Finite-generation, torsion, and p-primary Serre transfer

Subgroups, quotients, and extensions preserve finiteness, torsion, and $p$-primary torsion directly; for a torsion element in an extension, multiply first into the kernel and then kill it there. For finite generation, [F2] writes the ambient group as ${\mathbb{Z}}^r\oplus T$ with $T$ finite. A subgroup of $\mathbb{Z}$ is generated by its least positive element, and induction on coordinate projection shows every subgroup of ${\mathbb{Z}}^r$ is finitely generated; adjoining the finite intersection with $T$ proves the subgroup clause. Quotients preserve chosen generators, and lifts of finitely many quotient generators together with generators of the kernel prove extension closure. Thus each property defines a Serre class.

By [F2], tensor products of two finite or finitely generated groups are finite or finitely generated. Additivity and [F3] give the same conclusion for Tor, since a free cyclic summand contributes zero Tor and two finite cyclic summands contribute a finite cyclic group. Hence these two classes are Serre rings. They are not ideals: for $D={⨁}_{j\ge 0}\mathbb{Z}\mathrm{/}2$, the group $(\mathbb{Z}\mathrm{/}2)\otimes D\cong D$ is infinite, and $\mathbb{Z}\otimes D\cong D$ is not finitely generated.

If $A$ is torsion and $B$ arbitrary, every element of $A\otimes B$ is a finite tensor sum and is killed by a common positive integer. If $A$ is $p$-primary, a common power of $p$ works. Resolving $B$ and forming $A\otimes P_{\bullet }$ gives chain groups with the same property; their homology, including ${\mathrm{Tor}}_1(A,B)$, retains it by subgroup and quotient closure. Therefore the torsion and $p$-primary classes are Serre ideals.

Fix $0<i\le N$. For $s+t=i$, the axes of [F4] are the supplied base or fiber groups. If $s,t>0$, [F5] expresses $E_{s,t}^2$ as an extension of $H_s(B)\otimes H_t(F)$ by ${\mathrm{Tor}}_1(H_{s-1}(B),H_t(F))$. When $s=1$, the latter term is zero because $H_0(B)=\mathbb{Z}$; otherwise both factors have the selected property. Steps 1.1–1.3 put both terms in the associated Serre class, and the axes are supplied directly: $E_{s,0}^2=H_s(B)$ and $E_{0,t}^2=H_t(F)$ for $s,t>0$ by [F4]. Hence every $E_2$ term with $1\le p+q\le N$ belongs to that class. The transfer argument of [F6] applies on this positive-total-degree region: the isolated term $E_{0,0}^2$ has no outgoing differential in the first quadrant, so it contributes only to $H_0$, while every finite filtration quotient of $H_i(E)$ with $i\ge 1$ is a subquotient of an $E_2$ term of total degree $i$ and therefore lies in the class. Hence $H_i(E;\mathbb{Z})$ has property $\mathsf{P}$ for every $0<i\le N$.

For torsion and $p$-primary torsion, step 1.3 supplies precisely the Serre-ideal hypothesis of clause 1 in [F7]. Substitution gives sharp edge form 1 with only the fiber-range assumption. For all four classes, steps 1.1–1.3 supply the Serre-ring hypothesis of clause 2 in [F7]; substitution with its displayed base and fiber ranges gives sharp edge form 2. The counterexamples in step 1.2 explain why the finite and finitely generated cases are not inserted into the ideal conclusion.

For $N=0$ the positive-degree total-space assertion is empty and edge form 1 is the ordinary $H_0$ isomorphism. For $n=2$, the fiber range in edge form 2 is empty. The zero group is finite, finitely generated, torsion, and $p$-primary; the one-summand cyclic calculations and empty finite decomposition are covered by [F2]–[F3]. Degenerate tensor sums are zero. Both tensor and Tor terms, both axes of the $E_2$ triangle, both edge kernels and cokernels, all four properties, and the failed ideal endpoints are explicit. AC is used in [F3] and [F5], while all closure and finite-filtration deductions add none. No converse and no out-of-range conclusion is asserted.