A correction

This is about example 4 from this post. It turns out that you can repair what I said there to make it work. The mistake was pointed out by David Rydh in the comments of that post. Thanks for Bhargav Bhatt for explaning how to repair it. Any mistakes are mine (please tell me if there are any).

Let X be an algebraic space over a base S (not necessarily flat). Consider diagrams

Y ---> X
|      |
v      v
T ---> S

where f : Y —> T is proper and flat. In this situation let C’ ∈ D(Y) be the cone of the map

L_{X/S} ⊗ O_Y —> L_{Y/T}.

Then I claim there is a canonical map C’ —> f^*L_{T/S}[1] which controls the deformation theory of the diagram (i.e., we look at first order thickenings T’ of T over S and flat deformations of Y to T’ mapping into S).

This is much better than the original suggestion in the post but it works for the same reason and the obstruction group doesn’t depend on the thickening T’ only on the ideal I defining T in T’.

The “real” reason this works is the following observation: We can think of the cone C’ as the cotangent complex of Y over the derived base change of X to T. Hence it is clear that EXt(C’, f^*I) computes obstructions, infinitesimal deformations, infinitesimal automorphisms of the morphism of Y into the derived base change. But since Y is a usual scheme and flat over T any (flat) deformation of Y to T’ is still a usual scheme, and morphisms from a usual scheme to a derived scheme map through pi_0(the derived scheme).

Jason Starr has privately emailed me something similar for the Quot scheme, which I haven’t fully understood yet.

2 thoughts on “A correction

  1. Pingback: Openness of versality | Stacks Project Blog

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