A functorial control of integral torsion in homology

Abstract: We study the integral torsion of the values of strict polynomial functors defined over the integers. We interpret some classical homological invariants as values of strict polynomial functors and therefore obtain estimates of the integral torsion of these invariants, including cancellation results.

• The paper establishes a functorial framework that effectively bounds integral torsion in homology using strict polynomial functors.
• It introduces explicit torsion bounds based on the interplay between weight and degree, applicable to classical and derived homological invariants.
• The methodology simplifies computations in stable and unstable homology by providing clear categorical and combinatorial insights for further research.

Functorial Bounds for Integral Torsion in Homology

Overview

The paper "A functorial control of integral torsion in homology" (1310.2877) develops a functorial framework for analyzing and bounding the integral torsion present in homological invariants that arise from strict polynomial functors over the integers. By leveraging structures from the theory of strict polynomial functors and relating their combinatorial invariants (such as degree and weight), the paper establishes explicit and computable torsion bounds in a wide range of homological constructions, generalizing previously ad hoc or case-specific results for classical objects like Eilenberg-Mac Lane spaces. It provides categorical, representation-theoretic, and combinatorial tools for rigorously controlling torsion phenomena that have obscured the computation and structural understanding of stable and unstable homology in several settings.

Strict Polynomial Functors, Weight, and Degree

The essential objects of study are strict polynomial functors $F$ from finitely generated projective modules over a commutative ring $R$ to $R$-modules, endowed with a weight $s$ (linked to their polynomial “order” or “homogeneity” in the sense of Friedlander and Suslin) and Eilenberg–Mac Lane degree $\deg(F)$. The degree, defined via the vanishing of cross-effects, is a purely categorical measure of how the functor interpolates between being additive and exhibiting full nonlinearity. The category of homogeneous strict polynomial functors of weight $s$—denoted $\mathscr{P}_{s,R}$—admits rich algebraic and combinatorial structure, explicit decompositions, and strong vanishing and torsion phenomena.

A foundational result, Proposition 1.1, shows that if the degree of a strict polynomial functor is strictly less than its weight, then all values of the functor are torsion abelian groups. This criterion provides a direct functorial mechanism for distinguishing between those functors whose images may host free abelian or high-order torsion summands and those for which all homological content is torsion.

Main Theorems: Torsion Bound Classification

The core structural advance lies in Theorem 1.2, which provides a sharp functorial bound on the torsion present in the values of a $p$-primary strict polynomial functor $F$. Fixing integer weight $s$ and a prime $p$, the degree $d := \deg(F)$ of $F$ must belong to an explicitly described interval $I(p, s)$, governed by the $p$-adic expansion of $s$. Furthermore, if $d < s$, the order of torsion appearing in each value of $F$ is bounded above by $p^r$, where $r = \lceil (d+1 - E_p(s))/(p-1)\rceil$ and $E_p(s)$ is the sum of digits of $s$ in base $p$.

These bounds generalize previously sporadic results by Cartan, Dold, and Puppe and demonstrate that the interplay between weight and degree reveals deep obstructions to the existence of non-torsion elements in natural functorial homology theories. The method applies by expressing homological invariants (such as derived functors, Ext and Tor in functor categories, or stable homology groups of spaces and groups) as functorial values of strict polynomial functors, then transferring torsion estimates from the algebraic structure of the functor to the desired invariants.

Applications

The developed framework is applied to classical and contemporary topics as follows:

Taylor Towers of Functors

Functor-calculus approximations (the Taylor towers à la Johnson-McCarthy) are interpreted for strict polynomial functors, and the $i$-th homotopy group of the $n$-th Taylor approximation $P_nF$ inherits torsion bounds functorially (cf. Theorem 4.8). In particular, if $n < E_p(s)$, the $p$-primary part vanishes, and for larger $n$, the torsion is sharply bounded as above.

Stable Homology and Derived Functors

The results offer a clear, transparent explanation for why only $p$-torsion appears in the stable homology of Eilenberg–Mac Lane spaces, directly connecting the weights and degrees of functors representing these invariants. For example, for derived functors $L^iF(A; n)$, when $F$ is strict polynomial, the explicit bounds on torsion are established for the entire range $i < ns$, where the construction is effective.

Functor Cohomology (Ext and Tor)

Cohomology groups such as Ext and Tor between polynomial functors, particularly in categories of strict polynomial functors, are shown to inherit functorial degree restrictions, which translate to global torsion bounds (Theorem 4.17). Notably, the Ext groups $\mathrm{Ext}^\ast(\mathrm{Id}, F)$—the so-called Mac Lane cohomology—demonstrate complete vanishing or are explicitly computed as $\mathbb{F}_p$-vector spaces in cases where $s$ is a power of a prime $p$.

Cohomology of Algebraic Groups

The cohomology of representations of reductive algebraic groups with coefficients in values of strict polynomial functors is controlled functorially (Theorem 4.20). For free $R$-modules $M$ with good filtration under $G$, the torsion in $H^\ast(G, F(M))$ is bounded by the combinatorics of weight and $p$-adic structure, generalizing classical vanishing results.

Module-Theoretic Translation: Schur Algebras

The work provides module-theoretic translations of the main results via Schur algebras, connecting the combinatorics of weight and degree for strict polynomial functors with weights and length of weight spaces in modules. The result, Theorem A.4, gives an explicit criterion in terms of weights for when a module over a Schur algebra is torsion and gives explicit $p$-power bounds tied to weight length.

Implications and Directions

The functorial methods developed circumvent the need for intricate ad hoc calculations of torsion subgroups in unstable and stable homology, giving instead precise categorical and combinatorial criteria. They unify torsion phenomena across classical invariants (symmetric products, Lie functors, group (co)homology) and modern constructions (functor cohomology, Ringel duality, higher Ext and Tor in functor categories).

Practically, the paper offers effective bounds and vanishing criteria that facilitate explicit calculation with much reduced computational complexity, particularly over integral coefficients.

Theoretically, the results suggest that further structural phenomena in functor categories (e.g., for representations of other types of algebras or in derived settings) could be approached via similar combinatorial control of functor invariants. Extensions to non-strict polynomial or “generalized” polynomial functors and exploration of sharpness, or potential refinement, of the torsion bounds are natural lines of inquiry.

Conclusion

This work establishes a general and highly effective method for functorial control of integral torsion in homology of algebraic and topological objects expressible via strict polynomial functors. By relating categorical invariants (weight and degree) to arithmetic and combinatorial torsion bounds, it offers a unified perspective, strong vanishing criteria, and explicit calculations for a wide family of homological invariants, with significant implications for both representation theory and homotopical algebra.