Homotopical Algebraic Geometry II: Geometric Stacks and Applications

By Bertrand Toën, G. Vezzosi. 2004

562 citations recorded by the source corpus.

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17f1f5ca7e6dfa5078ea33ea0382bb7d15ed00f9

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Citations in the corpus

Algebraic K-theory and descent for blow-ups (2016, background): “me notation for derived schemes and to recollect some facts about their K-theory that we will need. 2.1. Simplicial rings and derived schemes. Basic references for the material of this subsection are [TV08] or [Lur16b]. An overview is [To¨e14]. Let sRing be the category of simplicial commutative rin”
Duality for topological modular forms (2011, background): “[TV08] , Homotopical algebraic geometry.”
Higher Hochschild Homology, Topological Chiral Homology and Factorization Algebras (2010, background): “For a (derived) stack Y, we denote OY its functions [32] (i.”
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (2008, methodology): “Trying to geometrically describe algebraic operations on ∞-categories of quasicoherent sheaves quickly takes us from ordinary algebraic geometry to an enhanced version which has been developed over the last few years known as derived algebraic geometry [L1, L3, L4, L5, ToVe1, ToVe2 ].”
The basic geometry of Witt vectors. II: Spaces (2010, unknown): “This follows from Toën–Vezzosi [40], Corollary 1.3.3.5, as does the fact that this category is closed under finite limits.”
Loop spaces and connections (2010, methodology): “For more details on derived algebraic geometry, we refer the reader to Toën’s extremely useful survey [31], and to the papers of Toën-Vezzosi [32, 33] and Lurie [21–23].”
On the algebraic K‐theory of higher categories (2012, methodology): “Much of what we will say can probably be done in other contexts of derived algebraic geometry as well, such as [71, 72]; we have opted to use Lurie’s approach only because that is the one with which we are least unfamiliar.”
The basic geometry of Witt vectors, I: The affine case (2008, methodology): “spaces and separatedness. Following To¨en–Vaqui´e [39], let us say a space is algebraic if it is geometric for the context (AﬀR,´et,´et). As remarked there, it follows from the work of To¨en–Vezzosi ([40], §1.3.3) that the full subcategory of SpS consisting of algebraic spaces is the closure of the ”
A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications (2013, methodology): “Classical K-schemes and Artin K-stacks will be written W,X, Y, Z, . . . , and derived K-schemes and derived Artin K-stacks in bold asW ,X,Y ,Z, . . . . Basic references for K-schemes are Hartshorne [12], for Artin K-stacks Laumon and Moret-Bailly [21], and for derived K-schemes and derived Artin Kst”
Sheaves of categories and the notion of 1-affineness (2013, background): “n X. We also consider the symmetric monoidal (∞,1)-category Sh(X)of k X-modules. • a derived Artin stack is a derived stack which is m-geometric (for some m) w.r.t. the class of smooth morphisms (see [19]) and which is locally of ﬁnite presentation (this is a bit more restrictive than the usual Arti”
Model structures on commutative monoids in general model categories (2014, background): “uction In recent years, the importance of monoidal model categories has been demonstrated by a number of striking results related to structured (equivariant) ring spectra, c.f. [9], [22], [17], [28], [41]. Commutative monoids played a key role in many of these applications, and it became important t”