---
abstract: 'We study two properties of two modules over a local hypersurface $R$: decency, which is close to proper intersection of the supports, and $\operatorname{Tor}$-rigidity. We show that the vanishing of Hochster’s function $\theta^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $\theta^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. Many applications are given.'
address: |
    Department of Mathematics\
    University of Kansas\
    Lawrence, KS 66045-7523 USA
author:
- Hailong Dao
title: 'Decency and Tor-rigidity for modules over hypersurfaces'
---

[^1]

Introduction
============

Throughout this paper we will deal exclusively with a local, Noetherian, commutative ring $R$ and finitely generated modules over $R$.

Two $R$-modules $M$ and $N$ such that $l(M{\otimes}_RN)<\infty $ are said to *intersect decently* if $\dim M + \dim N \leq \dim
R$. We say that $M$ is *decent* if for all $N$ such that $l(M{\otimes}_RN)<\infty $, $M$ and $N$ intersect decently. This property arises naturally from Serre’s work on intersection multiplicity ([@Se]), which shows that over a regular local ring, any two modules intersect decently. In fact, to have a satisfying local intersection theory, one needs modules to intersect decently as a minimum requirement. However, sufficient conditions for decent intersection become much more elusive in general, even when $R$ is a hypersurface. For example, one can look at a famous open question in Commutative Algebra:

Let $A$ be a regular local ring, and suppose $R$ is a local ring such that $A\subset R$ and $R$ is module-finite over $A$. Then $A$ is a direct summand of $R$ as an $A$-module.

The Direct Summand Conjecture could be proved if one could show a certain module over a local hypersurface is decent (see [@Ho1]). As another example, it was conjectured by Peskine and Szpiro (cf. [@PS]) that in general, a module of finite projective dimension is decent. This question is open even for hypersurfaces of ramified regular local rings.

A pair of modules $(M,N)$ is called *rigid* if for any integer $i\geq0$, $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$. Moreover, $M$ is *rigid* if for all $N$, the pair $(M,N)$ is rigid. Auslander studied rigidity in order to understand torsion on tensor products ([@Au]). He also observed that rigidity of $M$ implies other nice properties, such as any $M$-sequence must be an $R$-sequence. To further demonstrate the usefulness of rigidity, let us recall the:

([@HW1], 2.5) Let $R$ be a complete intersection. Let $M,N$ be non-zero finitely generated modules over $R$ such that $\operatorname{Tor}_i^R(M,N)=0$ for all $i\geq 1$. Then: $$\operatorname{depth}(M) + \operatorname{depth}(N) = \operatorname{depth}(R)+\operatorname{depth}(M{\otimes}_RN)$$

Thus, rigidity allows us to force a very strong condition on the depths of the modules by proving the vanishing of one single $\operatorname{Tor}$ module. Auslander’s work, combined with that of Lichtenbaum ([@Li]), showed that modules over regular local rings are rigid. Huneke and Wiegand ([@HW1],[@HW2]) continued this line to study rigidity over hypersurfaces.

The classical condition conjectured to be sufficient for both rigidity and decency was that one of the modules must have finite projective dimension. In general, this is open for decency and false for rigidity (see [@He]). In any case, having finite projective dimension is too restrictive for the most interesting applications, so a key question arises: *Are there more flexible sufficient conditions for rigidity and decency?* A good answer here would not only make applications easier, it would also shed some light on the behaviour of modules of finite projective dimension with respect to the two conditions.In this paper, we try to answer this question for modules over hypersurfaces using a mixture of results and techniques from Commutative Algebra and Intersection Theory. Perhaps not surprisingly, our answers often involve some conditions about the classes of the modules in the Grothendieck group of finitely generated modules over $R$.

Throughout this note we will assume that our hypersurface $R$ comes from an unramified or equicharacteristic regular local ring (we call such hypersurfaces “admissible", following Hochster). Since we need to apply results such as Serre’s Positivity and Non-negativity of $\chi_i$, which are open in general for the ramified case, this is necessary. In some particular instances, such as in low dimensions, this restriction can be relaxed, however we feel it may disrupt the flow of the paper to comment on every such case. The reader will lose very little by thinking of the equicharacteristic (containing a field) case.

Section \[2\] is a review of basic notation and some preliminary results. Of particular interest is Hochster’s theta function. For a local hypersurface $R$ and a pair of finitely generated $R$ modules $M,N$ such that $l(\operatorname{Tor}_i^R(M,N))<\infty$ for all $i\gg0$, we can define: $$\theta^R(M,N) = l(\operatorname{Tor}_{2e+2}^R(M,N)) - l(\operatorname{Tor}_{2e+1}^R(M,N))$$ Here $e$ is any sufficiently large integer. The theta function was first introduced by Hochster \[Ho1\] in his study of the direct summand conjecture. We recall the basic properties of $\theta(M,N)$ and prove a key technical result: [the vanishing of $\theta^R(M,N)$ implies rigidity of (M,N)]{}(proposition \[rg1\]).

In section \[3\] we study the vanishing of $\theta^R(M,N)$ when $R$ is a hypersurface with isolated singularity. The key advantage with such rings is that $\theta^R(M,N)$ is always defined for a pair $M,N$, so we can “move" the modules easily into favorable positions where vanishing of $\theta$ is more evident. We give a fairly complete picture when the dimension of the ring is at most $4$ and obtain many results in higher dimension (see \[dim1\] to \[dim2,3\]). One of them (\[moving\]) states that when $R$ contains a field and $\dim M + \dim N \leq \dim R$, then $\theta^R(M,N)=0$. This result partly answers a question raised to the author by Roger Wiegand. Our results also point to a conjecture that $\theta^R(M,N)$ should always vanish if $\dim R$ is [*even*]{}.

In section \[4\] we study rigidity over hypersurfaces in general. We prove new criteria for rigidity (theorem \[rig1.1\]), as well as a connection to decency when one of the modules is Cohen-Macaulay(see theorem \[rigidandproper\]).

In section \[5\] we apply our results from previous sections to give new proofs of seemingly different results in the literature, as well as new results. First, we show that if $char(R)=p>0$, then $^eR$ is rigid, where $^eR$ is $R$ with the module structure defined by the $e$-th iteration of the Frobenius homeomorphism (theorem \[AM\]). This is the hypersurface case of a theorem by Avramov and Miller ([@AM]). We also apply our rigidity results to gain understanding of depth of tensor products, following the same line of investigation done by Auslander, Huneke and Wiegand(see \[vanishingiso\], \[HWmain\]). Finally, we switch our attention to projective hypersurfaces and prove a result on intersection of subvarieties(theorem \[projhyper\]) using what we know about decency.

Finally, section \[lastsection\] discusses our attempts to generalize the theta function and some additional results we gain on the way which improve on previous works by Murthy and Jorgensen. We also give numerous examples to illustrate our results throughout the paper and show that they are optimal in certain senses.

After this preprint appeared online, a few works by the authors and other researchers have been focused on extensions and applications of the ideas and results in here. For example, [@Da2] uses the $\operatorname{Tor}$-rigidity results here to extend Auslander’s theorems on $\operatorname{Hom}(M,M)$ over regular local rings to hypersurfaces, and [@Da3] applies such results on understanding Van den Bergh’s definition of non-commutative crepant resolutions. Papers [@CeDa; @Ce; @Da] deal with various generalizations to complete intersections as well as analogous results for vanishing of $\operatorname{Ext}$ modules. Another new interesting development is [@MPSW] where the Conjecture \[DaoConj\] is studied and settled in the graded, characteristic $0$ situation using sophisticated geometric techniques.

Part of this paper was included in the author’s PhD thesis at the University of Michigan. The author would like to thank his advisor, Melvin Hochster, for numerous invaluable discussions and encouragements. It is virtually impossible for the author to get to many results here without learning various techniques and insights from the experts. It is a pleasure to thank Luchezar Avramov, Ragnar-Olaf Buchweitz, William Fulton, Craig Huneke, Mircea Mustaţǎ, Paul Roberts and Roger Wiegand for their incredible patience with the author’s questions. Special thanks must go to Greg Piepmeyer for his careful reading of an earlier version of this paper and many suggestions for improvements.

Notation and preliminary results {#2}
================================

Unless otherwise specified, all rings are Noetherian, commutative and local, and all modules are finitely generated. A ring $(R,m,k)$ is a *hypersurface* if its completion $\hat R$ has the form $T/(f)$, where $T$ is a regular local ring and $f$ is in the maximal ideal of $T$. We say that $T$ is *admissible* (as a regular local ring) if it is a power series ring over a field or a discrete valuation ring. If $T$ is admissible we also say that $R$ is admissible (as a hypersurface). Note that an admissible hypersurface may be a ramified regular local ring, and thus not admissible as a regular local ring.

For a ring $R$ and a non-negative integer $i$, we set $X^iR := \{p\in \operatorname{Spec}(R)| \dim(R_p)\leq i\}$. We denote by $Y(R)$ the set $X^{\dim(R)-1}$, the punctured spectrum of $R$. We denote by $G(R)$ the Grothendieck group of finitely generated modules over $R$ and by $\overline{G}(R):= G(R)/[R]$, the reduced Grothendieck group. Also, we let $\operatorname{Sing}(R) := \{p\in \operatorname{Spec}(R)| R_p \ \text{is not regular} \}$ be the singular locus of $R$. For an abelian group $G$, we let $G_{\mathbb{Q}} = G{\otimes}_{\mathbb{Z}}\mathbb{Q}$.

Let the torsion submodule of $M$, $t(M)$, be the kernel of the map $M\to K{\otimes}_RM$, where $K$ is the total quotient ring of $R$. The module $M$ is *torsion* provided $t(M)=M$ and *torsion-free* provided $t(M)=0$. Let $M^{*} := \operatorname{Hom}(M,R)$ be the dual of $M$. The module $M$ is called *reflexive* provided the natural map $M\to M^{**}$ is an isomorphism. The module $M$ is called *maximal Cohen-Macaulay* if $\operatorname{depth}_RM = \dim R$.

A pair of modules $(M,N)$ is called [*rigid*]{} if for any integer $i\geq0$, $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$.

