Let $(R,\mathfrak m,k)$ be a local ring of depth $d$ and $u:F_1\rightarrow F_0$ a homomorphism of finite free modules such that $\operatorname{Im}u\subset \mathfrak mF_0$. Then this map induces the zero map $$\mathrm{Ext}^d(k,F_1)\rightarrow\mathrm{Ext}^d(k,F_0).$$ Could you explain me why?
Why is zero this map between Exts?
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commutative-algebra
homological-algebra