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 $k$.


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.

  • $\begingroup$ That does help but I'm especially interested in the non-homogeneous case, so we can't use the machinery of the paper cited. $\endgroup$ – Jesse Burke Jan 16 '10 at 20:35
  • $\begingroup$ my example of generalized quadrics should work in the non-homogenous case, by choosing f,g appropriately. $\endgroup$ – Hailong Dao Jan 16 '10 at 20:42
  • $\begingroup$ I think the tag algebraic k-theory is appropriate $\endgroup$ – Hailong Dao Jan 16 '10 at 20:43
  • $\begingroup$ 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. $\endgroup$ – Jesse Burke Jan 16 '10 at 21:33
  • $\begingroup$ 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. $\endgroup$ – Hailong Dao Jan 16 '10 at 21:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.