Derived Algebraic Geometry III: Commutative Algebra

By Jacob Lurie. 2007

75 citations recorded by the source corpus.

This paper describes a higher-categorical version of the theory of colored operads, giving applications to the study of commutative ring spectra.

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

View Paper: https://www.semanticscholar.org/paper/fa819b391964e32d8a67ecba14b69f258f16bd1e

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Citations in the corpus

Central extensions of smooth 2–groups and a finite-dimensional string 2–group (2009, background): “f algebras for the same 2-monad restricted to the bicategory of groupoids. All the 2-morphisms of Gpd⊗ are invertible, hence it ﬁts into the formalism of (∞,1)-categories as considered in [Lur09] and [Lur07], and we may therefore bring to bear the sophisticated machinery developed in those sources i”
Character sheaves on unipotent groups in positive characteristic: foundations (2008, methodology): “ditions for the functors (B.1). We skip the precise list of the compatibilities mentioned in (iii) (the reader can easily guess it). Instead, in §B.2.4 we give a deﬁnition of pre-TFT in the format of [Lur07]; the idea is to combine data (i)-(iii) into a single 2-functor. Remark B.8. In Pre-deﬁnition”
Twists of K-theory and TMF (2010, background): “etric monoidal ∞-category of spectra. Lurie constructs such an ∞-category from scratch ([DAGI] introduces an ∞-category of spectra, which is shown to be monoidal in [DAGII], and symmetric monoidal in [DAGIII]). Lurie shows that his ∞-category is equivalent to the symmetric monoidal ∞-category arisin”
Units of ring spectra and Thom spectra (2008, methodology): “an algebra in Alg(Stab(Gpd∞)),so θ(R) is a coﬁbrant-ﬁbrant algebra in SΣ.Then we have R-mod as in §7.4, and may construct θ(R)-mod def= N(θ(R)–modules in SΣ)◦, as above. According to Theorem 2.5.3 of [DAGII] we have an equivalence of ∞-categories R-mod ≃θ(R)-mod, (7.40) as desired. We summarize the ”
On the Classification of Topological Field Theories (2009, background): “ptly ignore it and return to the somewhat informal approach of §1. In the interest of space, we will gloss over (b) and (c) (for an extensive discussion of (c) in the case n= 1 we refer the reader to [16]). In §2.3, we will address question (e) by studying various ﬁniteness conditions in the setting”
Mixed Weil cohomologies (2007, background): “One could also avoid any complication by working directly with symmetric monoidal ∞-categories [Lur07].”
The congruence criterion for power operations in Morava $E$-theory (2009, methodology): “We will need the following analogue of Lazard’s theorem on flat modules over a ring, which is a variant of an observation of Lurie [ Lur07 , §4.6].”
Model structures on commutative monoids in general model categories (2014, methodology): “Indeed, we hope that this can generalize further to the so-called excellent model categories introduced in [25].”
On the motivic spectra representing algebraic cobordism and algebraic $K$-theory (2007, methodology): “ic cobordism or algebraic K-theory. This is important because the category of modules over an E∞ motivic spectrum Rinherits a symmetric monoidal structure, at least in the higher categorical sense of [16]. As a result, there is a version of derived algebraic geometry which uses E∞ motivic spectra as”
On the equivalence between Lurie's model and the dendroidal model for infinity-operads (2013, background): “This was already stated explicitly in the early installments of Lurie’s DAG-series [26] and later in his Higher Algebra [28].”