One defines the finite length index of the pair $(M,N)$ as : $$f_R(M,N) := \min\{ i |\ l(\operatorname{Tor}_j^T(M,N))<\infty \ \text{for $j \geq i$} \}$$

If $M,N$ are modules over a regular local ring $T$, then for any integer $i\geq0$ such that $f_T(M,N) \leq i$, we can define : $$\chi_i^T(M,N) = \sum_{j\geq i} (-1)^{j-i} l(\operatorname{Tor}_j^T(M,N))$$ When $i=0$ we simply write $\chi^T(M,N)$ or $\chi(M,N)$. Serre ([@Se]) introduced $\chi^R(M,N)$ forty years ago as a homological definition of intersection multiplicity for modules over a regular local ring and showed that it satisfied many of the properties which should hold for intersection multiplicities:

(Serre)\[Serre\] Let $T$ be a regular local ring such that $\hat T$ is admissible. Then for any pair of $T$-modules $M,N$ such that $l(M{\otimes}_TN)<\infty$, we have:

1.  $\dim(M)+\dim(N) \leq \dim(T)$ (in other words, all modules are decent).

2.  (Vanishing) If $\dim(M)+\dim(N)<\dim(T)$, then $\chi^{T}(M,N)=0$.

3.  (Nonnegativity) It is always true that $\chi^{T}(M,N)\ge 0$.

4.  (Positivity) If $\dim(M)+\dim(N)=\dim(T)$, then $\chi^{T}(M,N)>0$.

The following “long exact sequence for change of rings" plays a vital role in our proof of rigidity criterion (\[rg1\]). It follows from the famous Cartan-Eilenberg spectral sequence ([@Av2], 3.3.2).

\[longexact\] Let $R=T/f$ such that $f$ is a nonzerodivisor on $T$, and let $M,N$ be $R$-modules. Then we have the long exact sequence of $\operatorname{Tor}$s :

$$\begin{array}{ll}
...\to \operatorname{Tor}_{n}^R(M,N) \to \operatorname{Tor}_{n+1}^T(M,N) \to \operatorname{Tor}_{n+1}^R(M,N)\\
\to  \operatorname{Tor}_{n-1}^R(M,N) \to \operatorname{Tor}_{n}^T(M,N) \to \operatorname{Tor}_{n}^R(M,N)\\
\to ...                                                         \\
\to \operatorname{Tor}_{0}^R(M,N) \to \operatorname{Tor}_{1}^T(M,N) \to \operatorname{Tor}_{1}^R(M,N) \to 0

\end{array}$$\

**The infinite projective dimension locus.**

Let $M$ be an $R$-module. One can define the infinite projective dimension locus of $M$ as $IPD(M):= \{ p\in \operatorname{Spec}(R) |\
\operatorname{pd}_{R_p}M_p =\infty \}$.

We gather some properties of this locus:

Let $R$ be a local hypersurface of dimension $d$. Let $M,N$ be $R$-modules. Let $\operatorname{Supp}_e(M,N)=\operatorname{Supp}(\operatorname{Tor}_{2d+2}^R(M,N))$ and $\operatorname{Supp}_o(M,N)=\operatorname{Supp}(\operatorname{Tor}_{2d+1}^R(M,N))$. Then we have :\
(1) $IPD(M)$ is a Zariski closed subset of $\operatorname{Spec}(R)$.\
(2) $IPD(M)\subseteq \operatorname{Supp}(M)\cap \operatorname{Sing}(R)$.\
(3) $\operatorname{Supp}_e(M,N) \cup \operatorname{Supp}_o(M,N) = IPD(M)\cap IPD(N)$.

\(1) Let $L = \operatorname{syz}_{d+1}^R(M)$. Let $F(M) = \{x\in R \ | \ L_{x} \ \text{is a free}
\ R_{x}\text{-module}\}$. For any prime $p$, $\operatorname{pd}_{R_p}M_p = \infty$ if and only if $L_p$ is not free if and only if $p \supset F(M)$. So $I(M) = V(F(M))$.\
(2) This is obvious.\
(3) Let $p\in \operatorname{Spec}(R)$ and localize at $p$. Then $R_p$ is a hypersurface. The assertion follows from the fact that $\operatorname{Tor}_{2d+2}^{R_p}(M_p,N_p)
= \operatorname{Tor}_{2d+1}^{R_p}(M_p,N_p)=0$ if and only if one of $M_p,N_p$ has finite projective dimension (see \[HW2\],theorem 1.9).

**The function $\theta^R(M,N)$.** \[\]

Let $R$ be a local hypersurface and assume $\hat R = T/(f)$ where $T$ is a regular local ring. The function $\theta^R(M,N)$ was introduced by Hochster (\[Ho1\]) for any pair of finitely generated modules $M,N$ such that $f_R(M,N)<\infty$ as: $$\theta^R(M,N) = l(\operatorname{Tor}_{2e+2}^R(M,N)) - l(\operatorname{Tor}_{2e+1}^R(M,N))$$ where $e$ is any integer $\geq (d+2)/2$. It is well known (see [@Ei]) that $\operatorname{Tor}^R(M,N)$ is periodic of period 2 after $d+1$ spots, so this function is well-defined. The theta function satisfies the following properties. First, if $M{\otimes}_RN$ has finite length, then: $$\theta^R(M,N) = \chi^T(M,N)$$ As a consequence of this fact and \[Serre\], we have :

\[decent\](Hochster) Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Then $(M,N)$ intersect decently if and only if $\theta^R(M,N)=0$

Secondly, $\theta^R(M,N)$ is biadditive on short exact sequences, assuming it is defined. Specifically, for any short exact sequence: $$0 \to N_1 \to N_2 \to N_3 \to 0$$ and any module $M$ such that $f_R(M,N_i)<\infty$ for all $i=1,2,3$, we have $\theta^R(M,N_2) = \theta^R(M,N_1) +
\theta^R(M,N_3)$. Similarly, $\theta(M,N)$ is additive on the first variable. Hochster exploited these properties to give a sufficient condition for an cyclic module in $R$ to intersect decently. We will give his result here, in a slightly different form (his result was stated in terms of ideals):

(\[Ho1\],theorem 0.1)\[hochster\] Let $R$ be an admissible local hypersurface and $M,N$ be $R$-modules. Assume that :\
(1) $\operatorname{Supp}(M) \supset \operatorname{Sing}(R)$.\
(2) $[M] = 0$ in $\overline G(R)_{\mathbb{Q}}$.\
(3) $l(M {\otimes}_R N) <\infty$.\
Then $\theta^R(M,N)=0$ and $\dim M + \dim N \leq \dim R$ (In other words, $M$ is decent).

(sketch) Note that (1) and (3) imply $IPD(N) \subseteq \operatorname{Sing}(R)\cap
\operatorname{Supp}(N) \subseteq \operatorname{Supp}(M)\cap \operatorname{Supp}(N) = \{m_R\} $. In other words, $N$ has finite projective dimension on the punctured spectrum of $R$. Hence $f(M',N)<\infty$ for any $R$-modules $M'$. This together with the fact that $[M]=0$ in $\overline
G(R)_{\mathbb{Q}}$ and biadditivity of $\theta$ shows that $\theta(M,N)=0$. But over $T$, this means $\chi^T(M,N)=0$, so by Serre’s theorem, $ \dim M + \dim N \leq \dim T-1 = \dim R$.

It is worth noting that the direct summand conjecture could be solved if we have a “sufficiently good" criterion for a certain module in a certain hypersurface to intersect decently (see \[Ho1\]). We also note that Hochster’s theta function is closely related to the notion of “Herbrand difference" defined using stable cohomology by Buchweitz in [@Bu]. In fact, they agree up to sign.

**The depth formula.**

A result by Huneke and Wiegand showed that when all the positive $\operatorname{Tor}$ modules vanish, the depths of the modules satisfy a remarkable equation:

([@HW1], 2.5) Let $R$ be a complete intersection. Let $M,N$ be non-zero finitely generated modules over $R$ such that $\operatorname{Tor}_i^R(M,N)=0$ for all $i\geq 1$. Then: $$\operatorname{depth}(M) + \operatorname{depth}(N) = \operatorname{depth}(R)+\operatorname{depth}(M{\otimes}_RN)$$

**Vanishing of $\theta^R(M,N)$ implies rigidity.**

The main technical result of this section says that, when $\theta^R(M,N)$ can be defined and vanishes, then $(M,N)$ is rigid:

\[rg1\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules such that $f_R(M,N)<\infty$ (so that $\theta^R(M,N)$ can be defined). Assume $\theta^R(M,N)=0$. Then $(M,N)$ is rigid.

Our main tool is a celebrated theorem first proved by Lichtenbaum (\[Li\]) except in a few cases. Those cases were completed by Hochster (\[Ho2\]):

(Lichtenbaum, Hochster)\[parchi\] Consider finitely generated modules $M,N$ over an admissible regular local ring $T$ and an integer $i$ such that $f_T(M,N)\leq i$ (so that $l(\operatorname{Tor}_j^T(M,N))<\infty$ for $j \geq i$). Then $\chi_i^T(M,N)\geq 0$ and it is $0$ if and only if $\operatorname{Tor}_j^T(M,N) = 0$ for all $j\geq i$.

In order to prove our rigidity result we will first need to prove a pivotal case, when all the relevant $\operatorname{Tor}$s have finite length, so we can apply theorem \[parchi\].

\[pivotal\] Let $R,M,N$ be as in \[rg1\]. Let $i$ be an integer such that $i
\geq f_R(M,N)$. Assume that $\theta^R(M,N)=0$ and $\operatorname{Tor}_i^R(M,N) =
0$. Then $\operatorname{Tor}_j^R(M,N) = 0$ for all $j\geq i$.

By completion we may assume $R=T/(f)$ where $T$ is an admissible regular local ring. We truncate the change of rings long exact sequence for $\operatorname{Tor}$s (\[longexact\]) as follows (note that all $\operatorname{Tor}^T(M,N)$ vanish after $d+1$ spots):\
$$\begin{array}{ll}

                                   0 \to \operatorname{Tor}^R_{2e+2}(M,N) \\
\to \operatorname{Tor}_{2e}^R(M,N) \to \operatorname{Tor}_{2e+1}^T(M,N) \to \operatorname{Tor}_{2e+1}^R(M,N)\\
\to  ...                                                          \\
\to \operatorname{Tor}_{i}^R(M,N) \to \operatorname{Tor}_{i+1}^T(M,N) \to \operatorname{Tor}_{i+1}^R(M,N) \to C \to 0
\end{array}$$\
It is easy to see that all the modules in this sequence have finite lengths. Therefore we can take the alternating sum of the lengths and get : $$l(C) + \chi_{i+1}^T(M,N) = (-1)^{2e+2-i}\theta^R(M,N) + l(\operatorname{Tor}_i^R(M,N))  = 0$$ This equation and theorem \[parchi\] forces $C=0$ and $ \operatorname{Tor}_j^T(M,N) =0$ for all $j\geq i+1$. The conclusion of the lemma now follows easily.

