Derived Algebraic Geometry: Foundations

Job: define derived rings and their spectra, the cotangent complex, and quasi-coherent complexes; locate Toën–Vezzosi HAG and Lurie DAG/SAG.

Core

Derived rings

Two standard affine inputs:

Model Objects Prefer when
Simplicial commutative rings \(A\in s\mathrm{CRing}\); \(\pi_0A\) commutative ring; \(\pi_nA\) \(\pi_0A\)-modules classical AG intuition; Toën–Vezzosi HAG contexts
Connective \(\mathbb{E}_\infty\)-ring spectra \(A\in\mathrm{CAlg}^{\mathrm{cn}}\) spectral AG; chromatic / topological methods
Commutative DG-algebras (char \(0\)) \(A^\bullet\) coconnective or connective conventions vary explicit obstruction cochains

A simplicial commutative ring is a simplicial object in commutative rings (equivalently: product-preserving functors from finite sets, etc.). Weak equivalences: maps inducing isos on all \(\pi_*\). The \(\infty\)-category \(\mathrm{SCR}\) is the localization.

Postnikov tower. \(A\to \pi_0A\) with fibers controlled by \(\pi_nA[n]\); square-zero extensions and André–Quillen cohomology classify lifts.

Spectra of derived rings

Affine derived scheme (simplicial setting): \(\mathrm{Spec} A\) as a functor $$ \mathrm{SCR}\to\mathcal{S},\qquad B\mapsto \mathrm{Map}_{\mathrm{SCR}}(A,B), $$ or as a ringed \(\infty\)-topos / locally ringed space with underlying \(|\mathrm{Spec}\pi_0A|\) and structure sheaf with \(\pi_0\) the classical structure sheaf.

Notation \(\mathrm{Spec}^h\) / \(\mathbf{R}\mathrm{Spec}\). Some authors write \(\mathrm{Spec}^h A\) for the functor-of-points on simplicial rings (homotopical spectrum) to distinguish from classical \(\mathrm{Spec}\pi_0A\). Étale topology on derived affines uses maps that are étale on \(\pi_0\) with vanishing relative cotangent (or equivalent HA characterizations).

Derived scheme. \(\infty\)-sheaf on \(\mathrm{SCR}\) (étale) locally equivalent to \(\mathrm{Spec} A\), with suitable \(0\)-truncated atlas conditions; alternatively glued along Zariski/étale from affines in the model category of simplicial ringed spaces. Derived stack: Artin-like conditions on diagonal and smooth/étale atlas in the derived sense (HAG).

Classical Derived upgrade
\(\mathrm{Spec} R\) \(\mathrm{Spec} A\), \(A\in\mathrm{SCR}\)
\(X\times_S Y\) homotopy pullback (derived fiber product)
\(\Omega_{B/A}\) cotangent complex \(\mathbb{L}_{B/A}\)
\(\mathrm{QCoh}\) abelian \(\mathrm{QCoh}\) stable \(\infty\)-category

Cotangent complex

For a map \(A\to B\) in \(\mathrm{SCR}\) (or \(\mathbb{E}_\infty\)-rings), the cotangent complex \(\mathbb{L}_{B/A}\in\mathrm{Mod}_B\) is the derived functor of Kähler differentials: left derived of \(\Omega_{-/A}\) via free/simplicial resolutions.

Properties.

\[ \pi_0\mathbb{L}_{B/A}\cong \Omega_{\pi_0B/\pi_0A}\quad\text{(under mild hyp.)} \]
  • Transitivity cofiber sequence: \(\mathbb{L}_{B/A}\otimes_B C\to\mathbb{L}_{C/A}\to\mathbb{L}_{C/B}\).
  • Étale \(\Rightarrow\) \(\mathbb{L}_{B/A}\simeq 0\); smooth \(\Rightarrow\) \(\mathbb{L}_{B/A}\) perfect of Tor-amplitude \([0,0]\) (classical smooth) or appropriate amplitude in derived setting.
  • Square-zero extension by \(M\in\mathrm{Mod}_{\pi_0B}\): classified by maps \(\mathbb{L}_{B/A}\to M[1]\) in the derived category (obstruction theory).

Illusie. Classical cotangent complex for discrete rings already needs simplicial resolutions; derived AG makes the ambient geometry match that homotopical algebra.

Quasi-coherent complexes

For a derived scheme/stack \(X\), \(\mathrm{QCoh}(X)\) is the limit (as \(\infty\)-categories) of \(\mathrm{Mod}_A\) over affines \(\mathrm{Spec} A\to X\). It is a presentable stable symmetric monoidal \(\infty\)-category. Perfect complexes \(\mathrm{Perf}(X)\subset\mathrm{QCoh}(X)\) are dualizable objects.

