I want to apply [EGA IV 4,Proposition 17.13.2.][1] to a cartesian diagram [![enter image description here][2]][2] of algebraic spaces over a fixed scheme $S$.
I know the relative dimensions $\mathrm{dim}(\mathfrak{X}'/S),\mathrm{dim}(\mathfrak{Y}/S)$ and $\mathrm{dim}(\mathfrak{X}/S)$ and I want to use the Proposition to determine the relative dimension of $\mathfrak{Y}'$ by showing that the map on tangent spaces is surjective.
The motivation for my question is [L.Lafforgue, Proposition 1, page 29][3] where he obviously applies the Proposition from EGA to a diagram of algebraic spaces.

The problem that I am concerned about is, that the Proposition is formulated and proven only for the case of schemes, so a priori it is not immediately applicable to the diagram.

I expect at least one of the following ideas to work:

**$\bullet$ State and prove the Lemma for algebraic spaces.**(If it still holds?)

How is the tangent space of an algebraic space even defined? Is this the tangent space of a functor as defined in deformation theory?

**$\bullet$ Use the fact, that every algebraic space has an atlas and apply the lemma to the resulting diagram!?**
I could not figure any of these ideas out yet and would appreciate, if someone could explain me what the author had in his mind.

$\DeclareMathOperator\dim{dim}$I want to apply EGA IV 4, Proposition 17.13.2 to a cartesian diagram of algebraic spaces over a fixed scheme $S$.

I know the relative dimensions $\dim(\mathfrak{X}'/S)$, $\dim(\mathfrak{Y}/S)$ and $\dim(\mathfrak{X}/S)$ and I want to use the Proposition to determine the relative dimension of $\mathfrak{Y}'$ by showing that the map on tangent spaces is surjective.

The motivation for my question is L. Lafforgue, Proposition 1, page 29 where he obviously applies the Proposition from EGA to a diagram of algebraic spaces.

The problem that I am concerned about is, that the Proposition is formulated and proven only for the case of schemes, so a priori it is not immediately applicable to the diagram.

I expect at least one of the following ideas to work:

• State and prove the Lemma for algebraic spaces. (If it still holds?) How is the tangent space of an algebraic space even defined? Is this the tangent space of a functor as defined in deformation theory?

• Use the fact that every algebraic space has an atlas and apply the lemma to the resulting diagram. I could not figure any of these ideas out yet and would appreciate, if someone could explain me what the author had in his mind.

I want to apply [EGA IV 4,Proposition 17.13.2.][1] to a cartesian diagram [![enter image description here][2]][2] of algebraic spaces over a fixed scheme $S$.
I know the relative dimensions $\mathrm{dim}(\mathfrak{X}'/S),\mathrm{dim}(\mathfrak{Y}/S)$ and $\mathrm{dim}(\mathfrak{X}/S)$ and I want to use the Proposition to determine the relative dimension of $\mathfrak{Y}'$ by showing that the map on tangent spaces is surjective.
The motivation for my question is [L.Lafforge, Proposition 1, page 29][3] where he obviously applies the Proposition from EGA to a diagram of algebraic spaces.

The problem that I am concerned about is, that the Proposition is formulated and proven only for the case of schemes, so a priori it is not immediately applicable to the diagram.

I expect at least one of the following ideas to work:

**$\bullet$ State and prove the Lemma for algebraic spaces.**(If it still holds?)

How is the tangent space of an algebraic space even defined? Is this the tangent space of a functor as defined in deformation theory?

**$\bullet$ Use the fact, that every algebraic space has an atlas and apply the lemma to the resulting diagram!?**
I could not figure any of these ideas out yet and would appreciate, if someone could explain me what the author had in his mind.

Thank You for Your efforts, SDIGR

I want to apply [EGA IV 4,Proposition 17.13.2.][1] to a cartesian diagram [![enter image description here][2]][2] of algebraic spaces over a fixed scheme $S$.
I know the relative dimensions $\mathrm{dim}(\mathfrak{X}'/S),\mathrm{dim}(\mathfrak{Y}/S)$ and $\mathrm{dim}(\mathfrak{X}/S)$ and I want to use the Proposition to determine the relative dimension of $\mathfrak{Y}'$ by showing that the map on tangent spaces is surjective.
The motivation for my question is [L.Lafforgue, Proposition 1, page 29][3] where he obviously applies the Proposition from EGA to a diagram of algebraic spaces.

The problem that I am concerned about is, that the Proposition is formulated and proven only for the case of schemes, so a priori it is not immediately applicable to the diagram.

I expect at least one of the following ideas to work:

**$\bullet$ State and prove the Lemma for algebraic spaces.**(If it still holds?)

How is the tangent space of an algebraic space even defined? Is this the tangent space of a functor as defined in deformation theory?

**$\bullet$ Use the fact, that every algebraic space has an atlas and apply the lemma to the resulting diagram!?**
I could not figure any of these ideas out yet and would appreciate, if someone could explain me what the author had in his mind.