Now we can prove our rigidity result :

(of \[rg1\]) We use induction on $d = \dim R$. If $d=0$, $M,N$ both have finite length, so the previous lemma applies. Now assume $d \geq 1$ and $\operatorname{Tor}_i^R(M,N)=0$. Localizing at any $p\in Y(R)$, the punctured spectrum of $R$, and using the induction hypothesis (note that $R_p$ is at worst a hypersurface with dimension less than d, and $\operatorname{Tor}_j^{R_p}(M_p,N_p)=0$ for $j\geq f_R(M,N)$), we may conclude that $f_R(M,N) \leq i$. Again lemma \[pivotal\] can be applied to finish the proof.

Hypersurfaces with isolated singularity {#3}
=======================================

In this section we will investigate the vanishing of $\theta^R$ when $R$ is a local hypersurface with isolated singularity. Then $\theta^R(M,N)$ is defined for all pairs $(M,N)$ (since all higher $\operatorname{Tor}$ modules have finite length). In this situation $\theta^R$ defines a bilinear map from $G(R)_{\mathbb{Q}}\times
G(R)_{\mathbb{Q}}$ to $\mathbb{Q}$, hence by theorem \[rg1\] it vanishes whenever one of the modules is equivalent to $0$ in $\overline
G(R)_{\mathbb{Q}} = G(R)_{\mathbb{Q}}/{\mathbb{Q}}[R]$ (since $\theta^R(R,-) = 0$). We record this here for convenience.

Let $R$ be an admissible hypersurface with isolated singularity. Then $\theta^R(M,N)$ is always defined and vanishes if $M=0$ in $\overline G(R)_{\mathbb{Q}}$.

However, our investigation shall show that there are many more cases when $\theta^R$ vanishes. Our methods and inspirations come mostly from intersection theory. One key point is that we can often “move" the modules in to favorable position to show vanishing of $\theta^R$. Since moving in the Grothendieck group is much harder than in the Chow group, we need to make use of the Riemann-Roch map between the two groups.

We first need to review some facts about Chow groups. Let $X$ be a Noetherian scheme. Let $Z_iX$ be the free abelian group on the $i$-dimensional subvarieties (integral, closed subschemes) of $X$. For any $i+1$-dimensional subscheme $W$ of $X$, and a rational function $f$ on $W$, we can define an element of $Z_iX$ as follows: $$[{\textup{div}}(f,W)] = \sum_{V} {\textup{ord}}_V(f)[V]$$ summing over all codimension one subvarieties $V$ of $W$. Then the $i$-Chow group $\operatorname{CH}_i(X)$ is defined as the quotient of $Z_iX$ by the subgroup generated by all elements of the form $[{\textup{div}}(f,W)]$. Let $\operatorname{CH}_*(X) = \oplus \operatorname{CH}_i(X)$ and $\operatorname{CH}_*(X)_{\mathbb{Q}} = \operatorname{CH}_*(X){\otimes}_{\mathbb{Z}}{\mathbb{Q}}$. An $i$-cycle (resp. cycle class) is an element in $Z_i(X)$ (resp. $\operatorname{CH}_i(X)$) (by a slight abuse of notation, we also talk about a cycle as an element of $Z_i(X)_{\mathbb{Q}}$). When $X= \operatorname{Spec}(R)$, where $R$ is a ring, we simply write $\operatorname{CH}_*(R)$. We write $\operatorname{CH}^i(X)$ for $\operatorname{CH}_{d-i}(X)$, here $d =\dim X$.

If $R$ is local and is a homomorphic image of a regular local ring $T$ we have the important notion of [*[Todd class]{}*]{}. For any $R$-module $M$, let $F_*$ be the minimal free resolution of $M$ over $T$. The [*[Todd class]{}*]{} of $M$ is defined as: $$\tau_{R/T}(L) := \operatorname{ch}(F_*)([M])$$ Here $\operatorname{ch}()$ denotes the local Chern character. For much more details about the definition and properties of the Todd class, we refer to \[Fu\] or \[Ro\]. The Todd class has been very useful to prove such results as Serre’s Vanishing Conjecture and the New Intersection Theorem (\[Ro\]). The Todd class gives an isomorphism of $\mathbb{Q}$-vector spaces: $$\tau_{R/T} : G(R)_{\mathbb{Q}}\to \operatorname{CH}_*(R)_{\mathbb{Q}}$$ When $R,T$ are clear we will simply write $\tau$ for $\tau_{R/T}$. Recall that the Todd class satisfies the top terms property: $$\tau(M) = \sum_{\dim R/p =\dim M} l(M_p)[R/p] + \text{terms of lower dimension}$$

We collect below a number of facts that will be used frequently:

\[factschow\] Let $R$ be a local ring. Let $M$ be an $R$-module and $d=\dim R$.

1.  If $d>0$ and $l(M)<\infty$ then $[M]=0$ in $G(R)_\mathbb{Q}$.

2.  If $R$ is regular, $\operatorname{CH}^i(R)_\mathbb{Q} = 0$ for $0 < i \leq d$ and $\operatorname{CH}^0(R)_\mathbb{Q} = \mathbb{Q}$.

3.  Assume that $R$ is a homomorphic image of a regular local ring.For any $P\in \operatorname{Spec}R$ we have $\tau^{-1}([R/P]) = [R/P] +  \text{terms of lower dimension} $.

<!-- -->

1.  It is easy to see that $[M] = l(M)[R/m]$ in $G(R)$. But take any $P\in \operatorname{Spec}R$ such that $\dim R/P =1$ and take $x \in m-P$, from the short exact sequence $$0 \to R/P \to R/P \to R/(P+(x)) \to 0$$ we then have $[R/(P+(x))] =0$ in $G(R)$. It follows that $[R/m]=0$ in $G(R)_\mathbb{Q}$, which gives what we want.

2.  Since $R$ is regular, $G(R) = \mathbb{Z}[R]$ since any module has a finite resolution by free modules. The claim follows because $\tau([R]) = [R]$.

3.  This follows immediately from the top term property.

Throughout the rest of this section we will assume that $R$ is a local hypersurface with isolated singularity. It is not hard to see that in all of our results we can safely pass to the completion of $R$ and the modules involved, so this is not a significant assumption.

\[dim1\] Assume $\dim R = 1$. Then $\theta^R(M,N)$ = 0 for all $N$ if and only if $M$ has constant rank,or equivalently, $[M]=0$ in $\overline{G}(R)$.

Let $p_1,p_2,...,p_n$ be the minimal primes of $R$. Then $G(R)_{\mathbb{Q}}$ has a basis consists of $[R/p_1],...,[R/p_n]$ by \[factschow\]. In particular, since $R$ has dimension 1 and is reduced, $[R] =
\sum_{1}^{n}[R/p_i]$. Let $\alpha_{ij} = \theta^R(R/p_i,R/p_j)$. For $i \neq j$, $p_i + p_j$ is $m$-primary, and it is not hard (for example, using \[ex2\]) to see that $\alpha_{ij} = l(R/(p_i+p_j)) >0$. Since $\theta^R(R,R/p_i) = 0$, we must have $\alpha_{ii} =
-\sum_{j\neq i} \alpha_{ij}$. Now, for a module $M$, let $[M] =
\sum a_i[R/p_i]$, here $a_i$ is the rank of $M_{p_i}$. If $a_1 =
a_2 = ... =a_n$, then $[M]=a_1[R]$, so $\theta^R(M,N) =
a_1\theta^R(R,N)= 0$. For the other direction, without loss of generality we may assume that $a_1$ is the largest among the $a_i$’s. Then since $0 = \theta^R(M,R/p_1) = \sum_{i=2}^{n}
\alpha_{1i}(a_i-a_1)$, we must have $a_i=a_1$ for all $i$.

Assume $\dim R =d > 1$ and $M,N$ are $R$-modules. Then $\theta^R(M,N)=0$ if $ \dim M \leq 1$.

Without affecting relevant issues we may complete and assume $R$ is a homomorphic image of a regular local ring. Since any module has a filtration by prime cyclic modules, we may assume that $M = R/P$ and $N=R/Q$ for some $P,Q \in \operatorname{Spec}R$. If $\dim R/P = 0$, so $P=m$, then $[R/P] = 0$ in $G(R)_\mathbb{Q}$, and $\theta$ vanishes. Also, we may assume $Q \neq 0$. If $Q$ is not contained in $P$, then $l(R/(P+Q)) <\infty$ because $\dim R/P
=1$, and since $\dim R/P +\dim R/Q \leq \dim R$ we have $\theta(R/P,R/Q) = 0$ by \[decent\]. So now we only need to consider the case $0 \neq Q \subset P$. We claim that there is cycle $\alpha = \sum l_i[R/Q_i] \in \operatorname{CH}^*(R)_\mathbb{Q}$ such that $\alpha = [R/Q]$ and $Q_i \nsubseteq P$. Consider the element $[R_P/Q] \in \operatorname{CH}^*(R_P)_\mathbb{Q}$. Since $R_P$ is regular, $[R_P/Q] = 0$. Therefore, formally, we have a collection of primes $q_i$ and elements $f_i$ and integers $n, n_i$ such that $n[R_P/Q]
= \sum {\textup{div}}(R_P/q_i,f_i)$. Now in $\operatorname{CH}^*(R)_\mathbb{Q}$ we will have $\sum {\textup{div}}(R/q_i,f_i) = n[R/Q] + \sum n_i[R/Q_i]$, with $Q_i
\nsubseteq P$, which proves our claim. The fact that $[R/Q] =
\sum_i l_i[R/Q_i]$ in $\operatorname{CH}^*(R)_\mathbb{Q}$ means that in $G(R)_\mathbb{Q}$, $[R/Q]=\sum_i l_i[R/Q_i]+ \text{terms of lower
dimension}$ (by part (3) of the proposition \[factschow\]). By the argument at the beginning of the proof, $\theta^R(R/P,R/Q_i)=0$ and we may conclude our proof by induction on $\dim R/Q$.

