I've become very confused about the tangent and cotangent spaces of a variety (read: finite type scheme over an algebraically closed field). Most of my problem is that most texts just refer to the vector spaces involved without giving any thought to the fields. Let's say $X, Y$ are varieties over $k$ with $k$ algebraically closed. We have a morphism over $k$, call it $f: X \rightarrow Y$ with $f(x) = y$. Note that there are three fields at play here: $k, \kappa(x), \kappa(y) $. Kappa is my notation for residue fields.
There are field extensions $k \hookrightarrow \kappa(y) \hookrightarrow \kappa(x)$ which just comes from the fact that the stalk morphism is a local homomorphism of local rings, right?
The tangent spaces are given by $\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2}$ and $\mathfrak{m}_{y}/\mathfrak{m}_{y}^{2}$. These are vector spaces over $\kappa(x)$ and $\kappa(y)$ respectively.
This is where things get murky. Every introduction to algebraic geometry says we have a morphism of cotangent spaces,
$$
\mathfrak{m}_{y} / \mathfrak{m}_{y}^{2} \longrightarrow \mathfrak{m}_{x}/\mathfrak{m}_{x}^{2}.
$$
But this is a morphism of what? Of $k$-vector spaces? Or are we performing a change of fields via a tensor product so that this is a morphism of $\kappa(f(p))$-vector spaces? The entire machinery of schemes has non-closed points built into it, so I would like to keep these vector spaces as being over the respective residue fields, but then in order to have a morphism between them there has to be a change of fields involved.
More confusing still, this is dualized. And the dualizing appear to happen over $k$, to obtain a morphism of tangent spaces. But then we would need some tensor-hom adjunction to turn them back into vector spaces over the respective residue fields.
Is someone able to explain what is going on here, or even just refer to an introductory text that includes the words "over [field name]" when they give a morphism of vector spaces?
Best Answer
First, people mostly talk about this when $\kappa(x)=\kappa(y)$ and there, everything is easy. For instance, if $x,y$ are closed $k$-rational points (all closed points are $k$-rational if $k$ is algebraically closed), then $k=\kappa(x)=\kappa(y)$ canonically and everything goes very well. In general, things can be trickier and less pleasant.
Your claim about $k\hookrightarrow \kappa(y)\hookrightarrow\kappa(x)$ is correct in general for any scheme over a field (not necessarily algebraically closed). The spaces $\mathfrak{m}_x/\mathfrak{m}_x^2$ and $\mathfrak{m}_y/\mathfrak{m}_y^2$ are the cotangent spaces, and the map $\kappa(y)\hookrightarrow\kappa(x)$ lets us consider any $\kappa(x)$-vector space as a $\kappa(y)$-vector space. So the map you're interested in is a map of $\kappa(y)$-vector spaces. Our viewing of $\kappa(x)$-vector spaces as $\kappa(y)$-vector spaces can be interpreted as applying the tensor functor $_{\kappa(y)}\kappa(x)_{\kappa(x)}\otimes_{\kappa(x)}-$ if you want, where the left $\kappa(y)$-module structure on $\kappa(x)$ is given by the map $\kappa(y)\hookrightarrow\kappa(x)$. This is easy and natural and requires no additional choices - this is part of what people mean when they say that cotangent spaces are more natural to consider in algebraic geometry.
Next, you appear to have some confusion about how to get the tangent space $T_{x}$ at $x$ from the cotangent space $\mathfrak{m}_{x}/\mathfrak{m}_{x}^2$. The dualization happens over $\kappa(x)$ here. If $\kappa(x)=\kappa(y)$, then thinking of this map as the dual of the map of cotangent spaces is easy and correct. If the residue fields are not equal, things get funny here. For instance, think about what should happen to the tangent space at the origin for the natural map $\Bbb A^1_\Bbb C\to\Bbb A^1_\Bbb R$, induced by the inclusion of coordinate rings, and then compare it to what actually happens: we kill all the tangent vectors in the imaginary direction. Things are even worse for the projection map $\Bbb A^2_\Bbb C\to \Bbb A^1_\Bbb C$ and the map of the generic point of a fiber over a closed point to that closed point. The map of contangent spaces is isomorphic to the inclusion $\Bbb C\to \Bbb C(t)$, and trying to dualize this is a huge mess.
Basically, the trip from cotangent spaces to tangent spaces is yucky unless all the points involved have the same residue field. What you really ought to do here if you're interested in the schemey perspective is to define the cotangent and tangent sheaves. The theory of these (especially the cotangent sheaf) is much better developed, and gives you back all the classical stuff you'd want at the closed points while playing better with all the schemey things you'd like to do. Most serious textbooks should spend a good bit of time on the cotangent sheaf - for example, Hartshorne chapter II section 8 and Vakil chapter 21 are two textbook references, and StacksProject has lots of content scattered in many different places.