# Bertini type result for torsion-freeness

Let $$R$$ be a local, regular $$\mathbb{C}$$-algebra and $$\mathfrak{m}$$ be the maximal ideal. Let $$M$$ be a finitely generated torsion-free $$R$$-module. Suppose there exists $$f \in \mathfrak{m}$$ such that $$M/fM$$ is $$R/(f)$$-torsion-free in the sense that $$M/fM$$ is torsion-free as a $$R/(f)$$-module. Is it true that for all but countably many $$g \in \mathfrak{m}$$ (or $$g \in \mathfrak{m}/\mathfrak{m}^2$$) we have $$M/gM$$ is $$R/(g)$$-torsion-free?

• What happens when $M=R/(f)$? Then it is torsion free mod $f$, but $M/gM = R/(f,g)$ is unlikely to be torsion free over $R/(g)$ if $(g)\neq (f)$. May 13 at 19:49
• @PiotrAchinger Thanks I have added the condition that $M$ is torsion-free as well.
– Chen
May 14 at 4:39