Interlude II: Derived Geometry (2000)
In Serre’s formula, the chain complex
is seen as a linear object and lacks a direct geometric interpretation within the framework of Grothendieck’s scheme theory.
Derived algebraic geometry is built from the following facts:
The chain complex given by the derived tensor product
\[ A=R/I_{Diag}\otimes _R^{\textcolor{purple}{\mathbb {L}}}R/I_{Axis} \]is a ring-object in the chain complexes (\(\mathbf{cdga}\)).
(Toën-Vezzosi, Lurie, 2002) Any \(\mathbf{cdga}\) has a geometric interpretation as the (derived) ring of functions on a space. More precisely, for any cdga A we can assign a geometric object, \(\textcolor{blue}{\mathsf{Spec}(A)} \) a derived affine scheme, such that \(\text{Functions}( \textcolor{blue}{\mathsf{Spec}(A)} )=A\).
(Toën-Vezzosi, Lurie, 2002) Affine derived schemes can be glued up to quasi-isomorphisms of cdga’s.
Any derived scheme has an underlying classical scheme, obtained by discarding the higher homology groups of the cdga.
The collection of all derived schemes forms an \((\infty ,1)-\text{category}\) \(\mathbf{dSch}\) which contains the category of Grothendieck’s schemes as a full subcategory.
By construction, intersections in \(\mathbf{dSch}\) automatically account for Serre’s formula.