| Résume | Work of Simpson, Drinfeld, and Bhatt–Lurie gives a "stacky" approach to several known cohomology theories for schemes, providing a parsimonious way of encoding these theories and their coefficients in terms of quasicoherent sheaves on stacks; specifically, each theory is determined by a single ring stack, via the formalism of "transmutation". This perspective has been particularly illuminating in the prismatic/syntomic setting, where it exposes subtle arithmetic information in terms of geometry. I'll explain joint work with Dhilan Lahoti showing that the stacks arising here are in fact moduli spaces of ring stacks, and even moduli spaces of ring structures on certain monoid stacks. Viewed through the lens of transmutation, this result states that the theory of syntomification is "defined over F_1".
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