The Stacks project

Situation 15.91.15. Let $A$ be a ring. Let $I = (f_1, \ldots , f_ r) \subset A$. Let $K_ n^\bullet = K_\bullet (A, f_1^ n, \ldots , f_ r^ n)$ be the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ viewed as a cochain complex in degrees $-r, -r + 1, \ldots , 0$. Using the functoriality of Lemma 15.28.3 we obtain an inverse system

\[ \ldots \to K_3^\bullet \to K_2^\bullet \to K_1^\bullet \]

compatible with the inverse system $H^0(K_ n^\bullet ) = A/(f_1^ n, \ldots , f_ r^ n)$ and compatible with the maps $A \to K_ n^\bullet $.

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