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The graded ring $\pi_\ast^s$ of stable homotopy groups of spheres is a horrible ring. It is non-Noetherian, and nilpotent torsion outside of degree zero.

Question: What are some "toy models" for the ring $\pi_\ast^s$?

By this, I mean examples which illustrate particular pathologies of this ring. For example, if $k$ is a field, then the ring of dual numbers $k[x]/x^2$ exhibits the pathology of being non-reduced. But that barely scratches the surface of the bad properties that $\pi_\ast^s$ has.

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    $\begingroup$ We don't know so many things about this ring. For example, it has not been proved that one can find elements $x_1, x_2, \dots$ in positive degree such that the product of the first $n$ of these is nonzero for all $n$. The rings John Palmieri mentions have this property, but maybe our ring doesn't. $\endgroup$ Dec 1, 2023 at 23:04

3 Answers 3

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My favorite warmup example to the stable homotopy groups of spheres is the following differential graded algebra.


Let $A$ have the underlying ring $$ \Bbb Z[y] \otimes \Lambda[x], $$ a ring with a polynomial generator $y$ and an exterior generator $x$. Give these the gradings $|x| = 1$ and $|y| = 2$, making it graded-commutative; give it differential $d(x) = 0$ and $d(y) = x$. Then $d(y^k) = k y^{k-1} x$ and $d(y^k x) = 0$ by the Leibniz rule.

The homology groups satisfy $$ H_k(A) = \begin{cases} \Bbb Z&\text{if }k = 0,\\ \Bbb Z/n&\text{if }k = 2n-1,\text{ generated by }y^{n-1} x\\ 0&\text{otherwise.}\\ \end{cases} $$


Here are some parallels with the stable homotopy groups of spheres.

Finiteness. The bottom group is $\Bbb Z$ and all the higher groups are torsion.

Nilpotence. All elements in positive degree are nilpotent. (For a silly reason: if $\alpha$ is any element in positive degree, then $\alpha^2$ is in positive even degree and so it is zero.)

Periodicity. The homology groups exhibit image-of-$J$-esque behavior. If we have a $p$-torsion element $\alpha$ in degree $k$, then we tend to have another $p$-torsion element "$y^p \alpha$" in degree $k+2p$. If we have a $p^2$-torsion element $\beta$ in degree $k$, then we tend to have another $p^2$-torsion element "$y^{p^2} \beta$" in degree $k + 2p^2$. (And so on.) This happens despite the fact that $y$ is not an element in the ring $H_* A$ and so multiplication by it is not a well-defined operation.

Periodic self-maps. Just as with the stable homotopy groups of spheres, we can explain periodicity by looking at $A$-modules. For any $n$ we have a differential graded module $A/p^n$ fitting into an exact triangle $A \to A \to A/p^n$ ($A$ was flat, and so we can represent this by the levelwise quotient). This module does have a self-map $y^{p^n}: A/p^n \to A/p^n$ inducing the periodicity we see on homology of any $A$-module.

Toda brackets. We can still correspondingly interpret these periodicity operators as brackets. If $\alpha$ is $p^n$-torsion, then there is a Massey product $$ \langle y^{p^n - 1} x, p^n, \alpha\rangle $$ in degree $|\alpha| + 2p^n$. This also implements this periodicity.

Analogues of chromatic spectra. There is an $A$-algebra $B = \Bbb Z[y]$ -- the quotient by $x$ -- and a $B$-algebra $K = \Bbb Z/p[y^{\pm 1}]$. These play similar roles to the Brown-Peterson spectrum $BP$ and a Morava $K$-theory $K(n)$ in detecting these periodic phenomena. The analogue of the Adams-Novikov spectral sequence is a long exact sequence $$ \dots \to H_{*-1} (B \otimes^{\Bbb L}_A M) \to H_*(M) \to H_*(B \otimes^{\Bbb L}_A M) \to \dots $$ which comes from the short exact sequence $0 \to (x) \to A \to A/(x) \to 0$, equivalent to $0 \to \Sigma B \to A \to B \to 0$.

Nilpotence theorem. If $R$ is an $A$-algebra, then the kernel of the map $H_* R \to H_*(B \otimes^{\Bbb L}_A R)$ consists of nilpotent elements. (All elements in the kernel of $H_* R \to H_*(B \otimes_A^{\Bbb L} R)$ are, in fact, representable by multiples of $x$, and so they square to zero.)

Telescopic localization. We can localize with respect to $K$. Localization with respect to $K$ is represented by the telescopic localization $\lim_n (y^{-1} A/p^n)$. (The category of $K$-local modules even has extra elements in its Picard group!)

Et cetera.


If nothing else, one of the things that this example illustrates is that there can be examples which are quite regular (the ring $A$ is simple) but whose visible structure (the ring $H_* A$) is pathological. The pathological appearance is partly because there is subtext (the generator $y$) which is not immediately visible to us and has to be detected in other ways.

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You could say that I've made a living out of looking at the stable module category of a finite group (or rather its slight enlargement, the homotopy category of complexes of injective modules, $\mathsf{KInj}(kG)$) as a toy model for the stable homotopy category, which makes $H^*(G,k)$ the analogue of the stable homotopy ring. The Balmer spectrum is a bit more complicated than that of stable homotopy theory, but the fact that $H^*(G,k)$ is Noetherian simplifies a lot of things. Please look at my work with Carlson and Rickard, with Iyengar and Krause, and with Greenlees, for various aspects of this.

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As Dave Benson says, the Noetherian condition simplifies a lot of things. The derived category of a commutative ring satisfies many of the properties of the stable homotopy category. The derived category of, say, $k[x]/(x^2)$ is boring from the point of view of stable homotopy category, but the derived category of $k[x_1, x_2, x_3, ...]/(x_i^2)$ is more interesting and complicated. You can look at my paper "The Bousfield lattice for truncated polynomial algebras" with Bill Dwyer: we looked at it hoping to see of the complications that arise when you pass to the non-Noetherian setting. Amnon Neeman looked at a similar derived category in his paper "Oddball Bousfield classes."

tl;dr: try $k[x_1, x_2, x_3, ...]/(x_i^2)$, or as Neeman did, $k[x_2, x_3, x_4, ...]/(x_i^i)$ for toy examples.

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