higher geometry / derived geometry
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For a scheme, analogous to how an -scheme is a scheme over , a -scheme is a scheme over the de Rham space of .
See also diffiety.
For a scheme, a -scheme is a scheme over the de Rham space of .
This definition makes sense in much greater generality: in any context of differential cohesion.
In the sheaf topos over affine schemes, an -affine -scheme is a commutative monoid object in the monoidal category of quasicoherent sheaves , which is equivalently the category of D-modules over :
This is (BeilinsonDrinfeld, section 2.3).
This is indeed equivalent to the above abstract definition
This appears as (Lurie, theorem, 0.6 and below remark 0.7)
The free -scheme on a given -scheme is the jet bundle of .
This is (BeilinsonDrinfeld, 2.3.2).
This fact makes -geometry a natural home for variational calculus.
The definition in terms of monoids in D-modules is in section 2.3 in
Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras
chapter 2, Geometry of D-schemes (pdf)
The observation that this is equivalent to the abstract definition given above appears in pages 5 and 6 of
Last revised on May 4, 2023 at 12:05:23. See the history of this page for a list of all contributions to it.