nLab
derived moduli stack of objects in a dg-category

Given any dg-category 𝒞\mathcal{C}, there is an associated derived moduli stack 𝒞\mathcal{M}_\mathcal{C} which parametrizes the pseudo-perfect? dg-modules over 𝒞 op\mathcal{C}^{op}. When 𝒞\mathcal{C} is smooth? and proper?, 𝒞\mathcal{M}_\mathcal{C} classifies the objects of 𝒞\mathcal{C}.

In the case where 𝒞\mathcal{C} is the dg-category enhancing the derived category of a smooth proper scheme XX, the derived Artin stack? X\mathcal{M}_X is called the derived moduli stack of perfect complexes on XX.

Proposition (Toen-Vezzosi, 3.4). The functor :Ho(DGCat(k)) opDSt(k)\mathcal{M}_- : Ho(DGCat(k))^{op} \to DSt(k) admits a left adjoint

Pf:DSt(k)Ho(DGCat(k)) opPf : \DSt(k) \to \Ho(\DGCat(k))^{op}

which associates to a derived stack its dg-category of perfect complexes.

References

Last revised on February 2, 2014 at 01:12:08. See the history of this page for a list of all contributions to it.