I find it really hard to understand the definition of flat. There are so many details, but I didn't find any "intuitive" geometric or algebraic approach to this type of functions.
To my problem: I have the simple inclusion morphism $i:T\simeq(\mathbb{C}^*)^n \to X_\Sigma$ between a torus $T$ and a $n$-dimensional toric variety $X_\Sigma$. Is it in general true that this morphism is flat? If not, does this hold for specific toric varieties (maybe only complete, smooth, simplicial or projective ones)?
I have no idea how to approach this question as the definition of flat requires looking at the stalks,... However, this seems a bit overkill and I think there should be a simple answer to such a "simple" morphism. Any suggestions?
An immersion of (locally Noetherian) schemes is flat if and only if it is an open immersion, or a closed immersion of connected components.
Open immersions are flat since the induced maps on stalks are isomorphisms, as pointed out by Zhen Lin in the comments.
Suppose $\phi\colon Z\to X$ is a closed immersion and assume it is flat. The question is local so we may assume that $X=\text{Spec}(R)$, $Z=\text{Spec}(R/I)$, and $\phi$ is induced by the quotient map $R\to R/I$. Consider the SES $$0\to I\to R\to R/I\to 0.$$ By assumption $R/I$ is a flat $R$-module, so we get an exact sequence $$0\to I/I^2\to R/I\to R/I\to 0$$ after tensoring with $R/I$. The last map is an isomorphism so $I^2=I$. Since $X$ is locally noetherian $I$ is finitely generated and Nakayama's lemma implies that $I=(e)$ where $e^2=e$. This means that $R\simeq R/(1-e)\times R/(e)$ and hence $Z$ is a union of connected components of $X$.
Note: You can remove the Noetherian hypothesis and only assume that $I$ is finitely generated. I'm not sure if the same holds when $I$ is infinitely generated, or if there are pathological examples.