nLab 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 $sComm^{op}$, the opposite of the category of simplicial commutative unital rings. The category of simplicial presheaves on $sComm^{op}$ has several model category structures. If a projective model structure is used, this category of simplicial presheaves is denoted $SPr(dAff)$ where weak equivalences and fibrations are defined levelwise. By $dAff^{\hat{}}$ one denotes the left Bousfield localization of the model category $SPr(dAff)$ with respect to the Yoneda images $h_X\to h_Y$ of equivalences in $dAff$; this model category $dAff^{\hat{}}$ is called the model category of prestacks over $dAff$. The fibrant objects in $dAff^{\hat{}}$ are the simplicial presheaves $F:dAff^{op}\to sSet$ such that

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

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

The homotopy category $Ho(dAff^{\hat{}})$ is naturally equivalent to the full subcategory of $Ho(SPr(dAff))$ 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)