用户: Scavenger/Fun Facts!/Automorphisms of the Category of Abelian Groups

Lemma 0.1. Let $A$ be a commutative ring with identity. Let $M$ and $N$ be $A$-modules such that $M{\otimes }_{A}N\cong A$. Then $M$ and $N$ are line bundles over $A$.

Proof. Let $h^M,h^N:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A$ denote the functors $\mathrm{Hom}(M,\bullet ),\mathrm{Hom}(N,\bullet )$, respectively. Then using the fact that tensor product is left adjoint to internal Hom, we see that $h^M\circ h^N,h^N\circ h^M$ are equivalent to $\text{id}$, thus they are equivalences, and we see that $M$ and $N$ are projective.By a similar reasoning we see that $\bullet \otimes M$ and $\bullet \otimes N$ are equivalences. Now choose a generator $u={\sigma }_{i=1}^n{m}_i\otimes n_i\in M{\otimes }_{A}N$ of $M{\otimes }_{A}N$ as an $A$-module, and let $M^{\prime }\subseteq M$ be the submodule generated by the elements $m_1,\cdots ,m_n$. Then $M^{\prime }\otimes N\to M\otimes N$ is surjective and injective (since $\bullet \otimes N$ is exact), and it is an isomorphism, which implies $M^{\prime }=M$ and we see that $M$ is finitely generated, thus finitely presented since it is projective. Similarly we see that $N^{\prime }$ is finitely presented. Now the lemma follows easily by localizing at prime ideals.

$\square$

Lemma 0.2 ([1] Theorem 7.7.2). Let $\varkappa$ be a strongly inaccessible cardinal. Let $A$ be a $\varkappa$-commutative ring. Let $f:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A$ be a functor that preserves $\varkappa$-small colimits and is $A$-linear. Then there is an $A$-module $P$ such that $f$ is equivalent to $\bullet \otimes P$.

Theorem 0.3. Let $\varkappa$ be a strongly inaccessible cardinal. Let $\mathrm{Ab}$ denote the category of $\varkappa$-small abelian groups. Let $f:\mathrm{Ab}\to \mathrm{Ab}$ be a functor that is an equivalence. Then $f$ is equivalent to ${\text{id}}_{\mathrm{Ab}}$.

Proof. Let $g$ be an inverse to $f$. We see that $f$ and $g$ are given as tensoring with abelian groups $M$ and $N$, respectively. Now composing $f$ and $g$, we see that tensoring with $M{\otimes }_{\mathbb{Z}}N$ is equivalent to $i{d}_{\mathrm{Ab}}$. Evaluating at the object $\mathbb{Z}$ we see that $M\otimes N\cong \mathbb{Z}$. So we see that $M$ and $N$ are line bundles over $\mathbb{Z}$, and thus are isomorphic to $\mathbb{Z}$.

$\square$

References

[1]	Wenwei Li, “代数学方法: 卷二”, available for download at https://wwli.asia/downloads/books/Al-jabr-2.pdf