Inside the thesis Jacob Lurie also developed fundamentals of derived algebraic geometry, when using the language of structured (infinity,1)-toposes where Toen -Vezzosi used model toposes. Also, he created a type of derived algebraic geometry that’s in your neighborhood modelled on E- rings. known as spectral algebraic geometry .
Terminology “derived” versus “ infty -”
The adjective “derived” means virtually such as the “ infty -” in -category. creating this greater algebraic geometry meaning being in your neighborhood symbolized by greater algebras. The term arises from using “derived” much like derived functor. This originated from the study into derived moduli problem. Namely to parametrize the moduli, one first examines some space of “cochains” that are candidates for structures to parametrize. another cuts individuals which indeed match the axioms (“equations”) for the structures. Satisfying these equations could be a limit-type construction, hence left exact the other is result in right derived functors to improve exactness across the right this leads to use cochain complexes. The acquired moduli is simply too big because there are many isomorphic structures, so you need to quotient using the automorphisms this really is frequently a colimit type construction hence right exact. The improved quotient may be the left derived functor, what’s acquired by passing to stacks.
structured ( . 1 ) (infty,1) -toposes
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Remark (derived plan are ( . 1 ) (infty,1) -toposes)
This definition draws on the observation that it’s inadequate the traditional idea of plan to demand that underlying an idea could be a topological space which a better definition is acquired by demanding it by getting a fundamental locale.
However a locale could be a -topos. This motivates your phrase a generalized plan as being a (in your neighborhood affine, structured ) (,1)-topos .
Comparison to its derived noncommutative geometry
Under some conditions, derived schemes X X idea of (Lurie, Structured Spaces ) are faithfully encoded by their stable (,1)-groups QCoh ( X ) QCoh(X) of quasicoherent sheaves. This can be truly the information of Tannaka duality for geometric stacks. (Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems ). So that you can change this around and think that a appropriate stable (,1)-category mathcal which isn’t within the form QCoh ( X ) QCoh(X) by having an actual derived plan X X represents a generalized, “noncommutative” derived plan. This resembles a couple of-category theory basically (,2)-category theory analog from the means by algebraic geometry the selection in the amount of monoids (algebras) is known as like every generalized spaces. This really is frequently (and possesses been) known as 2-algebraic geometry .
Accordingly, you are able to choose to regard the selection (,1)-quantity of appropriate (e.g. monoidal ) stable (,1)-groups like every “noncommutative derived schemes”. This is often effectively the perspective on noncommutative algebraic geometry that Maxim Kontsevich remains promoting.
In this way then in derived noncommutative algebraic geometry. a location is actually by definition a dg-category that’s smooth and proper in a appropriate sense.
The relation between noncommutative algebraic geometry and derived algebraic geometry is going to be summarized using the adjunction
Pf. DSt ( k ) op NCSp ( k ). Pf. DSt(k)^ rightleftarrows NCSp(k). mathcal_-