The above argument is true whenever $R_P$ is regular. This argument could be thought of as an algebraic “moving lemma" for $\operatorname{Spec}(R)$.

When the $R$ contains a field, we can of course apply the real moving lemma.

\[moving\] Suppose that $(R,m,k)$ contains a field. Then $\theta^R(R/P,R/Q)=0$ if $\dim R/P + \dim R/Q \leq \dim R$.

We first need to make some reductions. Assume we have a counterexample on $R$. We can first make a faithfully flat extension to replace $k$ by an algebraically closed field and then complete to get to the case of $R = k[[x_0,...x_d]]/(f)$, and $k$ is algebraically closed. The condition of isolated singularity is preserved by faithfully flat extension (cf Lemma 2.7, [@Wi]). Then by a theorem of Artin (see [@CS], theorem 1.6), $R =
\hat{S}$, where $S$ is local hypersurface with isolated singularity, and is essentially of finite type over $k$. But we can descend our example to $S^h$, the Henselization of $S$, by a standard argument (see [@Ho3] or [@Du1]). The only issue is how to descend the resolutions of the modules, which may be infinite. However, note that since the resolutions of our modules are eventually periodic, we only need to keep track of a finite part. Then since $S^h = \underrightarrow{\lim} S_n$, where each $S_n$ is an étale neighborhood of $S$, we have a counterexample in some $S_n$. Thus we may assume $R$ is essentially of finite type over an algebraically closed field $k$. Let’s say $R = A_m$, where $A$ is a finite $k$-algebra and $m$ is a maximal ideal in $A$. Since $R$ has isolated singularity, we have $\operatorname{Sing}(A)$ is a disjoint union of $\{m\}$ and some closed subset $Y \subset \operatorname{Spec}A$.

Let $X = \operatorname{Spec}(A) -\{m\}-Y$. Then $X$ is a smooth quasi-projective variety, so by the Moving Lemma (see [@Fu],11.4) one can find a cycle $\alpha = \sum_i n_i V(Q_i)$ in $\operatorname{CH}^*(X)$ such that for each $i$, $\dim (V(Q_i)\cup V(P))\leq 0$, that is the intersection consists only of points in $X$. When we restrict all the cycles and divisors to $\operatorname{Spec}(R)$ we will have $[R/Q] = \sum_j [R/Q_j]$ in $\operatorname{CH}^*(R)$ and for all $j$, $V(P)\cup V(Q_i) \subset \{m\}$. In $G(R)_{\mathbb{Q}}$ this means $[R/Q] = \sum_j [R/Q_j] +
\text{terms of lower dimension}$. Since $l(R/P+Q_j) <\infty$ and $\dim R/P + \dim R/Q_j \leq \dim R$ we have $\theta^R(R/P,R/Q_j)=0$ by \[decent\] and an induction on $\dim
R/Q$ finishes the proof.

\[dim2,3\] If $\dim R=2$ or $\dim R=3 $ and $\operatorname{CH}^1(R)_{\mathbb{Q}} = 0$ or $\dim R =4$ and $R$ contains a field, then $\theta^R(M,N)=0$ for all pairs $(M,N)$.

It suffices to assume that $M,N$ are cyclic prime modules, let’s say $M=R/P,N=R/Q$. Then by the previous theorems we only need to worry if both of them have dimension at least $2$. In the first case, they must be $R$ (note that since $R$ is normal and local, it is a domain), thus $\theta^R$ certainly vanishes. In the second case, assume that $\dim R/P=2$ (otherwise $R/P$ would be $R$). We can then complete $R$ without loss of generality. Then part (3) of \[factschow\] shows that in $G(R)$, $[R/P]$ is equal to a formal sum of cyclic primes of dimension $\leq 1$, forcing $\theta^R(R/P,R/Q)$ to be $0$. Finally, if $\dim R=4$ and $R$ contains a field we can apply \[moving\] and assume $\dim
R/P + \dim R/Q \geq 5$. Then one of the primes, say $P$ is height $1$ (if the minimal height is $0$ the assertion is trivial). We will be done if we can show that $\operatorname{CH}^1(R)=0$. But by the Grothendieck-Lefschetz theorem, the Picard group of $X = \operatorname{Spec}(R)-{m}$ is $0$. Since $X$ is regular, the Picard group of $X$ is the same as the $\operatorname{CH}^1(X)=
\operatorname{CH}^1(R)$.

Let $A=\mathbb{R}[x,y,z,w]/(F)$ where $F$ is a non-degenerate quadratic form of signature $2$ (for example, if $F=x^2+y^2+z^2+2w^2$). Let $m=(x,y,z,w)$ and $R=A_m$. By 10.3 and 11.7 of [@Fo], $Cl(R)=0$. So $\theta$ vanishes on any pair of modules over $R$.

Our next result is an algebraic Bertini type theorem for the vanishing of $\theta^R$. It is most useful when $M=N$ (so in a sense when moving them apart is the hardest).

Assume that $\dim R \geq 2$. Suppose $M,N$ are $R$-modules such that there is an element $x \in {\textup{Ann}}{M} \cap {\textup{Ann}}(N)$ such that $R/(x)$ is still a hypersurface with isolated singularity. Then $\theta^R(M,N)=0$.

Take $L = \operatorname{syz}_R^{2d}(N)$. Then $\theta^R (M,N) = \theta^R(M,L)$. Also, $L$ is maximal Cohen-Macaulay: in particular, $x$ is a nonzerodivisor on $L$. Thus $\operatorname{Tor}_i^R(M,L) = \operatorname{Tor}_i^{R/x}(M,L/(x))$, so it is enough to prove that $\theta^{R/(x)}(M,L/(x)) = 0$ (the assumption that $R/(x)$ is still a hypersurface with isolated singularity ensures that $\theta^{R/(x)}$ is well-defined). We define a map $\alpha : G(R) \to G(R/(x))$ as follows: $$\alpha([M]) = [\operatorname{Tor}_0^R(M,R/(x))] - [\operatorname{Tor}_1^R(M,R/(x))]$$ We need to show that $\alpha$ is well-defined. The only thing that needs to be checked is that if $0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of $R$-modules, then $\alpha([M_2]) =
\alpha([M_1]) + \alpha([M_3])$. But this follows from tensoring the exact sequence with $R/(x)$, and note that $\operatorname{Tor}_i^R(M,R/(x))
=0$ for all $R$-modules $M$ and $i>1$.

It is easy to see that $\alpha([N]) = [0]$ and $\alpha([L]) = [L/(x)]$ (since $x$ kills $N$ and is $L$-regular, respectively). But since $L = \operatorname{syz}^{2d}(N)$ we have $[L] = [N] + n[R]$ for some integer $n$. Applying $\alpha$ we get: $ [L/(x)] = n[R/(x)]$ in $G(R/(x))$. But then $\theta^{R/(x)}(M,L/xL) = 0$, which is what we want.

Now we will consider a special case, when $R$ is a local ring of an affine cone of a projective variety. That is, $R=A_{\mathfrak
m}$, where $A$ is a graded hypersurface over a field $k$ whose homogeneous maximal ideal is $\mathfrak m$. The following result by Kurano is very helpful (cf. [@Ku1],Thm 1.3).

(Kurano)\[kurano\] Let $A,R$ be as above. Let $X = \operatorname{Proj}(A)$. Assume that $R$ has isolated singularity.Then $X$ is smooth and $\operatorname{CH}^*(X)$ becomes a (graded) commutative ring with the intersection product. Let $h
\in \operatorname{CH}^1(X)$ represent the hyperplane section (alternatively, the first Chern class of the invertible sheaf ${\mathcal O}_X(1)$). Then there is a graded isomorphism of ${\mathbb{Q}}$-vector spaces from $\operatorname{CH}^*(X)/h\operatorname{CH}^*(X)$ to $\operatorname{CH}^*(R)$.

Let $X,R$ as above and $i\geq 0$ an integer. If $\operatorname{CH}^{i+1}(X)_{\mathbb{Q}} = \mathbb{Q}$ then $\operatorname{CH}^{i+1}(R)_{\mathbb{Q}} = 0$.

We only need to observe that in $\operatorname{CH}^*(X)_{\mathbb{Q}}$, multiplication by $h$ is a nonzero map from $\operatorname{CH}^i(X)_{\mathbb{Q}}$ to $\operatorname{CH}^{i+1}(X)_{\mathbb{Q}}$ (just take $h^{i+1}$). Hence if $\operatorname{CH}^{i+1}(X)_\mathbb{Q} = \mathbb{Q}$ then $ \operatorname{CH}^{i+1}(X)_\mathbb{Q} =h\operatorname{CH}^{i}(X)_\mathbb{Q}$, so $\operatorname{CH}^{i+1}(R)_\mathbb{Q}=0$ by Kurano’s result.

This allow us to exploit many results in the literature about the Chow groups of projective varieties. For example, we have the following:

([@ELV],2.3) Let $X \subset \mathbb{P}^d_k$ be an irreducible hypersurface of degree $s$. If: $${{i+s} \choose {i+1}} \leq d-i$$ Then $\operatorname{CH}^{d-1-i}(X)_{\mathbb{Q}} = \mathbb{Q}$.

Let $X$ be a smooth hypersurface of degree $s$ in $\mathbb{P}^d_k$. Let $R$ be the local ring at the homogenous maximal ideal of the affine cone of $X$. Let $n$ be the biggest integer such that: $${{n+s} \choose {n+1}} \leq d-n$$ Then $\theta^R(M,N) = 0$ if $\dim M \leq n+1$.

By the two previous results we have $\operatorname{CH}^{d-1-i}(R)_{\mathbb{Q}} = 0$ for $i\leq n$, in other words, $\operatorname{CH}_{i}(R)_{\mathbb{Q}} = 0$ for $i\leq n+1$. Thus $\tau([M]) = 0$, hence $[M] = 0$ in $G(R)_{\mathbb{Q}}$.

When $s=2$, we have $n+1 = \lfloor d/2 \rfloor $. This gives a strong version of \[moving\].

