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Let $f: X \to Y$ be a morphism of connected schemes over base field $k$ of characteristic zero.

By universal property of Kähler differentials & functoriality there are maps $f^*_j:\Omega_Y^j \to f_*\Omega_X^j$ for each $j \in \Bbb N$ inducing maps on cohomology $f^*_j:H^0(Y,\Omega_Y^j) \to H^0(X, f_*\Omega_X^j)$ pulling back global sections.

I'm looking for "standard references" discussing properties (primary , when injective, when surjective) of maps $f^*$ for "reasonable" class of morphisms $f$. (... to be concrete, i'm interested primary in 3 sames for $f$: (1) open & closed immersions, (2) (quasi-)finite & (3) proper maps.

What is known to me in this direction: In Hartshorne's AG there is a result for dominant maps $f:X \to Y$ between integral schemes of finite type with base field of char $0$ that $f$ becomes etale restricted to appropr non empty(!) open $U, W \subset f^{-1}(U)$.
This would boil down to check for which open $i_W:W \subset X$ the pullback map $i_W^*$ on differential forms become injective.

Can we conclude injectivity of $f^*$ from this? Also, I'm not sure if it isn't possible to weak up the assumptions on characteristic of $k$. For instance I expect (... but not have found a reference) that this shall also depend on if the extension of function field $K(X)/K(Y)$ is separable.

Although this kind of questions appears rather natural to me, up to now I haven't found a kind of "sample" or "standard references" treating rigorously the behaviour of maps between differential forms in context of algebraic geometry.

Update (added later): These notes helped to dealing partially with the smooth case.

user267839
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    Are you familiar with Matsumuras book "Commutative ring theory" - This has much material on modules of differentials. There is for every sequence $k \rightarrow A \rightarrow B$ an induced sequence $B\otimes_A \Omega^1_{A/k} \rightarrow \Omega^1_{B/k} \rightarrow \Omega^1_{B/A} \rightarrow 0$, etc. – hm2020 Jun 15 '25 at 08:58
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    ...there is a long exact sequence ending in the above sequence, hence the leftmost map is not injective in general - look for the "naive cotangent complex" and "Jacobi-Zariski sequence". – hm2020 Jun 15 '25 at 10:39

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Q: "Can we conclude injectivity of f∗ from this? Also, I'm not sure if it isn't possible to weak up the assumptions on characteristic of k. For instance I expect (... but not have found a reference) that this shall also depend on if the extension of function field K(X)/K(Y) is separable."

A: Given any pair of maps of commutative rings $k \rightarrow A \rightarrow B$ there is the "naive cotangent complex" which is an exact sequence of $B$-modules

$$ H_1(k,A,B) \rightarrow H_2(k,A,B) \rightarrow H_3(k,A,B) \rightarrow B \otimes_A \Omega^1_{A/k} \rightarrow^i \Omega^1_{B/k} \rightarrow \Omega^1_{B/A} \rightarrow 0$$

(this construction can be found on the "Stacks"-homepage and in the litterature), hence the canonical map $i$ is not injective in general. You find an introduction to the "module of Kahler differentials" in Matsumuras book. There is more advanced material on the stacks homepage. If $B/A$ is $0$-smooth it follows the sequence

$$0 \rightarrow B \otimes_A \Omega^1_{A/k} \rightarrow^i \Omega^1_{B/k} \rightarrow \Omega^1_{B/A} \rightarrow 0$$

is split exact (this is proved in Matsumuras book).

Note: There is a bijection

$$\psi: Hom_{A-mod}(\Omega^1_{A/k}, \Omega^1_{B/k}) \rightarrow Hom_{B-mod}(B\otimes_A \Omega^1_{A/k}, \Omega^1_{B/k}),$$

and there is for any map $j: A \rightarrow B$ a "canonical map"

$$\phi: \Omega^1_{A/k} \rightarrow \Omega^1_{B/k}$$

defined by $\phi(d_A(a)):=d_B(j(a))$. It follows the "adjoint map" is the map

$$\psi(\phi)(b\otimes d_A(a)):=bd_B(j(a)).$$

EX: If $k$ is a field (of char zero) and $A:=k[x], B:=k[x,y]$ you get the sequence

$$ B\{dx\} \rightarrow B\{dx,dy\} \rightarrow B\{dy\} \rightarrow 0$$

and

$$B\{dx,dy\} \cong A[y]\{dx,dy\}$$

and

$$B\{dy\} \cong A[y]\{dy\},$$

and both modules are finitely generated as $B$-module and "infinitely generated" as $A$-module.

