Tor and projective dimension Is it possible that $\mbox{Tor }^{r+1}(M,N)=0 \ \ \forall N$ yet $\mbox{proj. dim }M>r$?
What I do know is that if $(A,\mathfrak{m})$ is Noetherian local and $M$ is finitely generated over $A$ then $\mbox{Tor }^{r+1}(M,A/\mathfrak{m})=0 $ if and only if $\mbox{proj. dim }M\leq r$.
Generally speaking, is $\mbox{Tor }$ functor as good a tool to measure projective dimension as $\mbox{Ext }$ even when the ring/module is not Noetherian or local?
I suspect we can use $\mbox{Tor }$ to measure projective dimension when ring is Neotherian local and module is finitely generated because flatness and projectivity coincide in such case.
 A: There's another dimension, called flat dimension: $\mathrm{fd}\; M_R = n$ means that $n$ is the smallest integer such that there exists a resolution
$$0 \to F_n \to \cdots \to F_1 \to F_0 \to M \to 0$$
where each $F_i$ is a flat module. We have $\mathrm{fd}\; M_R \leq d$ if and only if $\mathrm{Tor}^R_{d+1}(M, N) = 0$ for all $_RN$. Since projective modules are flat, we have $\mathrm{fd}\; M_R \leq \mathrm{pd}\; M_R$, but as Richard Borcherds's answer shows, there are modules for which equality doesn't hold. However, if $R$ is right noetherian and $M_R$ is finitely generated, then $\mathrm{pd}\; M_R = \mathrm{fd}\; M_R$.
Here are some references: 


*

*Chapter 7 of Noncommutative noetherian rings by McConnell and Robson.

*Chapter 4 of An introduction to homological algebra by Weibel.

A: Since your question is really about projective dimension of flat modules, it is worth noting the following result  (see Raynaud-Gruson MR0308104, Cor 3.3.2 or Jensen MR0407091, Thm 5.8) which complements Richard's example:
The projective dimension of a flat module over a commutative Noetherian ring $R$ is bounded by $n+1$ if  the cardinality of $R$ is at most $\aleph_n$. 
EDIT: It looks like commutativity is not needed!
A: Over the integers, the rational numbers  Q are flat and so Tor^i(Q,M) = 0 for all M and all i>0. However Q is not projective so has projective dimension 1. 
