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3 fixed another error

Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more precisely, we should think of the category of quasi-coherent sheaves on the stack $M_{FG}$ as the "good approximation"). Every finite complex $X$ defines a module $MU_* X$ over the complex bordism ring $MU_*$, which is the Lazard ring classifying a universal formal group law; this is also a comodule over $MU_\ast MU$, which corresponds to strict automorphisms of a formal group law. Taking account of the grading lets one say "formal group" instead of "formal group law."

Let's say for instance that we have a two cell complex, with cells far away in dimensions from one another. Then this approach means that we get a two-dimensional vector bundle on $M_{FG}$. I'm wondering if we can understand these algebro-geometrically. For instance, line bundles on $M_{FG}$ can be understood (the Picard group is $\mathbb{Z}$, generated by the Lie algebra $\omega$ of a formal group), and perhaps 2-dimensional vector bundles are not too far off.

Here's the specific situation I have in mind. Let $\nu: S^3 \to S^0$ be the second (stable) Hopf map, which generates the 3-stem. Then the cofiber of $\nu$ is the desuspension $\Sigma^{-4} \mathbb{HP}^2$, and this (as an even, two-dimensional two cell complex) has free $MU$-homology. The homology of this corresponds to some vector bundle on $M_{FG}$, which is an extension of $\mathcal{O}$ and $\omega^4$. Alternatively, $\nu$ is detected in the 1-line of the ANSS as a class in $\mathrm{Ext}^1_{M_{FG}}(\omega^4, \mathcal{O})$, which has order twelve, I think. Is there a description of this class purely in terms of formal groups? I'd be interested as well in the pull-back of this bundle to $M_{1,1}$ (i.e., under the map $M_{1,1} \to M_{FG}$ sending an elliptic curve to its formal group.) More concretely, this describes cooperations in elliptic homology on $\mathbb{HP}^2$.

(As a simpler example, we can do this for $\eta$, the first Hopf map. Then we are looking at $\Sigma^{-2} \mathbb{CP}^2$, and the relevant bundle on $M_{FG}$ is the next order version of the Lie algebra. Given an even-periodic homology theory $E$, then $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to $\widetilde{E}^0(\mathbb{CP}^2)$, which corresponds to functions on the formal group of $E$ mod functions which vanish to degree 3 or higher at the origin. $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to this.)

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Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more precisely, we should think of the category of quasi-coherent sheaves on the stack $M_{FG}$ as the "good approximation"). Every finite complex $X$ defines a module $MU_* X$ over the complex bordism ring $MU_*$, which is the Lazard ring classifying a universal formal group law; this is also a comodule over $MU_\ast MU$, which corresponds to strict automorphisms of a formal group law. Taking account of the grading lets one say "formal group" instead of "formal group law."

Let's say for instance that we have a two-dimensional two cell complex, with cells far away in dimensions from one another. Then this approach means that we get a two-dimensional vector bundle on $M_{FG}$. I'm wondering if we can understand these algebro-geometrically. For instance, line bundles on $M_{FG}$ can be understood (the Picard group is $\mathbb{Z}$, generated by the Lie algebra $\omega$ of a formal group), and perhaps 2-dimensional vector bundles are not too far off.

Here's the specific situation I have in mind. Let $\nu: S^3 \to S^0$ be the second (stable) Hopf map, which generates the 3-stem. Then the cofiber of $\nu$ is the desuspension $\Sigma^{-4} \mathbb{HP}^2$, and this (as an even, two-dimensional complex) has free $MU$-homology. The homology of this corresponds to some vector bundle on $M_{FG}$, which is an extension of $\mathcal{O}$ and $\omega^4$. Alternatively, $\nu$ is detected in the 1-line of the ANSS as a class in $\mathrm{Ext}^1_{M_{FG}}(\omega^4, \mathcal{O})$, which has order twelve, I think. Is there a description of this class purely in terms of formal groups? I'd be interested as well in the pull-back of this bundle to $M_{1,1}$ (i.e., under the map $M_{1,1} \to M_{FG}$ sending an elliptic curve to its formal group.) More concretely, this describes cooperations in elliptic homology on $\mathbb{HP}^2$.

(As a simpler example, we can do this for $\eta$, the first Hopf map. Then we are looking at $\Sigma^{-2} \mathbb{CP}^2$, and the relevant bundle on $M_{FG}$ is the next order version of the Lie algebra. Given an even-periodic homology theory $E$, then $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to $\widetilde{E}^0(\mathbb{CP}^2)$, which corresponds to functions on the formal group of $E$ mod functions which vanish to degree 3 or higher at the origin. $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to this.)

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# Is there an algebro-geometric description of $\nu$?

Motivation: According to the "chromatic" picture of stable homotopy, we should think of the moduli stack $M_{FG}$ of formal groups as a "good approximation" to the stable homotopy category (more precisely, we should think of the category of quasi-coherent sheaves on the stack $M_{FG}$ as the "good approximation"). Every finite complex $X$ defines a module $MU_* X$ over the complex bordism ring $MU_*$, which is the Lazard ring classifying a universal formal group law; this is also a comodule over $MU_\ast MU$, which corresponds to strict automorphisms of a formal group law. Taking account of the grading lets one say "formal group" instead of "formal group law."

Let's say for instance that we have a two-dimensional complex, with cells far away in dimensions from one another. Then this approach means that we get a two-dimensional vector bundle on $M_{FG}$. I'm wondering if we can understand these algebro-geometrically. For instance, line bundles on $M_{FG}$ can be understood (the Picard group is $\mathbb{Z}$, generated by the Lie algebra $\omega$ of a formal group), and perhaps 2-dimensional vector bundles are not too far off.

Here's the specific situation I have in mind. Let $\nu: S^3 \to S^0$ be the second (stable) Hopf map, which generates the 3-stem. Then the cofiber of $\nu$ is the desuspension $\Sigma^{-4} \mathbb{HP}^2$, and this (as an even, two-dimensional complex) has free $MU$-homology. The homology of this corresponds to some vector bundle on $M_{FG}$, which is an extension of $\mathcal{O}$ and $\omega^4$. Alternatively, $\nu$ is detected in the 1-line of the ANSS as a class in $\mathrm{Ext}^1_{M_{FG}}(\omega^4, \mathcal{O})$, which has order twelve, I think. Is there a description of this class purely in terms of formal groups? I'd be interested as well in the pull-back of this bundle to $M_{1,1}$ (i.e., under the map $M_{1,1} \to M_{FG}$ sending an elliptic curve to its formal group.) More concretely, this describes cooperations in elliptic homology on $\mathbb{HP}^2$.

(As a simpler example, we can do this for $\eta$, the first Hopf map. Then we are looking at $\Sigma^{-2} \mathbb{CP}^2$, and the relevant bundle on $M_{FG}$ is the next order version of the Lie algebra. Given an even-periodic homology theory $E$, then $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to $\widetilde{E}^0(\mathbb{CP}^2)$, which corresponds to functions on the formal group of $E$ mod functions which vanish to degree 3 or higher at the origin. $\widetilde{E}_0(\mathbb{CP}^2)$ is dual to this.)