Universal thickenings

Let T be an algebraic space. A first order thickening of T is a closed immersion T —> T’ of algebraic spaces which is defined by an ideal of square zero. If T is over an algebraic space Y then we can talk about first order thickenings over Y. These form a category with an obvious notion of morphism.

Let X —> Y be a morphism of algebraic space. Consider first order thickenings T —> T’ over Y together with a morphism T —> X over Y. This gives a category of diagrams (T’ <— T —> X) over Y. Claim: If X —> Y is formally unramified then this category has a final object. Moreover, the universal object is actually a first order thickening X —> X’ of X over Y (i.e., T = X for the universal object). Let’s call this the universal first order thickening of X over Y.

Now, given X —> Y formally unramified we define the conormal sheaf of X over Y as the conormal sheaf of X in its universal first order thickening of X over Y. Notation C_{X/Y}. This construction is suitably functorial. For example if you have a morphism of arrows (f, g) : (X —> Y) —> (X’ —> Y’), and both arrows are formally unramified then you get a map f^*C_{X’/Y’} —> C_{X/Y}.

Why is this interesting? Well, I wanted to use this to clarify the notion of the module of differentials of a morphism of algebraic spaces. Namely, if f : X —> Y is an arbitrary morphism of algebraic spaces, then Δ : X —> X \times_Y X is not an immersion, just a monomorphism. Thus we need a slightly more general notion of a conormal sheaf in order to compare \Omega_{X/Y} to the conormal sheaf of Δ.

Note that a very natural definition of \Omega_{X/Y} is to define it as the module of differentials of the map of sheaves of rings f^{-1}O_Y —> O_X on the small etale site of X. (This is the one currently in the stacks project.) The result is that it is canonically isomorphic to the conormal sheaf of Δ. This can then be used to link with infinitesimal deformations of maps into X with \Omega_{X/Y}.

Moreover, as Jarod Alper pointed out, the material in the paragraph above should continue to work for morphisms of Deligne-Mumford stacks (as defined in the stacks project).

1 thought on “Universal thickenings

  1. Pingback: Update « Stacks Project Blog

Comments are closed.