Next we want to discuss a conjecture, attributed to Hartshorne (see [@Ha1], page 142), which could be relevant to our interest:

\[HarConj\](R.Hartshorne) Let $X$ be a smooth projective, complete intersection variety in $\mathbb{P}^n_k$. Then $\operatorname{CH}^i(X)_{\mathbb{Q}}= \mathbb{Q}$ for $i<{\dim X}/2$.

It is interesting to observe that Hartshorne’s conjecture, together with \[moving\], show that if $\dim R$ is even, when $R$ is the local ring at the origin of the affine cone of $X$, then $\theta^R$ always vanishes.

(of Hartshorne’s conjecture) Let $X$ be a smooth hypersurface in $\mathbb{P}^d_k$. Assume that $d$ is even. Let $R$ be the local ring at the origin of the affine cone of $X$. Then $\theta^R$ always vanishes.

Let $d=2n$. We have $\dim X = 2n-1$, so by Hartshorne’ conjecture with $Y=\mathbb{P}^d_k$ we have $\operatorname{CH}^i(X)_{\mathbb{Q}} ={\mathbb{Q}}$ for $i \leq n-1$. Thus $\operatorname{CH}^i(R)_{\mathbb{Q}} = 0$ for $i\leq n-1$, in other words, $\operatorname{CH}_i(R)_{\mathbb{Q}} = 0$ for $i \geq n+1$ . So in the Grothendieck group $G(R)_{\mathbb{Q}}$, any module can be represented as a sum of cyclic prime modules of dimension $\leq n$. But since $\dim R = 2n$, for any such pair of modules $(R/P,R/Q)$ we must have $\theta^R(R/P,R/Q)=0$ by \[moving\].

In view of this and our knowledge of dimensions $2$ and $4$, we feel it is reasonable to make:

\[DaoConj\] Let $R$ be a hypersurface with isolated singularity. Assume that $\dim R$ is even and $R$ contains a field. Then $\theta^R$ always vanishes.

Finally we present some classes of hypersurfaces on which the above conjecture can be readily verified. First we observe that the values of $\theta^R$ only depends on its values on pairs of maximal Cohen-Macaulay (MCM) modules (as one can replace the modules by their high syzygies). The group of isomorphism classes of indecomposable MCM modules has been a subject of intense study for a while, and in the case when it has been computed, we may exploit those results for our purpose. We note that the theta function is closely related to the notion of “Herbrand difference" defined using stable cohomology by Buchweitz in [@Bu]. In fact, it is not hard to see that they agree up to sign (We would like to thank Ragnar-Olaf Buchweitz for explaining to us the connection).

Finally, we want to discuss the case of simple singularities. We say that a complete local ring $R$ is a simple singularity if it is isomorphic to $T/(f)$, where $T=k[[x_0,x_1,...,x_d]]$, $k$ is algebraically closed of charateristic $0$, and $f$ has one of the following forms :\
$ (A_n)\ \ \ \ x_0^2+x_1^{n+1} + x_2^2+...+x_d^2 \ \ \ \ \ \ (n\geq 1)$\
$ (D_n)\ \ \ \ x_0^2x_1 + x_1^{n-1} +x_2^2+...+x_d^2 \ \ \ \ (n\geq 4)$\
$ (E_6)\ \ \ \ x_0^3 + x_1^4 +x_2^2+...+x_d^2 \ \ \ \ $\
$ (E_7)\ \ \ \ x_0^3 + x_0x_1^3 +x_2^2+...+x_d^2 \ \ \ \ $\
$ (E_8)\ \ \ \ x_0^3 + x_1^5 +x_2^2+...+x_d^2 \ \ \ \ $\
In the hypersurface case, simple singularity is the same as finite representation type, that is, the group of isomorphism classes of indecomposable MCM modules is finite. In this case, the Grothendieck group of MCM modules can be computed completely (see [@Yo], 13.10). One striking feature is that in even dimensions, all the Grothendieck groups are torsion after we kill the class of $[R]$. Thus we have the following result, which confirm Conjecture \[DaoConj\]:

\[MainConj\] Let $R$ be a hypersurface with isolated, simple singularity of even dimension. Then $\theta^R$ always vanishes.

Rigidity over hypersurfaces {#4}
===========================

By the virtue of theorem \[rg1\] and the results in the previous section, we have a lot of results about rigidity of modules when the hypersurface has an isolated singularity. We collect them in a theorem:

\[rigidiso\] Let $R$ be an admissible hypersurface with isolated singularity and $M,N$ be $R$-modules.

1.  If $[M] = 0$ in $\overline G(R)_{\mathbb{Q}}$, then $M$ is rigid.

2.  If $\dim R=1$ and $M$ has constant rank then $M$ is rigid.

3.  If $\dim R=2$ or $\dim R=3 $ and $\operatorname{CH}^1(R)_{\mathbb{Q}} = 0$ or $\dim R =4$ and $R$ contains a field, then any module is rigid.

4.  If $\dim M \leq 1$ then $M$ is rigid.

5.  If $R$ contains a field and $\dim M +\dim N \leq \dim R$, then $(M,N)$ are rigid.

6.  If there is an element $x \in {\textup{Ann}}{M} \cap {\textup{Ann}}(N)$ such that $R/(x)$ is still a hypersurface with isolated singularity. Then $(M,N)$ are rigid.

On general hypersurfaces, however, we need to be more careful about using the function $\theta^R$. Typically, we need some extra conditions to show that $\theta$ is defined for all the modules in the short exact sequences involved. We will give plenty of examples to show that these conditions are unavoidable. One very effective method is to use induction on dimension to show that the $\operatorname{Tor}$ modules become $0$ on the punctured spectrum of $R$, which means that they have finite lengths and we can use $\theta^R$.

We will now state two immediate corollaries of \[rg1\]. The first appeared implicitly in the work of Lichtenbaum, (\[Li\]) :

\[lichten\] Let $R$ be an admissible hypersurface and $M,N$ be finitely generated $R$-modules. If $M$ or $N$ has finite projective dimension, then $(M,N)$ are rigid.

\[rg2\] Let $R$ be an admissible hypersurface and $M,N$ be finitely generated $R$-modules. Assume that $\operatorname{pd}_{R_p}M_p <\infty$ for all $p\in Y(R)$ (the punctured spectrum of $R$) and $[N]=0$ in $\overline{G}(R)_{\mathbb{Q}}$. Then $(M,N)$ is rigid.

The first assumption ensures that $f_R(M,N)<\infty$ for all $N$, hence $\theta^R(M,N)$ can be defined for all $N$. Then the second assumption forces $\theta^R(M,N) = 0$.

Another immediate corollary of our result is the first “rigidity" theorem in a paper of Huneke and Wiegand (\[HW1\].

\[HWrg\] (\[HW1\],2.4). Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Assume:\
(1) $M{\otimes}_RN$ has finite length.\
(2) $\dim(M) + \dim(N) \leq \dim(R)$.\
Then $(M,N)$ are rigid.

Suppose $R=T/(f)$ where $T$ is regular local. In this case $\theta^R(M,N) =\chi^T(M,N)$, so by Serre’s vanishing theorem, it must be $0$.

The next result introduces a class of rigid modules not necessarily having finite projective dimension. To state it, recall the definition: $IPD(M):= \{ p\in \operatorname{Spec}(R) |\ \operatorname{pd}_{R_p}M_p =\infty \}$ (section 1)\

\[rig1.1\] Let $R$ be an admissible hypersurface, and $M$ be an $R$-module such that $[M] =0$ in $\overline G(R)_{\mathbb{Q}}$. Assume that $IPD(M)$ is either $\emptyset$ or is equal to $\operatorname{Sing}(R)$. Then $M$ is rigid.

If $IPD(M)=\emptyset$ then $\operatorname{pd}_RM<\infty$, so there is nothing to prove. Assume that $IPD(M)= \operatorname{Sing}(R) \neq 0$. We again use induction on $d = \dim R$. If $d=0$ the condition that $[M]=[0]$ in $\overline G(R)_{\mathbb{Q}}$ implies that $\theta^R(M,N)$ is defined and equal to $0$ for any $R$-module $N$. Suppose $d>0$ and $\operatorname{Tor}_i^R(M,N)=0$ for some $i$. We localize at any prime $p\in Y(R)$. Both conditions on $M$ localize, so by the induction hypothesis, $\operatorname{Tor}_j^{R_p}(M_p,N_p)=0$ for $j \geq i$. This forces either $\operatorname{pd}_{R_p}M_p$ or $\operatorname{pd}_{R_p}N_p$ to be finite (see theorem 1.9,\[HW2\]). But since $I(M) = \operatorname{Sing}(R)$, $N$ must has finite projective dimension on $Y(R) \cap \operatorname{Sing}(R)$. So $N$ has finite projective dimension on $Y(R)$, hence $\theta(M,N)=0$, finishing the proof.

The following will be useful for our application to torsion of tensor products.

\[vanishinglem\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules. Assume that:

1.  $\operatorname{Tor}_1^R(M,N)=0$

2.  $\operatorname{depth}(N)\geq 1$ and $\operatorname{depth}(M{\otimes}_RN)\geq 1$.

3.  $f_R(M,N)<\infty$.

Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.

The depth assumptions ensure that we can choose $t$ a nonzerodivisor for both $N$ and $M{\otimes}_RN$. Let $\overline{N} = N/tN$. Tensoring the short exact sequence : $$\xymatrix {0 \ar[r] &N \ar[r]^{t} &N \ar[r] &\overline{N} \ar[r] &0}$$ with $M$ and using (1) we get : $$\xymatrix{0 \ar[r] &\operatorname{Tor}_1^R(M,\overline{N}) \ar[r] &M{\otimes}_RN \ar[r]^{t} &M{\otimes}_RN \ar[r] &M{\otimes}_R \overline{N} \ar[r] &0}$$ which shows that $\operatorname{Tor}_1^R(M,\overline{N}) = 0$. But condition (3) is satisfied for both of the pairs $(M,N)$ and $(M,\overline{N})$ and so : $$\theta^R(M,\overline{N}) = \theta^R(M,N) - \theta^R(M,N)= 0$$ The conclusion then follows from theorem \[rg1\] and Nakayama’s lemma.

\[dim1rg\] Suppose $\dim R=1$ and $M$ is an $R$ module of constant rank. Then $\theta^R(M,N)$ is defined and equal $0$ for all $N$. Consequently, $M$ is rigid.

