Serre classes, Serre rings, ideals, and modulo-C morphisms

A Serre class $\mathcal{C}$ of abelian groups contains the zero group and is closed under subgroups, quotient groups, and extensions. Equivalently, for every short exact sequence $0⟶A⟶B⟶C⟶0,$ the middle group $B$ lies in $\mathcal{C}$ if and only if both end groups do. Membership is understood up to isomorphism.

A Serre class is a Serre ring if, whenever $A,B\in \mathcal{C}$, both $A{\otimes }_{\mathbb{Z}}B\in \mathcal{C}\text{and}{\mathrm{Tor}}_1^{\mathbb{Z}}(A,B)\in \mathcal{C}\mathrm{.}$ It is a Serre ideal if the same two conclusions hold whenever just one of $A,B$ belongs to $\mathcal{C}$ and the other is an arbitrary abelian group. Here ${\mathrm{Tor}}_1^{\mathbb{Z}}(A,B)$ denotes any supplied standard Tor group; the property is invariant under its canonical isomorphisms. This definition itself neither selects projective resolutions nor invokes a choice principle.

For a homomorphism $f:A\to B$:

• $f$ is a $\mathcal{C}$-monomorphism if $\ker f\in \mathcal{C}$;
• $f$ is a $\mathcal{C}$-epimorphism if $\mathrm{coker}f\in \mathcal{C}$;
• $f$ is a $\mathcal{C}$-isomorphism if it has both properties; and
• $A=0(\mathrm{m}\mathrm{o}\mathrm{d}\mathcal{C})$ means $A\in \mathcal{C}$.

These definitions include the zero class, the class of all abelian groups, zero homomorphisms, and maps with zero source or target.

For composable $A\stackrel{f}{\to }B\stackrel{g}{\to }C$, the standard kernel-cokernel sequence

is exact by the element construction in The Snake Lemma for modules. Subgroups, quotients, and extensions in (1) show that $\mathcal{C}$-isomorphisms are closed under composition and satisfy two-out-of-three. The same sequence proves composition closure separately for $\mathcal{C}$-monomorphisms and $\mathcal{C}$-epimorphisms.

The following qualifications are part of the convention:

• finite abelian groups and finitely generated abelian groups form Serre rings, but not Serre ideals;
• torsion abelian groups and $p$-primary torsion abelian groups form Serre ideals, hence Serre rings; and
• finite $p$-primary abelian groups form a Serre ring but not a Serre ideal.

For the finiteness assertions, finite presentations reduce tensor and Tor to kernels and cokernels of maps between finite or finitely generated groups. For torsion and $p$-primary torsion, every tensor is a finite sum of elementary tensors, so one common integer, respectively one power of $p$, kills it; the same statement holds in the homology of a supplied tensor-resolution complex. The ideal failures are witnessed explicitly: if $D={⨁}_{j\ge 0}\mathbb{Z}\mathrm{/}n$, then $(\mathbb{Z}\mathrm{/}n)\otimes D\cong D$ is infinite, so finite groups are not an ideal, while $\mathbb{Z}\otimes D\cong D$ is not finitely generated, so finitely generated groups are not an ideal. The zero, one-summand, empty direct-sum, and $n=1$ cases reduce to the zero group. These are closure statements, not biconditionals characterizing finite, torsion, or finitely generated groups.

Miller, Lecture 30, printed pp. 104–107, gives the short-exact-sequence definition, the modulo-$\mathcal{C}$ morphisms, Lemma 30.6, and the tensor-and-Tor definitions of Serre ring and ideal. Miller explicitly says that all listed examples are rings and those without finiteness conditions are ideals.