derived affine scheme

A derived affine scheme is a special kind of generalized scheme.

In a version of the theory of derived algebraic stack?s due to Toën, Vezzosi and Vaquie, the category of derived affine schemes is sComm opsComm^{op}, the opposite of the category of simplicial commutative unital rings. The category of simplicial presheaves on sComm opsComm^{op} has several model category structures. If a projective model structure is used, this category of simplicial presheaves is denoted SPr(dAff)SPr(dAff) where weak equivalences and fibrations are defined levelwise. By dAff ^dAff^{\hat{}} one denotes the left Bousfield localization of the model category SPr(dAff)SPr(dAff) with respect to the Yoneda images h Xh Yh_X\to h_Y of equivalences in dAffdAff; this model category dAff ^dAff^{\hat{}} is called the model category of prestacks over dAffdAff. The fibrant objects in dAff ^dAff^{\hat{}} are the simplicial presheaves F:dAff opsSetF:dAff^{op}\to sSet such that

  • (anodyne condition) for all XdAffX\in dAff, the simplicial set F(X)F(X) is fibrant; and

  • (prestack condition) for each equivalence XYX\to Y in dAffdAff, the induced morphism F(Y)F(X)F(Y)\to F(X) is a weak equivalence of simplicial sets.

The homotopy category Ho(dAff ^)Ho(dAff^{\hat{}}) is naturally equivalent to the full subcategory of Ho(SPr(dAff))Ho(SPr(dAff)) whose objects are the simplicial presheaves satisfying the above prestack condition.

  • B. Toën, Simplicial presheaves and derived algebraic geometry, lecture notes CRM, Barcelona 2008; (crm-2008.pdf)

Last revised on December 21, 2009 at 17:23:53. See the history of this page for a list of all contributions to it.