every positive natural number $n$, denote by $Rtotau_{leq n}R$ the $n$-truncation and by $tau_{geq n}R to R$ the analogous procedure killing all dimensions below n. These two rings may both be viewed as $R$-modules.
My question is the following: Is there a simple procedure computing $Tor_*^R(tau_{leq n}R, tau_{geq n+1}R)$ (in the category of graded modules)?
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