Descent spectral sequences through synthetic spectra

Christian Carrick, Jack Morgan Davies, Sven van Nigtevecht
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Abstract

The synthetic analogue functor $\nu$ from spectra to synthetic spectra does not preserve all limits. In this paper, we give a necessary and sufficient criterion for $\nu$ to preserve the global sections of a derived stack. Even when these conditions are not satisfied, our framework still yields synthetic spectra that implement the descent spectral sequence for the structure sheaf, thus placing descent spectral sequences on good footing in the $\infty$-category of synthetic spectra. As an example, we introduce a new $\mathrm{MU}$-synthetic spectrum $\mathrm{Smf}$.
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通过合成光谱获得后裔光谱序列
从光谱到合成光谱的合成类似函数 $\nu$ 并不保留所有极限。在本文中,我们给出了$\nu$保留派生堆栈全局截面的必要条件和充分条件。即使不满足这些条件,我们的框架仍然可以得到实现结构 sheaf 的下降谱序列的合成谱,从而使下降谱序列在$\infty$-类合成谱中站稳脚跟。作为一个例子,我们引入了一个新的$\mathrm{MU}$合成谱$\mathrm{Smf}$。
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