Derived algebraic geometry

Over characteristic 0 many of the derived geometries agree since the derived rings are the same. ${\displaystyle E_{\infty }}$ algebras are just commutative differential graded algebras over characteristic zero. We can then define derived schemes similarly to schemes in algebraic geometry. Similar to algebraic geometry, we could also view these objects as a pair ${\displaystyle (X,{\mathcal {O}}_{X}^{\bullet })}$ which is a topological space ${\displaystyle X}$ with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so ${\displaystyle {\mathcal {O}}_{X}^{n}=0}$ for ${\displaystyle n>0}$. The sheaf condition could also be weakened so that for a cover ${\displaystyle U_{i}}$ of ${\displaystyle X}$, the sheaves ${\displaystyle {\mathcal {O}}_{U_{i}}^{\bullet }}$ would glue on overlaps ${\displaystyle U_{ij}}$ only by quasi-isomorphism.

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

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.

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.

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 ${\displaystyle K(\mathbb {Q} ,n)}$.

• Toën, Bertrand (2014-01-06). "Derived Algebraic Geometry". arXiv:1401.1044 [math.AG].
• Toën, Bertrand; Vezzosi, Gabriele (2004). "From HAG to DAG: derived moduli stacks". In Greenlees, J. P. C. (ed.). Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 131. Dordrecht: Kluwer Academic Publishers. pp. 173–216. ISBN 1-4020-1833-9. Zbl 1076.14002.
• Vezzosi, Gabriele (2011). "What is ...a derived stack?" (PDF). Notices Am. Math. Soc. 58 (7): 955–958. Zbl 1228.14004.

• Eugster, J.; Pridham, J.P. (2021-10-25). "An introduction to derived (algebraic) geometry". arXiv:2109.14594 [math.AG].

• Spectral algebraic geometry - Rezk
• Operads and Sheaf Cohomology - JP May - ${\displaystyle E_{\infty }}$-rings over characteristic 0 and ${\displaystyle E_{\infty }}$-structure for sheaf cohomology
• Tangent complex and Hochschild cohomology of E_n-rings https://arxiv.org/abs/1104.0181
• Francis, John; Derived Algebraic Geometry Over ${\displaystyle {\mathcal {E}}_{n}}$-Rings

• Lowrey, Parker; Schürg, Timo. (2018). Grothendieck-Riemann-Roch for Derived Schemes
• Ciocan-Fontanine, I., Kapranov, M. (2007). Virtual fundamental classes via dg-manifolds
• Mann, E., Robalo M. (2018). Gromov-Witten theory with derived algebraic geometry
• Ben-Zvi, D., Francis, J., and D. Nadler. Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry.
• Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Invent. Math., 211 (2): 523–577, arXiv:1611.08466, Bibcode:2018InMat.211..523K, doi:10.1007/s00222-017-0752-2, MR 3748313, S2CID 119165673

• Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4