Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

(0 2-category 2-category-theory abelian-categories accessible adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry anomalies arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory derived-geometry descent differential differential-cohomology differential-geometry duality education elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra general topology geometric geometric-quantization geometry goodwillie-calculus gravity group-theory higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory hypercovers index-theory infinity integration-theory k-theory kan lie lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal-logic model model-category-theory monoidal monoidal-category monoidal-category-theory morphism motives motivic-cohomology n-groups newpage nonassociative noncommutative noncommutative-geometry number-theory of operator operator-algebra order-theory philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject supergeometry symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory web

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 23rd 2012

    There’s plenty about differential refinement of ordinary cohomology on nLab. Can one also have analytic or holomorphic refinement?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2012
    • (edited Apr 23rd 2012)

    Much of sheaf cohomology in the context of algebraic/analytic geometry is by default (and sort of by definition of subject) cohomology with coefficients in sheaves of holomorphic/analytic functions. You might call this “holomorphic” or “analytic” cohomology for emphasis, though this is not common usage of terms (even though maybe one could argue that it should be).

    It is noteworthy that deep at the historical roots, this is the very origin of the notion of sheaf and eventually of algebraic geometry in the first place: as opposed to smooth functions, there are very few holomorphic functions globally, and hence people who were interested in these much earlier had to pass from naive geometry to (what we now call) topos theory in order to glue local models to something interesting. This is at its heart the reason why today some people think that \infty-category theory is intrinsically a topic of algebraic geometry, while differential geometers are still managing to fight it by constructing ever more technology for generalized smooth manifolds.