By ([@HW1],1.3), we can find a homeormorphism : $\beta: M \to
F$ such that $F$ is free and the kernel $K$ and cokernel $C$ are torsion module. Since $\dim R=1$, both $K,C$ have finite length. Let $D = \operatorname{Im}\beta$. From the short exact sequence: $$0 \to D \to F \to C \to 0$$ we know that for any module $N$, $f_R(M,N) <\infty$ and $\theta^R(D,N) = \theta^R(F,N)-\theta^R(C,N)=0$ ($\theta^R(C,N)=0$ by \[HWrg\]). Now we look at the short exact sequence: $$0 \to K \to M \to D \to 0$$ and it follows that $f_R(M,N) <\infty$ and $\theta^R(M,N) = \theta^R(K,N)+ \theta^R(D,N) = 0$.

The next result shows some connection between rigidity, decency and a property of modules first studied by Auslander ([@Au]).

\[rigidandproper\] Let $R$ be an admissible hypersurface. For a Cohen-Macaulay $R$-module $M$, the following are equivalent:

1.  $(M,N)$ is rigid for all $N$ such that $l(M{\otimes}_RN)<\infty$.

2.  Every $M$-sequence is an $R$-sequence.

3.  $M$ is decent ($\dim M +\dim N \leq \dim R$ for all $N$ such that $l(M{\otimes}_RN)<\infty$).

4.  $\theta^R(M,N)=0$ for any $N$ such that $l(M{\otimes}_RN)<\infty$.

Assume (1). Then we can prove (2) by adapting the argument in Auslander paper ([@Au],4.1) (which assumed that $(M,N)$ is rigid for all $N$ but did not need $M$ to be Cohen-Macaulay). We give a sketch here. Let $X$ be the free resolution of $M$. Let $\bold x$ be a full $M$-sequence (so its length is $\dim M$ and $l(M/(\bold {x})) <\infty$). Let $Y$ be the Koszul complex on $\bold x$. Then the total complex $X{\otimes}_RY$ is acyclic. Filtering that complex by $F_p(X{\otimes}Y) = \sum_{p\geq r}\sum
_qX_r{\otimes}_RY_q$ ,we obtain a spectral sequence with $E^2_{p,q}
= H_p(X{\otimes}_RH_q(Y))$. By assumption $H_n(X{\otimes}_RY)=0$, so $E^{\infty}_{p,q}=0$ for $p,q>0$. We also have $E^2_{p,q}=0$ for $p,q<0$. Hence $E^2_{1,0}=E^i_{1,0}$ for $i>1$. But $E^{\infty}_{1,0}=0$ which implies that $H_1(X{\otimes}_RH_0(Y)) =
E^2_{1,0}=0$. Since $M{\otimes}_RH_0(Y) = M/(\bold x)$ has finite length, we must have $0 = H_p(X{\otimes}_RH_0(Y)) = E^2_{p,0}$ for all $p \geq 1$. By induction we will have: $$0 = E^2_{p,q} =  H_p(X{\otimes}_RH_q(Y))$$ for all $p\geq 1$ and $q\geq 0$ (note that since $\bold{x}R$ kills all the modules $H_q(Y)$, $M{\otimes}_RH_q(Y)$ has finite length so we can apply (1)). But since $E^{\infty}_{p,q}=0$ for $p,q>0$, we have: $$0 = E^2_{0,q} = H_0(X{\otimes}_RH_q(Y)) = M{\otimes}_RH_q(Y)$$ for each $q>0$. This forces $H_q(Y)=0$ for $q>0$, hence $\bold x$ is an $R$-sequence. If $\bold x$ is not a full $M$-sequence, we can always add more elements and reach the same conclusion.\
Assume (2). Let $N$ be an $R$ module such that $l(M{\otimes}_RN)<\infty$. Then we can find a full system of parameters $\bold x$ on $M$ such that $\bold x \subset {\textup{Ann}}(N)$. As $M$ is Cohen-Macaulay, $\bold x$ is also a full $M$-sequence. By assumption, $\bold x$ is an $R$-sequence. Thus: $$\dim N \leq \dim R/(\bold x)= \dim R -\dim M$$ Finally, $(3)\Rightarrow (4) \Rightarrow (1)$ is just the proof of \[HWrg\].

This result gives necessary conditions for rigidity that are easier to check then rigidity itself. The conditions (2) and (3) are quite familiar. They have played a vital role in a group of theorems and conjectures known as the “homological conjectures". Specifically we have (see [@Ho4], [@PS], [@Ro]):

\[FP1\] Let $R$ be a local ring and $M$ an $R$-module of finite projective dimension. Then every $M$-sequence is an $R$-sequence.

\[FP2\] Let $R$ be a local ring and $M$ an $R$-module of finite projective dimension. Then $\dim M +\dim N \leq \dim R$ for any $N$ such that $l(M{\otimes}_RN)<\infty$.

Theorem \[FP1\] is a consequence of the “intersection theorem", proved by Peskine and Szpiro for $R$ containing a field and by Roberts for the general case. Conjecture \[FP2\] is still wide open even for hypersurfaces in ramified regular local rings.

Applications {#5}
============

In this section we apply our results on a number of topics which involve decency or rigidity. Some of the results below are known, we just give a different proof. Some are new, to the best of our knowledge.\
**Rigidity of $^eR$.**\
Suppose $R$ is a hypersurface of characteristic $p>0$. Let $F:R\to
R$ be the Frobenius map. We denote by $F^e: R\to R$ the $e$-th iteration of $F$. Then $^eR$ is the $R$-module $R$ whose module structure is given by $F^e$. The following is a special case of Theorem 1 in [@AM] (is was proved for complete intersections):

\[AM\] $^eR$ is rigid. If $R$ is singular, and $M$ is an $R$-module such that $\operatorname{Tor}_i^R(M,^eR)=0$ for some $i>0$, then $pd_RM<\infty$.

To prove the first statement we will apply \[rig1.1\]. First, by making faithfully flat extensions we may assume that $R$ is complete and has a perfect residue field. Then we just need to make sure that the conditions in \[rig1.1\] are satisfied. Since $\operatorname{pd}_R{^eR}<\infty$ if and only if $R$ is regular, and $F$ commutes with localization, we may conclude that $IPD(^eR)=\operatorname{Sing}(R)$. And it is known that $[^eR] = p^{e\dim R}[R]$ in $G(R)$ (this is actually non-trivial, see [@Ku1], the fact that $R$ is complete with perfect residue field is needed here). The second statement follows because if $\operatorname{Tor}_i^R(M,^eR)=0$ for some $i>0$, then by rigidity all the higher $\operatorname{Tor}$s vanish. Thus one of the two modules $M,^eR$ has to have finite projective dimension, and since $R$ is singular, it must be $M$.

**Torsion on tensor products.**\
In this part we shall apply our rigidity results to show that tensor products rarely have good depths:

\[vanishingiso\] Let $R$ be an admissible hypersurface with dimension $d\geq2$. Let $M,N$ be $R$-modules. Assume that:\
(1) $R$ has isolated singularity.\
(2) $M{\otimes}_RN$ is torsion-free.\
(3) $\operatorname{depth}_R M{\otimes}_RN \geq 2$.\
Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.

Condition (1) makes sure that $\theta^R(M,N)$ is defined for any pair of modules $(M,N)$. We may assume that $M,N$ are torsion-free (This argument was basically in the proof of (2.4,\[HW1\]), we repeat it here for the reader’s convenience). Let $t(M)$ be the torsion part of $M$ and $\overline{M}=M/t(M)$. Tensoring the exact sequence : $$0 \to t(M) \to M \to \overline{M} \to 0$$ with $N$, we get: $$\xymatrix{ \operatorname{Tor}_1^R(\overline{M},N) \ar[r] &t(M){\otimes}_RN \ar[r]^{\alpha} &M{\otimes}_RN \ar[r]^{\beta} &\overline{M}{\otimes}_RN \ar[r] &0}$$ Since $t(M){\otimes}_RN$ is torsion and $M{\otimes}_RN$ is torsion-free, $\alpha$ is the $0$ map and $\beta$ is an isomorphism. So $\overline{M}{\otimes}_RN$ satisfies all the condition (2) and (3). By the torsion-free case,$\operatorname{Tor}_1^R(\overline{M},N)=0$, so $t(M){\otimes}_RN=0$ and that can only mean $t(M)=0$. By symmetry $N$ is also torsion-free. Now there is an exact sequence: $$0 \to M \to R^{\lambda} \to M_1 \to 0$$ Here $\lambda = \lambda(M^*)$. This exact sequence is called the *pushforward* of $M$ (see [@HJW]). By tensoring the pushforward exact sequence of $M$ with $N$ , we get: $$0 \to \operatorname{Tor}_1^R(M_1,N) \to M{\otimes}_RN \to N^{\lambda} \to M_1{\otimes}_RN \to 0$$ By condition (1) $N$ is generically free, so $\operatorname{Tor}_1^R(M_1,N)$ is torsion, and it must be $0$ since $M{\otimes}_RN$ is torsion-free. Since $\operatorname{depth}_R M{\otimes}_RN \geq 2$, $\operatorname{depth}_RN\geq1$ (since $N$ is torsion-free and $d\geq2$), we must have $\operatorname{depth}_R M_1{\otimes}_RN \geq 1$. Now the theorem follows from lemma \[vanishinglem\].

In the dimension $2$ case, we can do a little bit better:

\[dim2normal\] Let $R$ be an admissible hypersurface of dimension $2$. Assume further that $R$ is normal. Let $M,N$ be $R$ modules such that $M{\otimes}_R N$ is torsion-free. Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq 1$.

We may assume $M$ is torsion-free. Now, let $M_1$ be the pushforward of $M$. We have $\operatorname{Tor}_1^R(M_1,N)=0$. By the fact that $R$ is an isolated singularity of dimension 2, every module is rigid, so $\operatorname{Tor}_i^R(M_1,N)=0$ for $i>1$, which gives the desired conclusion.

It was asked in [@HW1] (4.1 and the discussion before 5.3), whether or not there are two non-free reflexive modules over a hypersurface of dimension $2$ such that their tensor product is torsion-free. In general, such pairs of modules exist. For example, let $R=k[[x,y,z]]/(xy)$ and $M=N=R/(x)$. But with the extra assumptions of the above, such modules can not exist. For by the conclusion, one of them must have finite projective dimension, and, being maximal Cohen-Macaulay, must be free.

