nLab moduli stack of tori




Inside the compactified moduli stack of elliptic curves, at the cusp point corresponding to the nodal cubic curve sits the moduli stack of one dimensional tori 𝔾 m\mathcal{M}_{\mathbb{G}_m} (Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2

𝔾 mB 2 \mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2

(Lawson-Naumann 12, prop. A.4). Here the 2\mathbb{Z}_2-action is the inversion involution on abelian groups.

Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf 𝒪 top\mathcal{O}^{top} (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a 2\mathbb{Z}_2-∞-action. This is KU with its involution induced by complex conjugation, hence essentially is KRKR.

Accordingly, the global sections of 𝒪 top\mathcal{O}^{top} over 𝔾 m\mathcal{M}_{\mathbb{G}_m} are the 2\mathbb{Z}_2-homotopy fixed points of this action, hence is KO. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).

As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at p=2p = 2) the inclusion KOKUKO \to KU as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.

moduli spaces


Discussion of KUKU with its 2\mathbb{Z}_2-action as the E-∞ ring-valued structure sheaf of the moduli stack of tori is due to

which is reviewed and amplified further in

Last revised on November 13, 2020 at 18:28:02. See the history of this page for a list of all contributions to it.