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

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

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

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