hm2020
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  • About the splitting property you mentioned in last sentence: haven't found it on Stacks, but Bosch's AG contains it in Chap 8.5 on smooth morphisms, see Prop. 10 (actually stalkwise the splitting happens already for smooth $B/A$ and Cor. 12 the etale case. To relate it to the initial question: So for smooth $X \to Y$ we can characterize $f^\Omega_Y \to \Omega_X$ by this mentioned stalkwise split. What can we say in turn about corresponding $\Omega_Y \to f_\Omega_X$ via the $(f^,f_)$ adjunction? As its the latter which induces map $ f^:H^0(\Omega_Y) \to H^0(f_\Omega_X)$ I'm interested in – user267839 Jun 19 '25 at 10:07
  • @user267839 - This must be checked, but a first guess is that the corresponding map $\Omega_Y→f_∗\Omega_X$ also splits. There is an isomorphism of abelian groups $Hom_X(f^∗G,F)≅Hom_Y(G,f_∗F)$. If this isomorphism is "functorial" in a suitable sense it follows the corresponding morphism is split. – hm2020 Jun 19 '25 at 11:54
  • not sure if this is actually the case (...but maybe I'm overseeing something). Explicitly, the natural iso $Hom_X(f^∗G,F)≅Hom_Y(G,f_∗F)$ maps a $f^{}G\xrightarrow{\phi} F$ to composition $G\xrightarrow{u}f_f^{}G\xrightarrow{f_\phi} f_F$ with unit map $u$ of the adjunction. The problem I see so far is that if $\phi$ is split, ie there exist $s: F \to f^G$ with $\phi \circ s=id_F$ then indeed by functoriality we get splitting $f_* \circ f_\phi= id_{f_F}$ but we a priori not have a map splitting $u$. Any idea how to overcome that? – user267839 Jun 19 '25 at 12:45
  • @user267839 - I have a question: Why do you need to study the "adjoint map"? Usually one does not do this. – hm2020 Jun 19 '25 at 12:48
  • Well, my goal is to attain insights about maps $f^:H^0(\Omega_Y) \to H^0(f_\Omega_X)$ on coho and this is induced by $\Omega_Y→f_∗\Omega_X$, the adjoint of $f^\Omega_Y→\Omega_X$. Now, splitting at stalk of latter you observed (for $f$ smooth) is a very strong interesting property of the latter, so its natural to ask which implications about the structure of the former it gives, as its the former* which induce the map on cohomology I'm finally interested in – user267839 Jun 19 '25 at 12:55
  • @user267839 - Another question: In which examples are you able to explicitly calculate the induced map $f^*$ (at the cohomology level)? – hm2020 Jun 19 '25 at 12:59
  • Till now the examples I'm mainly familar with are the affine cases, eg $R[X_1,.., X_n] \to R[X_1,.., X_n]/(P)$ for some poly $P$ more general of type $R[X_1,.., X_n]/I$ for some idel $I$ ; my main "examples source" is Liu's AGAC Chap 6 – user267839 Jun 19 '25 at 13:08
  • @user267839 - In the above situation (when the schemes are affine) you get the following: The map $f^$ is the canonical map $f^: H^0(X, f^*\Omega^1_{Y/k}) \cong B \otimes_A \Omega^1_{A/k} \rightarrow H^0(X, \Omega^1_{X/k}) \cong \Omega^1_{B/k }$. You must then take the "adjoint" of this map. – hm2020 Jun 19 '25 at 13:19
  • @user267839 - People usually study the canonical map $B\otimes_A \Omega^1_{A/k} \rightarrow \Omega^1_{B/k}$ since this map is a map of $B$-modules. The adjoint map is a map of $A$-modules and one studies the $A$-module structure of $\Omega^1_{B/k}$. – hm2020 Jun 20 '25 at 10:26
  • The module $\Omega^1_{B/k}$ may be finitely generated as $B$-module and "infinitely generated" as $A$-module. This is why you should study the map of $B$-modules. – hm2020 Jun 20 '25 at 10:51
  • @user267839 - There are many problems arising when you are studying "infinite rank projective modules" and "infinite rank vector bundles": The Grothendieck group becomes trivial. Hence if you are studying Chern classes in the Grothendieck group you want to avoid studying vector bundles of infinite rank. – hm2020 Jun 22 '25 at 13:14