T^i functors are isomorphic for analytically isomorphic isolated singular points

2012-08-10T20:46:21Z

I've been having trouble proving the following: Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are analytically isomorphic, i.e. have isomorphic completions, then the deformation theory modules $T^i_{B/k}$ and $T^i_{B'/k}$ are isomorphic for $i=1,2$ as modules over the completions of $B$ and $B'$.

A good answer would only use the definitions and basic properties of these functors, and/or would give a reference where this is worked out in some amount of detail, i.e. not where it is just stated as a fact.

Thanks!

Answer by Damian Rössler for T^i functors are isomorphic for analytically isomorphic isolated singular points

2012-08-13T09:43:36Z

Here is a stab at an answer, but it is incomplete.

Let $S:={\rm Spec}\, B$ and let $\widehat{S}:={\rm Spec}\,\widehat{B}$ be the completion of $B$ along its maximal ideal $m$. Let $\phi:\widehat{S}\to S$ be the natural morphism (which is faithfully flat). The composition of morphisms $\widehat{S}\to S\to{\rm Spec}\,k$ gives rise to a triangle of cotangent complexes and hence to an exact sequence $$ \dots\to T^1_{\phi}\to T^1_{\widehat{S}/k}\to \phi^*T^1_{S/k}\to T^2_{\phi}\to T^2_{\widehat{S}/k}\to \phi^*T^2_{S/k}\to T^3_\phi\to\dots {\rm (*)} $$ so what you need to show is that $T^i_\phi$ vanishes for $i=1,2,3$. You would then get isomorphisms $T^i_{\widehat{S}/k}\to \phi^*T^i_{S/k}$ and since you can repeat this for $B'$ instead of $B$, you would get the required isomorphism.

Let $L_\phi$ be the cotangent complex of $\widehat{S}/S$. This complex $L_\phi$ is concentrated in degree $0$. A quick way to see this is to notice that

the formation of the cotangent complex is compatible with direct limits of rings (see Quillen, "Cohomology of commutative rings", eq. (4.11))
the ring $B$ is excellent because it is essentially of finite type over a field (Grothendieck) and thus the fibres of $\phi$ are geometrically regular;
a (deep...) result of Popescu (see for instance Th. 1.3 in "Approximations of versal deformations", by B. Conrad and AJ de Jong) then implies that $\widehat{B}$ is a direct limit of smooth $B$-algebras, and for the latter the cotangent complex is clearly concentrated at $0$.

By considering the sequence analogous to (*) for the modules $T_i$ instead of $T^i$, this proves the analog of your assertion for the $T_i$. Furthermore, we see that $T^i_\phi={\rm Ext}^i(L_\phi,\widehat{B})={\rm Ext}^i(\Omega_\phi,\widehat{B})$.

So the issue is to show that ${\rm Ext}^i(\Omega_\phi,\widehat{B})=0$ for $i>0$. Maybe your hypothesis on the fact that the singularity of $S$ is isolated at the closed point plays a role here.

T^i functors are isomorphic for analytically isomorphic isolated singular points - MathOverflow

T^i functors are isomorphic for analytically isomorphic isolated singular points

Answer by Damian Rössler for T^i functors are isomorphic for analytically isomorphic isolated singular points