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Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where $H^i((x^n,y^n),R)$ is Koszul Homology with respect to $(x^n,y^n).$

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Yes, but you don't need a graduation, nor $R$ being noetherian, nor having only 2 generators. This is SGA 2, Exposé 2, Proposition 5 (see here).

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