Let $(R, \mathfrak{m})$ be a regular complete local ring, let $f \in \mathfrak{m}^2$, and let $b \in \mathfrak{m}$ such that $b$ is a non zero divisor of both $R$ and $R/\langle f \rangle$. Denote $S=R/\langle b \rangle$ and the image of $f$ in $S$ by $\overline{f}$.
A matrix factorization of $f$ over $R$ is a pair of matrices $(\varphi, \psi)$ over $R$, viewed up to change of variables, such that $\varphi \psi = f I$ and $\psi\varphi=fI$, where $I$ is the identity matrix. Similarly we define matrix factorizations of $\overline{f}$ over $S$.
Denote the set of matrix factorizations of $f$ over $R$ by $MF_R(f)$, and similarly denote $MF_S(\overline{f})$. Then there is a natural map $MF_R(f) \to MF_S(\overline{f})$ , which is going$\mod b$. Is this map surjective? Injective? Neither?
Can we say that $MF_R(f)$ and $MF_S(\overline{f})$ have the same cardinality?
