By Bertrand Toen, Gabriele Vezzosi
This can be the second one a part of a sequence of papers known as "HAG", dedicated to constructing the rules of homotopical algebraic geometry. The authors begin through defining and learning generalizations of ordinary notions of linear algebra in an summary monoidal version type, reminiscent of derivations, etale and delicate morphisms, flat and projective modules, and so on. They then use their conception of stacks over version different types to outline a basic concept of geometric stack over a base symmetric monoidal version class $C$, and turn out that this inspiration satisfies the predicted homes.
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Extra info for Homotopical Algebraic Geometry II: Geometric Stacks and Applications
The natural point ∗ −→ S 1 induces a natural morphism in Ho(Comm(C)) A −→ T HH(A) making T HH(A) as a commutative A-algebra, and as a natural object in Ho(A − Comm(C)). 1. Let A be a commutative monoid in C. The topological Hochschild homology of A (or simply Hochschild homology) is the commutative Aalgebra T HH(A) := S 1 ⊗L A. 2. PRELIMINARIES ON LINEAR AND COMMUTATIVE ALGEBRA IN AN HA CONTEXT More generally, if A −→ B is a morphism of commutative monoids in C, the relative topological Hochschild homology of B over A (or simply relative Hochschild homology) is the commutative A-algebra T HH(B/A) := T HH(B) ⊗LT HH(A) A.
3) The model category M is compactly generated if it satisfies the following conditions. (a) The model category M is cellular (in the sense of [Hi, §12]). 8]) and ω-small with respect to the whole category M . (c) Filtered colimits commute with finite limits in M . The following proposition identifies finitely presented objects when M is compactly generated. 5. Let M be a compactly generated model category, and I be a set of generating cofibrations whose domains and codomains are cofibrant, ω-compact and ω-small with respect to the whole category M .
In other words, B⊕M can be seen as an object of the double comma model category A−Comm(C)/B. 1. Let A −→ B be a morphism of commutative monoids, and M be a B-module. The simplicial set of derived A-derivations from B to M , is the object DerA (B, M ) := M apA−Comm(C)/B (B, B ⊕ M ) ∈ Ho(SSetU ). Clearly, M → DerA (B, M ) defines a functor from the homotopy category of Bmodule Ho(B − M od) to the homotopy category of simplicial sets Ho(SSet). 2. PRELIMINARIES ON LINEAR AND COMMUTATIVE ALGEBRA IN AN HA CONTEXT lifting the previous functor on the homotopy categories.
Homotopical Algebraic Geometry II: Geometric Stacks and Applications by Bertrand Toen, Gabriele Vezzosi
Categories: Algebraic Geometry