nLab formally unramified morphism

Idea

A space $X$ is called formally unramified if every morphism $Y \to X$ into it has for every infinitesimal thickening of $Y$ at most one infinitesimal extension.

(If all thickenings exist it is called a formally smooth morphism. If the thickening exist uniquely, it is called a formally etale morphism.)

Traditionally this has been considered in the context of geometry over formal duals of rings and associative algebras. This we discuss in the section (Concrete notion). But generally the notion makes sense in any context of infinitesimal cohesion. This we discuss in the section General abstract notion.

The concept of formally unramified morphisms is the infinitesimal version of that of unramified morphisms.

General abstract notion

Definition

Let

$$\mathbf{H} \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}} \mathbf{H}_{th}$$

be a triple of adjoint functors with $u^*$ a full and faithful functor that preserves the terminal object.

We may think of this as exhibiting infinitesimal cohesion (see there for details, but notice that in the notation used there we have $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$).

We think of the objects of $\mathbf{H}$ as cohesive spaces and of the objects of $\mathbf{H}_{th}$ as such cohesive spaces possibly equipped with infinitesimal extension.

As a class of examples that is useful to keep in mind consider a Q-category $(cod \dashv \epsilon \dashv dom) : \bar A \to A$ of infinitesimal thickening of rings and let

$$((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := ([dom,Set] \dashv [\epsilon, Set] \dashv [codom,Set] : [\bar A, Set] \to [A,Set])$$

be the corresponding Q-category of copresheaves.

For any such setup there is a canonical natural transformation

$$u^* \to u^! \,.$$

Details of this are in the section Adjoint quadruples at cohesive topos.

From this we get for every morphism $f : X \to Y$ in $\mathbf{H}$ a canonical morphism

(1)$$u^* X \to u^* Y \prod_{u^! Y} u^! X \,.$$

Definition

A morphism $f : X \to Y$ in $\mathbf{H}$ is called formally unramified if (1) is a monomorphism.

This appears as (KontsevichRosenberg, def. 5.1, prop. 5.3.1.1).

The dual notion, where the morphism is required to be an epimorphism is that of formally smooth morphisms. If both conditions hold, hence if the morphism is in fact an isomorphism, one speaks of formally etale morphisms.

Definition

An object $X \in \mathbf{H}$ is called formally unramified if the morphism $X \to *$ to the terminal object is formally unramified.

Proposition

The object $X$ is formally unramified precisely if

$$u^* X \to u^! X$$

is a monomorphism.

This appears as (KontsevichRosenberg, def. 5.3.2).

Properties

Proposition

Formally unramified morphisms are closed under composition.

This appears as (KontsevichRosenberg, prop. 5.4).

Concrete notion

For the moment see the discussion at unramified morphism.

Related concepts

• unramified morphism

• ramification of ideals

• formally smooth morphism and formally unramified morphism $\Rightarrow$ formally etale morphism.