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Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W, \ldots, \partial_n W)$ is $(x_1, \ldots, x_n)$-primary. For finitely generated $R$-modules $M$ and $N$ define the function $\theta$ to be $\theta(M, N) = \lambda( \operatorname{Tor}^R_{2i} (M,N)) - \lambda( \operatorname{Tor}^R_{2i-1}(M,N))$ for any $i \gg 0$. Here $\lambda$ denotes the length of a module. The definition makes sense because all modules over $R$ have eventually 2-periodic resolutions (since $R$ is a hypersurface) and the $\operatorname{Tor}$'s have finite length for $i \gg 0$ (since $R$ is an "isolated singularity"). Hochster made this definition in his 1981 paper "The dimension of an intersection in an ambient hypersurface."

I'm looking for examples (or preferably a family) of isolated hypersurface singularity rings with $n \geq 4$ and modules $M$ over such rings with $\theta(M, - )$ non-zero. See Hailong's answer below for an equivalent formulation of this in terms of certain Chow groups when $k = \mathbb{C}$ and $W$ is homogeneous. I would prefer if $W$ were not homogeneous but am interested in all cases.

Of course any insight as to when $\theta$ is non-zero would be great but in general this is hard. For instance in the paper above Hochster showed that the direct summand conjecture is true if $\theta$ is non-zero for an explicit family of modules and rings. It is conjectured that $\theta$ is zero when $n$ is odd (this is known when $W$ is homogeneous, see http://arxiv.org/abs/0910.1289v1), and it is known that when $n=4$ the function $\theta$ is nonzero if and only if the class group of $R$ is nonzero.

EDIT: There is a physical interpretation of the above in the spirit of this post: Matrix factorizations and physics. My knowledge of physics is limited so I apologize in advance for any mistakes.

D-branes in a B-twisted topological Landau-Ginzburg models with potential $W$ are given by matrix factorizations of $W$. We only care about values of $\theta$ on maximal Cohen-Macaulay (MCM) $R$-modules, and all such modules are given by matrix factorizations of $W$. Thus MCM modules over $R$ can be thought of as D-branes. Now physicists talk about the BRST-cohomology of two branes $M,N$ (which I don't understand) but it seems that it is given by $\operatorname{Ext}_R^2(M \oplus \Omega M, N \oplus \Omega N)$ (or equivalently the stable homomorphisms between these modules) where $\Omega( - )$ denotes the first syzygy; see for instance http://arxiv.org/abs/0802.1624. It is not hard to see, viewing the modules as matrix factorizations, that for two MCM modules $M$ and $N$ we have $\theta(M, N) = \lambda( \operatorname{Ext}^1_R( M^*, N) ) - \lambda( \operatorname{Ext}^2_R(M^*, N) )$, where $M^*$ is the MCM module given by $\operatorname{Hom}_R(M, R)$. Thus to find an example of modules with nonzero $\theta$ is equivalent to finding branes whose "even" and "odd" BRST cohomology have different dimensions over $k$.

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up vote 6 down vote accepted

Assume $k= \mathbf C$ and $W$ homogeneous. Let $X=Proj (k[x_1,\cdots,x_n]/(W))$. $X$ is then a smooth hypersuraface in $\mathbb P_{n-1}$.

Assume $n=2d$ is even. Corollary 3.10 of the paper you quoted says that $\theta=0$ for all pairs iff the homological Chow group $CH^{d-1}_{hom}$ modulo $[h]^{d-1}$ is not torsion (here $[h]$ is the class of the hyperplane section). So your question, in this case, is equivalent to ($l=d-1$):

Examples of smooth hypersurfaces of dimension $2l$ such that $CH^{l}_{hom}/([h]^{l})$ is not torsion ?

(By the way, I think if you phrased your question this way, it probably would become more popular, consider how many geometry-inclined people visit this site! So if you want more and better answers, consider changing the title.)

Now, a cheap way to get examples you want is to take $W= x_1x_{d+1} + \cdots + x_dx_{2d}$. Then the cycle defined by $(x_1,...,x_d)$ will not be a multiple of a power of the hyperplane section. Why? Because, I am waving my hand a bit here, if it is then the intersection with the cycle $(x_{d+1},\cdots, x_{2d})$ would be positive. But they are disjointed in $X$!

The same trick works for generalized quadrics, i.e. if $W = f_1g_1 +\cdots +f_dg_d$.

EDIT: Let me give more details here. In this situation you can easily make $W$ non-homogeneous as you desire. But the trouble is you can't use my argument above as there is no longer a projective variety $X$. But one can get around this. Let $S=k[x_1,\cdots,x_{2d}]_{m}$

here $m$ is the irrelevant ideal. Suppose $W = f_1g_1 +\cdots +f_dg_d$ and assume that $(f_1,\cdots, f_d, g_1,\cdots, g_d)$ is a full system of parameters in $S$. Let $R=S/(W)$, $P=(f_1,\cdots,f_d)$ and $Q=(g_1,\cdots,g_d)$. I claim that $\theta^R(R/P,R/Q) \neq 0$.

The reason is that $\theta^R(R/P,R/Q) = \chi^S(S/P,S/Q)$, the Serre's intersection multiplicity (see Hochster's original paper). Because $dim S/P + dim S/Q = d+d =dim S$, we must have $\chi^S(S/P,S/Q)>0$ by Positivity, which is known in this case.

More exotic examples should be abound, and I am sure people who know more intersection theory can provide some, once they are aware of what this question is about. I would be interested in hearing more answers along that line.

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That does help but I'm especially interested in the non-homogeneous case, so we can't use the machinery of the paper cited. –  Jesse Burke Jan 16 '10 at 20:35
    
my example of generalized quadrics should work in the non-homogenous case, by choosing f,g appropriately. –  Hailong Dao Jan 16 '10 at 20:42
    
I think the tag algebraic k-theory is appropriate –  Hailong Dao Jan 16 '10 at 20:43
    
What conditions (if any) do you assume on the f and g for W to be a generalized quadric? The results of the paper still apply if W is homogeneous with respect to some grading of the ring. So I guess we could pick f and g so that this holds, but I'm not sure if this is what you meant...I (re)added the tag. –  Jesse Burke Jan 16 '10 at 21:33
    
Obviously you need to make sure {W=0} has isolated singularity. I think (at least in char 0) requiring that the ideal generated by f_is,g_is is (x_1,...x_n)-primary would be enough, but I have not checked carefully. –  Hailong Dao Jan 16 '10 at 21:39
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