Kernel of morphism of Kähler modules

Question

Let $f:X\to Y$ be a morphism of $k$-schemes. Then we have a natural morphism of associated Kähler modules $\Omega_{Y}\to f_*\Omega_X$ derived by adjunction from the canonical morphism $f^*\Omega_Y\to \Omega_X$. If $f$ is smooth, then this morphism is injective. My question is if we can express the kernel of this morphism in general, that is when $f$ is not smooth. You would assume that this is measured by some deformation-theoretic object, presumably heuristically measuring how "unliftable" some unliftable morphisms are, but I'm unaware of such a thing.

Answer 1

Question: "My question is if we can express the kernel of this morphism in general, that is when f is not smooth."

Answer: If $Y:=Spec(A/I), X:= Spec(A)$ and $f:Y \rightarrow X$ is the canonical map and $I \neq (0)$ it follows $f$ is not smooth. There is an exact sequence ($B:=A/I$)

$$ I/I^2 \rightarrow^{\delta} B \otimes \Omega^1_{A} \rightarrow^{\phi} \Omega^1_B \rightarrow 0$$

where $\delta(\overline{x}):=1\otimes d(x)$ with $d: A \rightarrow \Omega^1_{A}$ the universal derivation. This gives a description of the kernel $ker(\phi):=Im(\delta)$.

Dually there is the Exalcomm functors of Grothendieck. For any homomorphisms of commutative rings $A → B → C$ and a $C$-module $L$ there is a long exact sequence

$$ 0\rightarrow \operatorname{Der}_{B}(C,L)\rightarrow \operatorname{Der} _{A}(C,L)\rightarrow \operatorname{Der}_{A}(B,L)\rightarrow $$

$$\operatorname{Exalcomm}_{B}(C,L)\rightarrow \operatorname{Exalcomm}_{A}(C,L)\rightarrow \operatorname{Exalcomm}_{A}(B,L) \rightarrow $$

I believe this sequence has a continuation. Keywords: Cotangent complex, Andre-Quillen cohomology and deformation theory. Ref: EGA IV Publ-IHES no. 20. The Exalcomm functors are defined using extensions by bimodules. If $A$ is an arbitrary (commutative ring) and $B$ is a (commutative) $A$-algebra with $I$ a $(B,B)$-module, define $\operatorname{Exalcomm}_A(B,I)$ as the set of equivalence classes of extensions

$$0 \rightarrow I \rightarrow E \rightarrow B \rightarrow 0$$

where $\pi:E \rightarrow B$ is an extension satisfying various criteria. You find details in the EGA IV-book. The construction is "category theoretical" in the sense that you view the groups as extensions in a suitable category. If all rings are over a field, there is an explicit construction of these groups in terms of Hochschild cohomology.

More generally: If $k \rightarrow A \rightarrow B$ is a sequence of maps of commutative rings there is for every $B$-module $M$ a 9 term exact sequence

$$T_1(A/k,M) \rightarrow T_1(B/k,M) \rightarrow T_1(B/A,M) \rightarrow^d $$

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

and $Im(d)$ is your sought after kernel.

(this follows from the paper on the cotangent compex of Schlessinger-Lichtenbaum, TAMS 128 1967 - this paper is recommended). This may answer your question. This construction is "more explicit".

You fins a discussion of this exact secuence for fields in Matsumura, "Commutative ring theory", in the sectionon on modules of differentials.