Reading Liu, I got quite confused about what the "canonical map" $T_{f,x}:T_{X,x}\to T_{Y,f(x)}\otimes_{k(f(x))}k(x)$ should be for a map of general schemes $f:X\to Y$. Reading several MSE posts, I get the impression this map is only well-defined if $T_{Y,f(x)}$ is finite dimensional over $k(f(x))$ (or if $k(x)$ is a finite extension of $k(y)$, but I'd guess condition is harder to deal with. I may be wrong)
My question is very simple: looking around, I haven't been able to find anything comprehensible about a general definition for the instances where we have infinite dimensional tangent spaces. Is there such a definition?
Here is my understand of the definition in the finite case:
For brevity let $V:=\mathfrak{m}_x/\mathfrak{m}_x^2$ and $W:=\mathfrak{m}_y/\mathfrak{m}_y^2$, $K:=k(f(x)),L:=k(x)$. We have maps $K\hookrightarrow L$ and $W\to V$ which could both be called by $f$. $V$ is an $L$-vector space, $W$ a $K$-vector space. We want to get a $L$-linear map $\hom_L(V;L)\to\hom_K(W;K)\otimes_KL$. We have a canonical map $\hom_K(W;K)\otimes_KL\to\hom_L(W\otimes_KL;L)$ which is adjunct to: $$(\hom_K(W;K)\otimes_KL)\otimes_L(W\otimes_KL)\cong(\hom_K(W;K)\otimes_KW)\otimes_K(L\otimes_LL)\\\to K\otimes_K L\cong L$$Explicitly this map is given by $\phi\otimes\lambda\mapsto(w\otimes\mu\mapsto\lambda\mu\cdot f(\phi(w)))$. If we have a finite dimensionality then we can check this is an isomorphism. Using this, it then suffices to find a map $\hom_L(V;L)\to\hom_L(W\otimes_KL;L)$, and we just need a map $W\otimes_KL\to V$ to get that. Easily enough, we just take $w\otimes\lambda\mapsto\lambda\cdot f(w)$.
It's a little unfortunate we can't explicitly describe $T_{f,x}$ without first fixing a basis but whatever; as far as I can tell, though the map $\hom_K(W;K)\otimes_KL\to\hom_L(W\otimes_KL;L)$ is natural and coordinate-free, whether or not it is invertible depends on dimension and all constructions of the inverse are the same but require choosing a basis. Namely, if $(\beta_j)_{j=1}^n$ is a basis for $W$ over $K$ then we take some $L$-linear $\psi:W\otimes_KL\to L$ and assign to it $\sum_{k=1}^n\pi_k\otimes\psi(\beta_k\otimes1)$ where the $\pi_k:W\to K$ are the $k$th coordinate functions w.r.t the basis $\beta_\bullet$. This overall gives me: $$T_{f,x}(\phi)=\sum_{k=1}^n\pi_k\otimes\phi(f(\beta_k)),\,\phi:\mathfrak{m}_x/\mathfrak{m}_x^2\to k(x)$$
What can we do if $\dim_{k(f(x))}T_{Y,f(x)}\ge\aleph_0$?