Let $X$ be an algebraic variety and $W:X\rightarrow\mathbb{A}^1$ a regular map, then the triangulated category of matrix factorizations $D^b(X,W)$ is defined to be


$D^b(X,W)=\bigsqcup\_{t\in\mathbb{A}^1}D^b\big(W^{-1}(t)\big)/\mathrm{Perf}\big(W^{-1}(t)\big)$


Let $Y=\mathrm{Crit}(W)$ be the critical locus of $W$, then the triangulated equivalence


$D^b(Y)\cong D^b(X,W)$


is known to hold in many cases. For example, when $X=\mathbb{C}^2$ and $W=xy$ is the product of coordinates.


Assume $W$ has only isolated non-degenerate critical points (a singular fiber of $W:X\rightarrow\mathbb{A}^1$ may have several singularities), my question is will $D^b(Y)\cong D^b(X,W)$ always be true with these assumptions?



A question on triangulated category of matrix factorizations
If W has a single critical value, then Y = Crit(W) is a finite collection of points, hence $D^{b}(Y) \cong \oplus\_{p \in Crit(W)} D^{b}(\mathbb{C}-vect)$. This will usually not be equivalent to the triangulated category of matrix factorizations of W.


For example, say that $W = x^{3} + y^{3} + z^{3}$ on $\mathbb{C}^{3}$. This has a single critical value at 0, and $W^{-1}(0)$ is the affine cone over a plane elliptic curve E. Its unique critical point is $(0,0,0) \in \mathbb{C}^{3}$, the vertex of the cone, hence $D^{b}(Y) \cong D^{b}(\mathbb{C}-vect)$. However, $D^{b}(\mathbb{C}^{3}, W)$ is known to be equivalent to $D^{b}(E)$. This holds for Calabi-Yau hypersurfaces by a theorem of Orlov.


