Weak $E_2$-Morita equivalences via quantization of the 1-shifted cotangent bundle

Abstract

In this paper, we investigate the structure of the convergent quantization of the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. The quantization is an $E_2$-algebra over the Frobenius twist $S^{\prime }$ of the 1-shifted cotangent bundle which restricted to the zero section $X^{\prime }\to S^{\prime }$ is weakly $E_2$-Morita equivalent to the structure sheaf ${\mathcal{O}}_{X^{\prime }}$ of the Frobenius twist $X^{\prime }$ of $X$. Explicitly, we show that the $(\infty ,2)$-category of coherent (left-)modules over ${\mathcal{O}}_{X^{\prime }}$ is equivalent to the full subcategory of the $(\infty ,2)$-category of coherent (left-)modules over the quantization restricted to the zero section generated by ${\mathcal{O}}_X$ are equivalent.

References in corpus (2)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• Shifted cotangent stacks are shifted symplectic