tmf Is Not a Ring Spectrum Quotient of String Bordism (1312.2440v4)

Abstract: This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $\pi_\mathrm{MO}\langle8\rangle$, the $\pi_\mathrm{MO}\langle8\rangle$-module $$\pi_\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)$$ is not (even abstractly) isomorphic to $\pi_\mathrm{tmf}{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $\pi(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $$\pi_(R/X)\otimes\mathbf{Q} \cong \mathrm{H}*(\mathrm{Tot}(\mathrm{K}(X)))\otimes\mathbf{Q},$$ where the right-hand side is the rational homology of the (total) Koszul complex of $X$, which is strictly bigger than $\pi(R)/(X)\otimes\mathbf{Q}$ unless $X$ is a $\pi_(R)\otimes\mathbf{Q}$-quasi-regular sequence. The result then follows from the fact that the kernel of the $p$-local Witten genus cannot be generated by a $\pi_*\mathrm{MO}\langle8\rangle\otimes\mathbf{Q}$-quasi-regular sequence.