By Bertrand Toen, Gabriele Vezzosi
This is often the second one a part of a sequence of papers known as "HAG", dedicated to constructing the principles of homotopical algebraic geometry. The authors begin by means of defining and learning generalizations of normal notions of linear algebra in an summary monoidal version class, resembling derivations, etale and delicate morphisms, flat and projective modules, and so forth. They then use their thought of stacks over version different types to outline a normal concept of geometric stack over a base symmetric monoidal version class $C$, and turn out that this suggestion satisfies the predicted houses.
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Extra info for Homotopical Algebraic Geometry II: Geometric Stacks and Applications
4. We will reproduce it for the reader convenience. We first consider the Quillen adjunction −⊗A B : A−Comm(C)/B −→ B−Comm(C)/B A−Comm(C)/B ←− B−Comm(C)/B : F, where F is the forgetful functor. This induces an adjunction on the level of homotopy categories − ⊗LA B : Ho(A − Comm(C)/B) −→ Ho(B − Comm(C)/B) Ho(A − Comm(C)/B) ←− Ho(B − Comm(C)/B) : F. e. non-unital commutative monoids in B − M od). The functor I takes a diagram of p s / / B to the kernel of p computed in the catcommutative monoids B C egory of non-unital commutative B-algebras.
The adjunction (LK, RI) is an equivalence. Proof. 1. Indeed, it implies that for any fibration in C, f : X −→ Y , which has a section s : Y −→ X, the natural morphism i s:F Y −→ X, where i : F −→ X is the fiber of f , is an equivalence. It also implies that the homotopy fiber of the natural morphism id ∗ : X Y −→ X is naturally equivalent to Y . These two facts imply the lemma. 1. DERIVATIONS AND THE COTANGENT COMPLEX 27 Finally, we consider a third adjunction Q : B − Commnu (C) −→ B − M od B − Commnu (C) ←− B − M od : Z, where Q of an object C ∈ B − Commnu (C) is the push-out of B-modules C ⊗B C µ /C / Q(C), • and Z sends a B-module M to the non-unital B-algebra M endowed with the zero multiplication.
We fix, once for all an object SC1 ∈ C, which is a cofibrant model for S(1) ∈ Ho(C). For any commutative monoid A ∈ Comm(C), we let 1 SA := SC1 ⊗ A ∈ A − M od be the free A-module on SC1 . It is a cofibrant object in A − M od, which is a model 1 is cofibrant in A − M od, but not in C unless for the suspension S(A) (note that SA A is itself cofibrant in C). The functor 1 SA ⊗A − : A − M od −→ A − M od has a right adjoint 1 , −) : A − M od −→ A − M od. 2 implies that SA ⊗A − is a left Quillen functor.
Homotopical Algebraic Geometry II: Geometric Stacks and Applications by Bertrand Toen, Gabriele Vezzosi