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--- examples:matrix-factorization [2023/09/12 06:39] – [section 2.2] pzhou
+++ examples:matrix-factorization [2023/09/13 23:14] (current) – pzhou
@@ Line 24: / Line 24: @@
 Why? What is perf? What is perfect complex? bounded complex of finite projective B-modules. But, what is projective?
-=== Example ===
+++++ Example: $A = \C[x,y], W = xy, B = \C[x,y]/(xy). $ |
-$A = \C[x,y], W = xy, B = \C[x,y]/(xy). $
 Q1: is $M = B / (x)$ a projective $B$-module? A sufficent condition is that $\Hom(M, -)$ should have no higher cohomology, but, look at the free resolution of $M$, we get
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 So, is $(x+1)\C[x]$ a free module? I think yes! And, in fact, if $(x+1)$ is not a zero-divisor, then $(x+1) B$ is a free module.
+So, many objects in $Coh(X)$ is not in $Perf(X)$. For example, $M_1 = B/(x), M_2 = B/(y), M_0 = B/(x,y)$.
+The $y$-axis, $x$-axis, and the origin.
+I think, the tail of the infinite resolution of $M_1$ and $M_2$ are the same, we have $[M_1] = [M_2] [1]$.
+On the other hand, what about $M_0$? We have resolution
+$$ B^2 \xto{y,x} B^2 \xto{x,y} B \to M_0 $$
+So, it is like $[M_0] = [M_1] \oplus [M_2]$. That's pretty fun!
+So, what's the A-model picture? I think $Coh(X)$ is the Fuk of
+++++
+===== MF example =====
+consider the simplest matrix factorization example on toric 3-folds. They are both gluing two copies of $\C^3$ together.
+The first one is given by the fan with ray generator $(0,0,1), (1,0, 1), (0,1,1), (1,1,1)$. yes, a square in $x,y$ direction on level $z=1$. It is the total space of two line bundles on $\P^1$, but which two? Let's compute the face conormal, we have
+  * $a=(-1, 0, 0), b=(0, -1, 0) $ and $c=(1,0, -1), d=(0, 1, -1)$.
+one can see that $a+c =b+d$. That means the coordinate ring of the affine space is given by $AC = BD$, wher $A,B,C,D\in \C^4$. Now, we can do blow-up, I guess we can do two patches with coordinate $A, B, u = C/B=D/A$ and the other with $C, D, v = B/C = A/D$. OK, great! Now, how does the fiber coordinate tranform? We have $u B = C, uA = D$, so it is like, $Tot[O(-1) \oplus O(-1)]$.
+There is another way to resolve. Let's not go there.
+The second one is given by the fan on generators $(0, 0, 1), (1, 0, 1), (-1, 0, 1), (0, 1, 1)$. with dual cone generator being $a=(0, -1, 0), b=(1, 1, -1), c=(-1,1,-1)$. The problem with this set of generator is that, $b,c$ span a unsaturated sublattice, we have $d=(b+c)/2 = (0, 1, -1)$, which should be added. so the relation is $2d = b+c$. Then the coordinates are $A,B,C,D$ with $D^2 = BC$, and we do $\P^1$ coordinate like $u=B/D = D/C$ and $v = 1/u = D/B = C/D$. So $v^2 = C/B$, the local charts are
+  * $(A, B, v)$ and $(A, C, u)$, with change of coordinates like $u = 1/v$ and $C = v^2 B$. We have $Tot[O(0) \oplus O(-2)]$
+Now, for the superpotential, we know we want the coordinate $(0,0,-1)$.
+  * In the first case, we have function $ABu = AB(C/B) = AC = AB(D/A)=BD$, similarly, we have $CDv=CD(B/C)=BD..$, so $u = c-b=(1,1,-1)$ and $a+b+u= a+c=(0,0,-1)$ hmm, it works.
+  * For the other case, we have $a+d = (0,0,-1)$, so the function is $AD = ABv = ACu$.
+OK, great. Now, how do we compute its matrix factorization? We are no longer on an affine scheme, although, the function $W$ still is a well-defined function on the affinization.
 ===== References =====