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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2013
    • (edited Dec 9th 2013)

    I am wondering about some basic issues in higher geometry and specifically in E-infinity geometry (β€œspectral geometry”), maybe somebody has a hint for me:

    • in Structured Spaces we get the very general formulation of what it means to have a structure sheaf π’ͺX. In Deformation Theory we get the very general formulation of what it means to be a module. One would hope that in this generality there is a definition of quasicoherent π’ͺX-modules, too. But at least in Quasi-Coherent Sheaves and Tannaka Duality Theorems the definition of QCoh is given in a more traditional way. Is there anything on QCoh away from E∞-geometry in a more general context as in Structured Spaces?

    • What can we say about sufficient conditions for morphisms f:Xβ†’Y in E∞-geoemtry to induce closed monoidal f*:QCoh(Y)β†’QCoh(X)? I don’t care for the moment so much about X and /or Y being regular (schemes, algebraic spaces, DM-stacks,…) but am looking for answers for X,Y generally ∞-presheaves on CRingop∞.

    • for E an E∞-algebra and X an ∞-groupoid, Pic(QCoh(Xβ‹…SpecE)) is closely related to the twisted generalized E-cohomology of the homotopy type of X. Joost and I are currently playing with rephrasing the constructions in his thesis in terms of six operations yoga in E∞-geometry under this identification. Part of this is at least implicit in A Survey of Elliptic Cohomology. Has this been developed/spelled out further anywhere?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 5th 2013

    I don’t know the answers, but I would like you just to have in mind that in generalized algebraic geometry of Nikolai Durov thesis where the local model is based on a finitary monad in Set (very fundamental and basic setup, everything commutative), the categories of quasicoherent sheaves of π’ͺ-modules are not forming an abelian category, hence the infinity analogue would not form a stable (infinity,1)-category. Sufficiently general treatment should have this in mind.

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