Derived Algebraic Geometry

Job: extend schemes and stacks so that intersections, quotients, and deformation problems remember higher derived data (cotangent complexes, quasi-coherent complexes) — Toën–Vezzosi and Lurie frameworks.

Core

Classical schemes glue ordinary commutative rings. Derived AG glues derived rings (simplicial commutative rings, or connective \(\mathbb{E}_\infty\)-rings) so that, e.g., non-transverse intersections acquire a derived structure whose \(\pi_*\) records Tor. The result fixes formal deformation theory, virtual fundamental classes, and many moduli problems.

Layer Content
Affine \(\mathrm{Spec}\) of simplicial commutative / \(\mathbb{E}_\infty\)-rings
Geometry derived schemes / stacks; étale, smooth morphisms
Quasi-coherent \(\mathrm{QCoh}(X)\) stable presentable \(\infty\)-category
Calculus cotangent complex \(\mathbb{L}_{X/S}\); obstruction theory

Reading map

Page Focus
Foundations derived rings; \(\mathrm{Spec}^h\); cotangent complex; QCoh; TV / Lurie pointers
Ind-Coherent Sheaves \(\mathrm{IndCoh}\) vs \(\mathrm{QCoh}\); Grothendieck–Serre duality; crystals / \(D\)-modules via \(\mathrm{IndCoh}(X_{\mathrm{dR}})\); Orlov singularity / MF
Derived Moduli & Perfect \(\mathrm{Perf}_X\); perfect stacks / self-duality; Betti / de Rham / étale realization / six functors; mapping stack / loop / Hochschild
Shifted Symplectic & DT \(p\)-shifted symplectic (PTVV); \((-1)\)-shifted / DT / DT–PT–GV / MNOP; shifted Poisson / \(E_n\); mirror as shifted symplectic
D-Modules & Crystals de Rham stack / \(D\)-modules via \(\mathrm{IndCoh}(X_{\mathrm{dR}})\); Riemann–Hilbert / six functors; crystalline / prismatic; prismatic \(D\)-modules / Fargues–Fontaine

Dependency order

  1. Algebraic geometry (schemes, QCoh).
  2. Homological algebra + spectral sequences.
  3. Higher categories (presentable \(\infty\)-cats; \(\mathbb{E}_\infty\)).
  4. Affine derived geometry \(\to\) stacks \(\to\) applications (moduli, shifted symplectic).

Interfaces

Area Contact
AG foundations classical truncation \(t_0(X)\)
AG schemes underlying scheme / ringed space
Homological algebra \(D_{\mathrm{qc}}\) as \(h\mathrm{QCoh}\)
Higher categories language of \(\infty\)-sheaves
K-theory algebraic \(K\) of perfect complexes
Arithmetic geometry derived enhancements of moduli of bundles / shtukas (research frontier)
Prismatic cohomology \(\delta\)-rings / prisms as derived (animated) rings with Frobenius
Condensed mathematics analytic rings extend animated rings by a condensed structure sheaf (solid/liquid)

Notation (paper-standard)

  • \(s\mathrm{CRing}\), \(\mathrm{SCR}_k\); \(\pi_0A\), \(\pi_nA\) as \(\pi_0A\)-modules
  • \(\mathbb{L}_{B/A}\), \(\mathbb{T}_{B/A}=\mathrm{RHom}(\mathbb{L}_{B/A},B)\)
  • \(\mathrm{QCoh}(X)\), \(\mathrm{Perf}(X)\); \(t_0(X)\), \(\pi_i\mathcal{O}_X\)
  • HAG = Toën–Vezzosi; SAG = Lurie spectral AG

See it

Derived intersection. For closed immersions \(Y,Z\subset X\) of schemes, classical scheme-theoretic intersection is \(\mathrm{Spec}\) of \(\mathcal{O}_Y\otimes_{\mathcal{O}_X}\mathcal{O}_Z\). Derived intersection uses derived tensor \(\mathcal{O}_Y\otimes^{\mathbb{L}}_{\mathcal{O}_X}\mathcal{O}_Z\); if \(Y,Z\) are regularly immersed of complementary dimension, the quasi-smooth derived intersection carries a virtual fundamental class in Chow / \(K\)-theory matching Behrend–Fantechi / Li–Tian style constructions. Excess intersection formulas become statements about \(\mathrm{Tor}\) sheaves \(=\pi_i\) of the structure sheaf.

Check

When does the derived tensor product of rings reduce to the ordinary tensor?

When one factor is flat over the base (Tor-independent): \(\mathrm{Tor}_i^A(B,C)=0\) for \(i>0\) \(\Rightarrow\) \(\pi_0(B\otimes^{\mathbb{L}}_A C)\cong B\otimes_A C\) and higher \(\pi_i\) vanish if both discrete.

What is the classical truncation \(t_0\) of an affine derived scheme \(\mathrm{Spec} A\)?

\(\mathrm{Spec}(\pi_0 A)\) with the induced map \(\mathrm{Spec} A\to\mathrm{Spec}\pi_0 A\) realizing \(t_0\) as a closed immersion on underlying topological spaces (in the simplicial commutative setting).

Pitfalls

  • Simplicial commutative vs \(\mathbb{E}_\infty\): agree rationally and in char \(0\) for many purposes; diverge in char \(p\) (power operations, flatness notions).
  • Derived \(\neq\) spectral \(\neq\) DG: three related toolkits; check which reference uses which.
  • Underived moduli functors may be non-representable while derived enhancements are representable by derived stacks.

Refs

  • Toën–Vezzosi, Homotopical Algebraic Geometry II
  • Lurie, Spectral Algebraic Geometry; Derived Algebraic Geometry series
  • Toën, Derived algebraic geometry (EMS Surv.); Gaitsgory–Rozenblyum
  • Stacks Project (classical background)

Next

Foundations · Ind-Coherent Sheaves · Derived Moduli & Perfect · Shifted Symplectic & DT · D-Modules & Crystals · Algebraic geometry · Higher categories · Homological algebra · K-theory