Consider $X$ and $Y$ varieties inside a smooth variety $M$. I say that $X$ and $Y$ intersect transversally at $m\in M$ if the tangent spaces of $X$ and $Y$ span the whole tangent space of $M$ at $m$. I see, for example just from Serre's Tor-formula, that intersection theory often deals with $\operatorname{Tor}$, but I cannot see the link between the definition of transversal intersection and it. Some question I have: has transversal intersection something to do with the condition $\operatorname{Tor}_{>0}(\mathcal{O}_{X,m},\mathcal{O}_{Y,m})=0$? Both are "good intersection conditions" but does one implies the other?
Feel free to consider intersection of Cohen-Macaulay varieties, even to take $X$ hypersurface. I'm trying to have an insight.
Thank you!
Assume $X$ is a locally complete intersection and $Y$ is Cohen--Macaulay. Then loally $\mathcal{O}_X$ has a free resolution given by the Koszul complex. Then $Tor$'s are computed by the restriction of this resolution to $Y$. If the dimension of the intersection is the expected one (this is a weaker condition than the condition about the tangent spaces) then this restriction is exact (by the CM property), hence $Tor$ vanish.