To illustrate the efficiency of using $\theta^R$ for rigidity, we will give a short proof of one of the key results of [@HW1]:

\[HWmain\]\[[@HW1],2.7\] Let $R$ be an admissible hypersurface and $M,N$ be $R$-modules, at least one of which has constant rank. If $M{\otimes}_RN$ is reflexive, then $\operatorname{Tor}_i^R(M,N)=0$ for $i>0$.

We will use induction on $d = \dim R$. If $d=0$, the constant rank condition means one of the modules must be free, and the conclusion follows trivially. Now assume $d\geq 1$. By the induction hypotheses, $l(\operatorname{Tor}_i^R(M,N))<\infty$ for $i>0$. We can assume both $M,N$ are torsion free by standard arguments (see \[vanishingiso\]). In particular, they must have depth at least $1$. Let $M_1$ be the pushforward of $M$: $$0 \to M \to F \to M_1 \to 0$$ Then by the same reason as in proof of \[vanishingiso\], we have $\operatorname{Tor}_1^R(M_1,N)=0$. So we have: $$0 \to M_1 {\otimes}_RN \to F{\otimes}_RN \to M{\otimes}_RN \to 0$$ By the depth lemma, we get $\operatorname{depth}(M_1{\otimes}_RN)\geq 1$. Finally, since $l(\operatorname{Tor}_i^R(M,N))<\infty$ for $i>0$ we must have $f_R(M,N)<\infty$. Applying \[vanishinglem\] for $M_1$ and $N$, we get $\operatorname{Tor}_i^R(M_1,N)=0$ for $i>1$, which implies $\operatorname{Tor}_i^R(M,N)=0$ for $i>0$.

**Hypersurfaces in Projective spaces.**\
Our study so far has made use of some results from projective geometry. It is perhaps fair to try to give some thing back, so we will use our result to gain some information on projective hypersurfaces. We have:

\[projhyper\] Let $k$ be a field. Let $X \subset {\mathbf{P}}_k^n$ be a smooth hypersurface. Let $U,V$ be subvarieties of $X$ such that $\dim U +
\dim V \geq \dim X$. Assume that $[U] = h.[U']$ in $\operatorname{CH}^*(X)_{\mathbb{Q}}$, here $h$ is the hyperplane section. Then $U\cap V \ne \emptyset$.

Let $X = Proj(A)$ where $A = k[x_0,...,x_n]/(F)$. Let $R$ be the local ring at the origin of $A$. Suppose $P,Q \in \operatorname{Spec}(R)$ define $U,V$ respectively. Our assumption becomes : $R$ is a hypersurface with isolated singularity of dimension $n$, $\dim R/P +\dim
R/Q\geq n+1$, and $[R/P] = 0$ in $\operatorname{CH}^*(R)_{\mathbb{Q}}$ (by Kurano’s theorem \[kurano\]). We need to show that $R/P$, as a *module*, is decent. Now, by \[hochster\] we are done if we can show: there exist a module $M$ such that $M =0$ in $\overline
G(R)_{\mathbb{Q}}$, and $\operatorname{Supp}(M) = \operatorname{Supp}(R/P)$. We first pick $M
= R/P$. This may not guarantee that $M =0$ in $\overline
G(R)_{\mathbb{Q}}$, because by Riemann-Roch:

$$\tau(M) = [R/P] + \sum_{i}n_i[R/p_i]$$ Here the $p_i$s are in $\operatorname{Supp}(R/P)$, but have smaller dimensions. Our strategy will be to replace them one by one by elements of even smaller dimensions. Let’s look at $p_1$. Replacing $M$ by a multiple of $M$ if necessary, we may assume $n_1 \in {\mathbf{Z}}$. Next, we replace $M$ by $ M' = {p_1}^aM\oplus (R/p_1)^{\oplus{b}}$ for some $a,b \in {\mathbf{Z}}$. Then $$\tau(M') = \tau(M) - \tau(M/p_1^aM) + b \tau(R/p_1)$$ We now choose $a,b$ such that $b- l(M_{p_1}/{p_1}^aM_{p_1}) =
-n_1$. Then $p_1$ is replaced in the representation of $M$ by some elements of smaller dimension. Repeating this process, we will get to dimension $0$, which is $0$ in $\operatorname{CH}^*(R)_{\mathbb{Q}}$. So we get a module $M$ such that $\tau(M) = n[R/P] = 0$, which is what we need.

Miscellaneous results and examples {#lastsection}
==================================

In this section we will discuss our attempts to generalize the $\theta^R$ function and extend our class of rigid modules. We will end with a host of examples to illustrate our results.

An obvious drawback of our crucial rigidity theorem \[rg1\] is the requirement that all higher $\operatorname{Tor}$ modules have finite lengths. This prevents us from proving rigidity for a bigger class of modules (unless we impose extra conditions like isolated singularity on $R$). That raises a question: can we replace the length function in the definition of $\theta^R(M,N)$ by other functions with bigger domains? One possibility is taking the length at the minimal primes of the module. For any $R$-module $L$ , we define the class of $L$ in $Z_*(R)$ as : $$cl(L) := \sum_{p \in {\textup{Min}}(L)} l_{R_p}(L_p)[R/p]$$ and for a pair of modules $(M,N)$, let: $$\alpha^R(M,N):= cl(\operatorname{Tor}_{2e+2}^R(M,N)) - cl(\operatorname{Tor}_{2e+1}^R(M,N))$$ Here $e$ is any integer bigger than the dimension of $R$. Note that if the $\operatorname{Tor}$s have finite length then $\alpha^R(M,N)=\theta^R(M,N)[R/m_R]$. Then we have the following corollary of \[rg1\] :

Let $R$ be an admissible hypersurface, and $M,N$ be $R$-modules. Assume that $\alpha^R(M,N)=0$. Then $(M,N)$ is rigid.

We use induction on $d = \dim R$. If $d=0$ then $\alpha^R(M,N)$ coincides with $\theta^R(M,N)$. Suppose $d>0$ and $\operatorname{Tor}_i^R(M,N)=0$ for some $i>0$. Then by localizing at primes on the punctured spectrum and induction hypotheses $l(\operatorname{Tor}_j^R(M,N))<\infty$ for $j \geq i$. Then the vanishing of $\alpha$ again implies the vanishing of $\theta^R$, and we are done by \[rg1\].

Let $R$ be an admissible hypersurface, and $M$ be an $R$-modules. Assume that the minimal resolution of $M$ over $R$ is eventually periodic of period 1. Then $M$ is rigid.

Let $R$ be an admissible hypersurface, and $M,N$ be $R$-modules. Assume that there are integers $0<a<b$ such that $b-a$ is an odd integer and $\operatorname{Tor}_a^R(M,N)=\operatorname{Tor}_b^R(M,N)=0$. Then $\operatorname{Tor}_i^R(M,N)=0$ for $i\geq a$.

Let $M' = \operatorname{syz}_{a-1}M\oplus\operatorname{syz}_{b-1}M$. Since $b-a$ is odd and the resolution of $M$ is eventually periodic of period 2, it follows that the resolution of $M'$ is eventually periodic of period 1. Because $\operatorname{Tor}_i^R(M',N)=\operatorname{Tor}_{a+i-1}^R(M,N)\oplus\operatorname{Tor}_{b+i-1}^R(M,N)$, for $i\geq1$, the conclusion follows from the rigidity of $M'$.

The case $b=a+1$ is a well-known result by Murthy (\[Mu\]), and an asymptotic version (i.e when $a,b$ are big enough) was proved in (\[Jo2,3.1\]).

We could not get better results with $\alpha$ because it is not additive on short exact sequences. A much more refined version can be obtained replacing the function $cl$ in the definition of $\alpha$ by the Todd class $\tau$. We recall that $\tau$ is a map from $G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(R)_{\mathbb{Q}}$. It is additive, so if we let : $$\beta^R(M,N) = \tau(\operatorname{Tor}_{2e+2}^R(M,N)) - \tau(\operatorname{Tor}_{2e+1}^R(M,N))$$ then we have a bilinear map from $G(R)_{\mathbb{Q}}\times G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(R)_{\mathbb{Q}}$. Indeed, it is a bilinear map from $\overline G(R)_{\mathbb{Q}}\times \overline G(R)_{\mathbb{Q}}$ to $\operatorname{CH}_*(\operatorname{Sing}(R))_{\mathbb{Q}}$. The problem is that in $\operatorname{CH}_*(\operatorname{Sing}(R))_{\mathbb{Q}}$, the element $[R/m_R]$ is $0$ if $\dim \operatorname{Sing}(R)>0$. So when the $\operatorname{Tor}$s have finite length, the vanishing of $\beta^R(M,N)= \theta^R(M,N)[R/m_R]$ does not give you the vanishing of $\theta^R(M,N)$. To make this work, one possibility is to develop a version of “asymptotic chern character".

Another natural question is to extend the results in this paper to complete intersections. Typically, we will need more $\operatorname{Tor}$ modules to vanish. While for many results in this paper could be extended using some induction arguments with repeated use of the long exact sequence \[longexact\], the most efficient way is to define a generalized theta function for pair of modules over complete intersections. Since this required a bit of efforts, it will be dealt with in a separate paper (\[Da\]).

Finally, we will give some examples. The purpose is to demonstrate that many of our technical conditions can not be removed, and some statements can not be reversed.

\[ex2\] Let $R$ be a hypersurface and $M$ a Cohen-Macaulay $R$-module. Assume there exists a Cohen-Macaulay $R$-module $N$ such that:\
(1) $l(M{\otimes}_RN)<\infty$.\
(2) $\dim M + \dim N = \dim R + 1$.\
Then $\operatorname{Tor}_i^R(M,N)=0$ if and only if $i$ is an odd integer.

Suppose $R=T/(f)$, where $T$ is regular local. Then $\dim M + \dim N = \dim T$. Since both $M,N$ are Cohen-Macaulay, we have $\operatorname{depth}M + \operatorname{depth}N = \operatorname{depth}T$ as well. By \[HW2,2.2\] we have $\operatorname{Tor}_i^T(M,N)=0$ for all $i\geq 1$. A glance at the change of rings exact sequence gives $\operatorname{Tor}_1^R(M,N)=0$ and $\operatorname{Tor}_{i+2}^R(M,N)\cong \operatorname{Tor}_i^R(M,N)$ for $i\geq0$. But then $\operatorname{Tor}_2^R(M,N)\cong M{\otimes}_R N \neq 0$.

