Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 1

For $A\to B\to C=B/I$, defineandThe first map is well-defined because $d(ij)=idj+jdi$ becomes zero after tensoring with $C$ when $i,j\in I$. The second is induced by the universal derivation $B\to C\stackrel{d}{\to }{\Omega }_{C/A}$ and is surjective because the elements $d\bar{b}$ generate ${\Omega }_{C/A}$.

The composite is zero since $\bar{i}=0$. Conversely, quotienting ${\Omega }_{B/A}{\otimes }_{B}C$ by the $di$ for $i\in I$ imposes exactly the relations needed for the derivation of $B$ to descend to $C$. The universal property of the Module of Kähler differentials therefore identifies that quotient with ${\Omega }_{C/A}$, proving the Conormal exact sequence for Kähler differentials

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Every $k$-derivation $D:K[t]\to M$ is uniquely determined by its restriction to $K$ and by $D(t)$. Conversely, a $k$-derivation $K\to M$ and an arbitrary element $m\in M$ extend uniquely byRepresenting this natural decomposition of derivations gives

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Present $L=K[t]/(f)$ by sending $t$ to $b$. Applying the Conormal exact sequence for Kähler differentials and part ii givesWriting $f(t)={\sum }_i{a}_i{t}^i$, the image of $1$, represented by $f$, isThis has the required form $(\ast ,f^{\prime }(b)dt)$.

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If $L/K$ is finite separable, the primitive element theorem writes it as $K(b)$ with $f^{\prime }(b)\ne 0$. The relation in part a has a nonzero $dt$ component, so projection along that relation gives an isomorphism

For an arbitrary simple finite extension, part a presents ${\Omega }_{L/k}$ as a quotient of a vector space of dimension ${\dim }_K{\Omega }_{K/k}+1$ by the image of a space of dimension at most one. Its dimension is therefore at least ${\dim }_K{\Omega }_{K/k}$. Applying this one generator at a time through a finite tower proves the same inequality for every finite extension.

Strict inequality occurs in characteristic $p$. Take $k=K={\mathbb{F}}_p(u)$ and $L=K(u^{1/p})$. Then ${\Omega }_{K/k}=0$, while the relation $b^p-u=0$ has zero differential relative to $k$, sohas dimension one.

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