I am reading paper "Standard systems of parameters and their blowing-up rings", J. Reine Angew. Math. 344 (1983), 201--220 of Peter Schenzel. In proof of Theorem 3.9, page 209-the second diagram, he use a short exact sequence of Koszul cohomology with out any explain. It maybe a standard property of Koszul cohomology but I can not prove it. Could you help me or give a reference of it?
In fact, I expect a general result as follows.
Proposition. Let $M$ be a finitely generated $R$-module.Let $x$ be an element such that $0:_Mx = 0:_M x^2 $ and $a_1, \ldots, a_t$ a sequence of elements. The multiple map $M \overset{x}{\to} M$ reduces the map $\widetilde{x}: \overline{M} \to M$, where $\overline{M} = M/0:_Mx$. Then the induced homomorphism of Koszul cohomology $$ H^i( x, a_1, \ldots, a_t; \overline{M}) \to H^i( x, a_1, \ldots, a_t; M)$$ is vanish for all $i \ge 0$.
