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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?

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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.

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  • $\begingroup$ I think maybe you are thinking of some version of Orlov's theorem involving graded matrix factorizations here, although the point still stands. $\endgroup$
    – Daniel Pomerleano
    Commented Jan 13, 2015 at 16:39
  • $\begingroup$ Yes, you are correct, I was thinking of graded matrix factorizations. In the non-graded case, there is even a simpler answer: the suspension functor is doubly periodic ([2] = id), which never happens in any derived category. $\endgroup$
    – Vincent Gelinas
    Commented Jan 13, 2015 at 23:54
  • $\begingroup$ @VincentGelinas But in any case, the split-closed triangulated category of singularity $D_{sing}^\pi(W^{-1}(0))$ is split-generated by skyscraper sheaves at those singular points. So if we assume the singularities of the central fiber is good enough, say nodal singularities, then Knorrer periodicity should hold in this case, am I right? $\endgroup$
    – YHBKJ
    Commented Jan 14, 2015 at 9:20
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What you want to be true is false for a number of different reasons.

1) One point is whether you take idempotent completions or not in your triangulated category of matrix factorizations? Even if all of your isolated critical points are non-degenerate and in an even dimensional space X, you will only get what you want if take idempotent completions of the categories.

2) Even if what you consider is a non-degenerate quadratic function on an odd dimensional vector space (e.g. $(\mathbb{C}[x],x^2))$, the category is not equivalent to $D^bCoh(\mathbb{C})$. This is some form of "Bott periodicity."

3) If the critical points are not non-degenerate this is very false, independent of how you interpret "Crit(W)." If you interpret it as reduced subscheme then this is false as Vincent noted. It isn't helped by taking any other reasonable version of critical locus (e.g. scheme theoretic) either.

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