In his really nice thesis, Tobias Dyckerhoff proved the following theorems about matrix factorizations(of possibly infinite rank) over a regular local k-algebra R with a function w and residue field k such that the Tyurina algebra, T= $R/(w,dw)$ is finite dimensional. This last condition says that w has an isolated singularity. For further reference, let S denote the ring R/(w).

1. The homotopy category of matrix factorizations has a compact generator as a triangulated category, which he denotes as $k^{stab}$.

2. As a consequence of 1), he derives that there is a natural complex which represents the identity functor thought of as an element of MF($R\otimes R,1\otimes w-w\otimes 1)$ which he denotes as the stabilization of the diagonal $R^{stab}$.

3. MF(R,w) is a Calabi Yau dg-category.

Now my question is how much of the above remains true for when the singularity is non-isolated? In some writings, Kontsevich, while not explicitly saying so, writes as if the homotopy category always has a compact generator and that the category is there by "dg-affine", e.g. equivalent to D(A), the derived category of modules over a dg-algebra. Is this indeed known to be true or false? If not, is there a way to prove 2) without making reference to 1)? I'm asking because I haven't found anything about this stuff in the literature, but a lot of things in this field are not written or written in physics literature that I'm not familiar with.

I've checked a few examples with non-isolated singularities and it appears that for example in the category of factorizations $(k[[x,y]], xy^2)$, that while $k^{stab}$ doesn't generate as Dyckerhoff proves, one has $(k\oplus k[[x]])^{stab}$, which I think does generate. The way I want to argue this is Dyckerhoff's theorem 3.6 that it is enough to show that $Tor_S(k\oplus k[[x]], M)$ implies that $Tor_S(N,M)=0$, where N is a finitely generated T module and M is any S module.Then one does an analysis of finitely generated modules over T(I didn't think about the characteristic 2 case) and does some devissage with short exact sequences. Please let me know if this sounds off. I also think that with a bit more calculation one can prove similarly that in (k[[x,y,z]], xyz) the module $(k[[x]]\oplus k[[y]]\oplus k[[z]]\oplus k)^{stab}$ is a compact generator.

Added: I think the right generalization of the above two examples is the following, in http://websupport1.citytech.cuny.edu/faculty/hschoutens/pdf/finiteprojdim.pdf, the author introduces the notion of a "net". The above method should give a compact generator, whenever the net of finitely generated modules over T is generated as a net by finitely many modules. This happens for example when T has finitely prime ideals. The modules A/p, where p is a prime, generate the net of finitely generated projective modules over T, which is enough to prove the vanishing above. In particular, this should take care of the case when T has dimension 1. A question is what are some conditions on (R,w) which lead to the net of finitely generated modules over T being generated by finitely many objects?

Assuming that this right, I think that to derive 2 and 3 for these examples becomes a formality in view of Dyckerhoff's section 5. One just replaces his compact generator with the new one.