Flat resolutions of DG-schemes

Recall that a DG-scheme is a pair (X,OX), where (X,O0X) is a scheme, OX is a sheaf of commutative DG-algebras over (X,O0X), and each OiX is a quasi-coherent O0X-module. I would like to know if in this generality (without assuming things like X being quasi-projective over a field of characteristic zero), is it known that … Read more

Reference request: category of sheaves of O-modules with coherent cohomology

Suppose X is a smooth algebraic variety (say, in characteristic 0). It’s a folklore result that DbCoh(X) is equivalent to the derived category of complexes of sheaves of OX-modules whose cohomology is coherent. The only reference I could find are these notes, which are written in some exotic language I can’t parse. Does anyone know … Read more

Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety X of dimension n, let Hi(X) denote its integral Hodge structure of weight i. Define ~H0(X)=⨁H2i(X)⊗Z(i) and ~H1(X)=⨁H2i+1⊗Z(i), respectively. It is known that any derived equivalence Φ:Db(X)→Db(Y) induces isomorphisms of rational Hodge structures ~H0Q(X)≅~H0Q(Y) and ~H1Q(X)≅~H1Q(Y), however these isomorphisms are defined by characteristic classes whose coefficients aren’t necessarily integral. … Read more

Motivic Interpretation of Rationally Trivial Cycles

The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects this to be an isomorphism after tensoring appropriately. On the other hand, there seems to be no clear way to … Read more

Why do motivic stacks make sense?

In the paper “Motivic model categories and motivic derived algebraic geometry”, Yuki Kato, whose email-address I sadly couldn’t find out, describes a procedure to “motivy” the objects of any (∞,1)-category by doing the construction used to get motivic spaces over an arbitrary base. He then applies this construction to (∞,1)-Cat itself to get motivic categories … Read more

The lisse-etale site and derived algebraic geometry

If one reads say Olsson’s book on algebraic stacks or Laumon-Moret-Bailly. The lisse-etale topology is used to define quasi-coherent sheaves and the cotangent complex (or rather cutoff’s of the cotangent complex). Now it could well be that I did not read close enough, but my impression is that in Toen-Vezzosi’s Homotopical Algebraic Geometry II and … Read more

Global functions algebra of formal (infinity) groupoid associated to Lie (infinity) algebroid

I was wondering if there is a smooth (sophisticated) way to associate the algebra of global functions of formal groupoid associated to Lie-Rinehart algebra (considered as 1-stack) to its Chevalley-Eilenberg complex, that could be generalized to Lie algebroids and even Lie infinity algebroids. So far, (at least when a Lie-Rinehart A-algebra L is free as … Read more

Homotopy colimit description of stacks

Let F be an Artin stack. If p:X→F is an atlas for F, can we express F, in the ∞-category Shvet´ of higher stacks, as a homotopy colimit over the simplicial diagram … \rightrightarrows X \times_{F} X \rightrightarrows X in {\rm Shv} ^{\acute{et}}(k)? I ask this because we know that if U_{\bullet} \to X is … Read more

Examples of non-hypercomplete sheaves on affine schemes

Let A be a commutative ring and let O be a sheaf of E∞-ring spectra on SpecA such that π0O=OSpecA. Lurie provides a criterion when (SpecA,O) coincides with SpecO(SpecA), namely if the homotopy groups πnO are quasi-coherent sheaves on SpecA and O is hypercomplete (Spectral Algebraic Geometry, Proposition 1.6.1.1). To get a better understanding why … Read more

Does the ∞-category of Derived/Spectral schemes admit all colimits over constant diagrams?

In the case of ordinary schemes, all coproducts exist, so given any constant diagram $D_S:C\to \operatorname{Sch}$, the colimit over $D_S$ is isomorphic to the coproduct of $S$ over the connected components of $C$. In the homotopical setting, colimits over constant diagrams are much more interesting, representing tensors by homotopy types. Do all colimits over constant … Read more