Derived algebraic geometry

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ${\displaystyle \mathbb {Q} }$), simplicial commutative rings or ${\displaystyle E_{\infty }}$-ring spectra from

Introduction

Basic objects of study in the field are

${\displaystyle A\otimes ^{L}B}$, whose higher homotopy is higher Tor, whose

Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact ${\displaystyle d[x^{p}]=pd[x^{p-1}]}$[1]. This can be overcome by using simplicial algebras.

Derived geometry over arbitrary characteristic

Derived rings over arbitrary characteristic are taken as simplicial commutative rings because of the nice categorical properties these have. In particular, the category of simplicial rings is simplicially enriched, meaning the hom-sets are themselves simplicial sets. Also, there is a canonical model structure on simplicial commutative rings coming from simplicial sets.^[3] In fact, it is a theorem of Quillen's that the model structure on simplicial sets can be transferred over to simplicial commutative rings.

Higher stacks

It is conjectured there is a final theory of higher stacks which model homotopy types. Grothendieck conjectured these would be modelled by globular groupoids, or a weak form of their definition. Simpson^[4] gives a useful definition in the spirit of Grothendieck's ideas. Recall that an algebraic stack (here a 1-stack) is called representable if the fiber product of any two schemes is isomorphic to a scheme.^[5] If we take the ansatz that a 0-stack is just an algebraic space and a 1-stack is just a stack, we can recursively define an n-stack as an object such that the fiber product along any two schemes is an (n-1)-stack. If we go back to the definition of an algebraic stack, this new definition agrees.

Spectral schemes

Another theory of derived algebraic geometry is encapsulated by the theory of spectral schemes. Their definition requires a fair amount of technology in order to precisely state.^[6] But, in short, spectral schemes ${\displaystyle X=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}})}$ are given by a spectrally ringed ${\displaystyle \infty }$-topos ${\displaystyle {\mathfrak {X}}}$ together with a sheaf of ${\displaystyle \mathbb {E} _{\infty }}$-rings ${\displaystyle {\mathcal {O}}_{\mathfrak {X}}}$ on it subject to some locality conditions similar to the definition of affine schemes. In particular

1. ${\displaystyle {\mathfrak {X}}\cong {\text{Shv}}(X_{top})}$ must be equivalent to the ${\displaystyle \infty }$-topos of some topological space
2. There must exist a cover ${\displaystyle U_{i}}$ of ${\displaystyle X_{top}}$ such that the induced topos ${\displaystyle ({\mathfrak {X}}_{U_{i}},{\mathcal {O}}_{{\mathfrak {X}}_{U_{i}}})}$ is equivalent to a spectrally ringed topos ${\displaystyle {\text{Spec}}(A_{i})}$ for some ${\displaystyle \mathbb {E} _{\infty }}$-ring ${\displaystyle A_{i}}$

Moreover, the spectral scheme ${\displaystyle X}$ is called connective if ${\displaystyle \pi _{i}({\mathcal {O}}_{\mathfrak {X}})=0}$ for ${\displaystyle i<0}$.

Examples

Recall that the topos of a point ${\displaystyle {\text{Sh}}(*)}$ is equivalent to the category of sets. Then, in the ${\displaystyle \infty }$-topos setting, we instead consider ${\displaystyle \infty }$-sheaves of ${\displaystyle \infty }$-groupoids (which are ${\displaystyle \infty }$-categories with all morphisms invertible), denoted ${\displaystyle {\text{Shv}}(*)}$, giving an analogue of the point topos in the ${\displaystyle \infty }$-topos setting. Then, the structure of a spectrally ringed space can be given by attaching an ${\displaystyle \mathbb {E} _{\infty }}$-ring ${\displaystyle A}$. Notice this implies that spectrally ringed spaces generalize ${\displaystyle \mathbb {E} _{\infty }}$-rings since every ${\displaystyle \mathbb {E} _{\infty }}$-ring can be associated with a spectrally ringed site.

This spectrally ringed topos can be a spectral scheme if the spectrum of this ring gives an equivalent ${\displaystyle \infty }$-topos, so its underlying space is a point. For example, this can be given by the ring spectrum ${\displaystyle H\mathbb {Q} }$, called the Eilenberg–Maclane spectrum, constructed from the Eilenberg–MacLane spaces ${\displaystyle K(\mathbb {Q} ,n)}$.

Applications

• Derived algebraic geometry was used by Kerz, Strunk & Tamme (2018) to prove Weibel's conjecture on vanishing of negative K-theory.
• The formulation of the Geometric Langlands conjecture by Arinkin and Gaitsgory uses derived algebraic geometry.^[7]

See also

• Derived scheme
• Pursuing Stacks
• Noncommutative algebraic geometry
• Simplicial commutative ring
• Derivator
• Algebra over an operad
• En-ring
• Higher Topos Theory
• ∞-topos
• étale spectrum

Notes