Now we are ready to give some examples:

Corollary \[dim2,3\] shows that one of the modules is $0$ in $\overline G(R)_{\mathbb{Q}}$ is not necessary for $\theta^R(-,-)$ to vanish because in an admissible hypersurface, normal domain of dimension $2$, $\theta^R$ always vanish. However, if we take our hypersurface $R$ to be the local ring at origin of $A=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$, then $X=\operatorname{Proj}(A)$ is an elliptic curve, therefore $\operatorname{CH}_*(R)_{\mathbb{Q}}$, hence $G(R)_{\mathbb{Q}}$ and $\overline G(R)_{\mathbb{Q}}$ , are infinite dimensional $\mathbb{Q}$-vector spaces.

In this example we will give a module $M$ such that $\theta^R(M,-)$ does not always vanish, but $M$ is still rigid and decent. Let $R=k[[x,y,u,v]]/(xu-yv)$ and $P=(x,y),Q=(x,v)$. Let $M
= R/P\oplus R/P\oplus R/Q$. It is easy to check that there is an exact sequence: $$0 \to Q \to R^2 \to P \to 0$$ which shows that $R/P+R/Q=0$ in $G(R)$. So $\theta^R(M,-) =
\theta^R(R/P,-)$. Clearly $\theta^R(R/P,-)$ is not always $0$, because $\theta^R(R/P,R/Q)=1$. It remains to show that $M$ is rigid and decent. Let $M'=R/P\oplus R/Q$. Then by the argument above $M'=0$ in $G(R)$, so it is rigid and decent since $R$ is an isolated singularity. So for any module $N$, $\operatorname{Tor}_i^R(M,N)=0$ $\Rightarrow$ $\operatorname{Tor}_i^R(R/P,N)=\operatorname{Tor}_i^R(R/Q,N)=0$ $\Rightarrow$ $\operatorname{Tor}_i^R(M',N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(M',N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(R/P,N)=\operatorname{Tor}_{i+1}^R(R/Q,N)=0$ $\Rightarrow$ $\operatorname{Tor}_{i+1}^R(M,N)=0$. As for decency, observe that $M$ and $M'$ have the same support, and as decency only depends on the support, $M$ must be decent as well.

In this example we will give concrete examples of isolated hypersurface singularities of dimension 3 such that $\theta$ always vanishes. $R=\mathbb{C}[[x,y,z,w]]/(x^2+y^2+z^p+w^q)$, where $p,q$ are coprime integers. Then $R$ is factorial and it follows from corollary \[dim2,3\] that $\theta^R$ must vanish. So over this hypersurface, rigidity and decency always hold.

Let $R=k[[x,y,u,v,t]]/(xu-yv)$ and let $M=R/(x,y,t)$. Then $M$ is not rigid (let $N=R/(u,v)$ and use \[ex2\]). However, the exact sequence: $$\xymatrix {0 \ar[r] &R/(x,y) \ar[r]^{t} &R/(x,y) \ar[r] &\overline{M} \ar[r] &0}$$ shows that $[M]=0$ in $\overline G(R)$. It is easy to check that $\operatorname{Sing}(R)=V((x,y,u,v))$ and $IPD(M) = \{(x,y,u,v,t)\} = \{m_R\}$. This example shows that the technical requirements for rigidity in theorem \[rig1.1\] can not be relaxed.

Let $R=k[[x,y,u,v]]/(xu-yv)$, $M=(x,y), N=(u,y)$. Then $M{\otimes}_RN \cong (x,y,u,v)$ is torsion free and has depth $1$. Also, $R$ is an isolated singularity. However, $\operatorname{Tor}_1^R(M,N)\neq 0$. This shows that the condition (3) of \[vanishingiso\] is critical.

[BroSh]{}

M. Auslander, *Modules over unramified regular local rings*, Ill. J. Math. **5** (1961), 631-647.

L.L Avramov, *Modules of finite virtual projective dimension*, Invent.Math.**96** (1989),71-101.

L.L Avramov, *Infinite free resolutions*, Six lectures in commutative algebra (Bellaterra, 1996), Progress in Math. **166**, Birkhausër, Boston, (1998), 1-118.

L.L Avramov, R-.O Buchweitz, *Support varieties and cohomology over complete intersections*, Invent. Math. **142** (2000), 285-318.

L.L Avramov, C. Miller, *Frobenius powers of complete intersections*, Math. Res. Lett. **8** (2001), 225-232.

W. Bruns, J. Herzog, *Cohen-Macaulay rings*, Cambridge Univ. Press, Cambridge (1996).

R-.O Buchweitz, *Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings*, Preprint, Unive. Hannover, (1986).

, *Vanishing pattern of Ext for modules of finite complete intersection dimensions*.

O. Celikbas, *Vanishing of Tor over complete intersections*, J. Commutative Alg., to appear, arXiv: math.AC/ 0904.1408v.

S.D. Cutkosky, H. Srinivasan, *Equivalence and finite determinacy of mappings*, J. Algebra **188**, (1997), 16-57.

H.L Dao, *Asymptotic behaviour of Tor over complete intersections and applications*, preprint.

H. Dao, *Some observations on local and projective hypersurfaces*, Math. Res. Let. 15 (2008), no. 2, 207–219.

H. Dao, *Remarks on non-commutative crepant resolutions of complete intersections*, Advances in Math., to appear.

S. Dutta, *A Theorem on Smoothness-Bass-Quillen, Chow groups and Intersection Multiplicity of Serre*, Tran. Amer. Math. Soc.**352** (2000) 1635-1645.

S. Dutta, *On negativity of higher Euler characteristics*, Amer. J. Math. **126** (2004), 1341-1354.

D. Eisenbud, *Homological algebra on a complete intersection ,with an application to group representations*, Tran. Amer. Math. Soc.**260** (1980), 35-64.

E.G Evans, P. Griffith, *Sygyzies*, Lond. Math. Soc. Lect. Notes **106** (1985).

H. Esnault, M. Levine, E. Viehweg, *Chow groups of projective varieties of very small degree*, Duke Math. J. **87** (1997), 29-58.

, R. Fossum, *The Divisor Class Group of a Krull Domain*, Springer-Velag, New York 1973.

W. Fulton, *Intersection Theory*, Springer-Verlag, Berlin (1998).

A. Grothendieck, *Eléments de géometrique algébrique, Chapage IV* Publ. Math. I.H.E.S. **24** (1965).

A. Grothendieck, *Cohomologie locale des faisceaux cohérents et théorèms de Lefschetz locaux et globaux* North Holland, Amsterdam (1968).

T.H Gulliksen, *A change of ring theorem with applications to Poincaré series and intersection multiplicity*, Math. Scan **34** (1974), 167-183.

R. Hartshorne, *Equivalence relations on algebraic cycles and subvarieties of small codimension*, Proc. Symp. Pure Math. **29** (1975), 129-164.

R. Hartshorne, *Algebraic Geometry*, Graduate Text in Mathematics, Springer-Verlag, New York, (1977).

R. Heitmann, *A counterexample to the rigidity conjecture for rings*, Bull. Am. Math. Soc. **29** (1993), 94-97.

M. Hochster, *The dimension of an intersection in an ambient hypersurface*, Proceedings of the First Midwest Algebraic Geometry Seminar (Chicago Circle,1980), Lecture Notes in Mathematics **862**,Springer-Verlag, 1981, 93-106.

M. Hochster, *Euler characteristics over unramified regular local rings*, Ill. J. Math. **28** (1984), 281-288.

M. Hochster, *Nonnegativity of intersection multiplicities in ramified regular local rings following Gabber/De Jong/Berthelot*, unpublished notes.

M. Hochster, *Topics in the Homological theory of modules over commutative rings*, Regional Conference Series in Mathematics **24**, Amer. Math. Soc. Providence, RI (1975).

C. Huneke, R. Wiegand, *Tensor products of modules and the rigidity of Tor*, Math. Ann. **299** (1994), 449-476.

C. Huneke, R. Wiegand, *Tensor products of modules, rigidity and local cohomology*, Math. Scan. **81** (1997), 161-183.

C. Huneke, R. Wiegand, D. Jorgensen, *Vanishing theorems for complete intersections*, J. Algebra **238** (2001), 684-702.

D. Jorgensen, *Tor and torsion on a complete intersection*, J. Algebra **195** (1996), 526-537.

D. Jorgensen, *Complexity and Tor on a complete intersection*, J. Algebra **211** (1999), 578-598.

K.Kurano, *A remark on the Riemann-Roch formula for affine scheme associated with Noetherian local ring*, Tohoku Math J. **48** (1996), 121-138.

K.Kurano, *Numerical equivalence defined on Chow groups of Notherian local rings*, Invent. Math. **157** (2004), 575-619.

S. Lichtenbaum, *On the vanishing of Tor in regular local rings*, Ill. J. Math. **10** (1966), 220-226.

F. Moore, G. Piepmeyer, S. Spiroff, M. Walker, *Hochster’s theta invariant and the Hodge-Riemann bilinear relations*, arXiv math.AC/0910.1289.

M.P Murthy, *Modules over regular local rings*, Ill. J. Math. **7** (1963), 558-565.

C. Peskine, L. Szpiro, *Dimension projective finie et cohomologie locale. Applications $\grave{a}$ la d$\acute{e}$monstration de conjectures de M. Auslander, H. Bass et A. Grothendieck*, Inst. Hautes tudes Sci. Publ. Math. **42** (1973), 47–119.

P. Roberts, *Multiplicities and Chern classes in Local Algebra*, Cambridge Univ. Press, Cambridge (1998).

J.P. Serre,*Algèbre locale. Multiplicités*, Lect. Note in Math. **11**, Springer-Verlag, Berlin, New York, 1965.

T. Shioda, *On the Picard number of a complex projective variety*, Annales Scientifiques de l’cole Normale Suprieure Sr. 4 **14**, no. 3 (1981),303-321.

Y. Yoshino, *Cohen-Macaulay modules over Cohen-Macaulay rings*, Lond. Math. Soc. Lect Notes **146** (1990).

R. Wiegand, *Local rings of finite Cohen-Macaulay type*, J. Algebra **203** (1998), 156-168.

[^1]: The author is partially supported by NSF grant 0834050