Operation Meaning
\(f^*\) pullback of modules (monoidal)
\(f_*\) pushforward (right adjoint; derived)
\(\otimes\) derived relative tensor
\(\mathrm{RHom}\) internal Hom

On classical underived \(X\) with mild hypotheses, \(h\mathrm{QCoh}(X)\simeq D_{\mathrm{qc}}(X)\) recovers the usual derived category of quasi-coherent sheaves — see AG cohomology and derived categories.

Deformation and obstruction (package)

To extend a map \(f_0:T_0\to X\) along a square-zero extension \(T_0\to T\) with ideal \(M\):

  • Obstruction in \(\pi_0\,\mathrm{Map}(f_0^*\mathbb{L}_X, M[1])\) (global version: \(H^1(T_0,f_0^*T_X)\) classically).
  • Lifts form a torsor under \(\pi_0\,\mathrm{Map}(f_0^*\mathbb{L}_X,M)\) when the obstruction vanishes.
  • Higher \(\pi_*\) encode automorphisms and higher coherences — essential for derived moduli stacks of sheaves, maps, and complexes.

Toën–Vezzosi / Lurie pointers

Source Emphasis
Toën–Vezzosi HAG I–II model categories of prestacks; geometric stacks; Weil restrictions; \(D^-\)-stacks
Lurie DAG I–XIV (preprints) quasicategorical foundations; structured spaces
Lurie SAG spectral Deligne–Mumford stacks; proper base change; etc.
Gaitsgory–Rozenblyum ind-coherent sheaves; D-modules; \(\infty\)-categorical AG toolkit

Choosing a language. For algebro-geometric moduli in char \(0\), simplicial commutative / DG approaches dominate. For chromatic or brave-new inputs, use spectral AG. Results often compare after rationalization or on connective covers.

See it

Derived critical locus. For \(f:X\to\mathbb{A}^1\) smooth, the classical critical locus \(\mathrm{Crit}(f)=V(df)\) may be highly singular. The derived critical locus is the derived zero locus of \(df:X\to T^*X\), i.e.\ homotopy fiber product. Its structure sheaf’s \(\pi_*\) relates to the Koszul complex on partial derivatives; in \((-1)\)-shifted symplectic geometry (Pantev–Toën–Vaquié–Vezzosi) this object carries a natural shifted symplectic form — the foundation of Donaldson–Thomas virtual classes in a derived package.

Cotangent complex of a local complete intersection. If \(B=A/(f_1,\ldots,f_r)\) regular sequence, \(\mathbb{L}_{B/A}\simeq (I/I^2)[1]\) with \(I=(f_i)\) — recovers the classical Illusie computation; higher derived structure vanishes beyond amplitude \([0,1]\).

Check

Prove that for \(B=A[x_1,\ldots,x_n]\), \(\mathbb{L}_{B/A}\) is free on \(dx_i\) in degree \(0\).

Polynomial algebras are free in \(\mathrm{SCR}\) / free commutative; Kähler differentials are free on \(dx_i\), and free objects are cofibrant so underived \(\Omega\) computes \(\mathbb{L}\). (Precisely: a free simplicial resolution can be taken discrete.)

How does the derived fiber product fix Serre’s intersection multiplicity?

Serre’s \(\chi(\mathrm{Tor}_*^A(B,C))\) is \(\sum (-1)^i\mathrm{length}\,\pi_i(B\otimes^{\mathbb{L}}_A C)\) when \(\mathrm{Spec} B,\mathrm{Spec} C\subset\mathrm{Spec} A\) are complementary-dimensional local complete intersections — i.e.\ the Euler characteristic of the derived intersection’s structure sheaf, not of the classical \(\otimes\) alone.

Pitfalls

  • Connectivity conventions: connective vs coconnective DG-rings flip Postnikov vs Whitehead towers; match your reference.
  • Flatness: derived flatness \(\neq\) \(\pi_0\)-flatness automatically in all settings; check definitions (Lurie HA/SAG).
  • Artin conditions for derived stacks require cotangent complex of finite presentation / perfect amplitude — not every prestack is “geometric.”
  • \(t_0(X)\) non-reduced vs derived thickenings: nilpotents in \(\pi_0\) are classical; higher \(\pi_{>0}\) are homotopical and invisible to classical scheme theory.

Refs

  • Illusie, Complexe cotangent et déformations I–II
  • Quillen, Homotopical algebra; André, Homologie des algèbres commutatives
  • Toën–Vezzosi, HAG II; Toën EMS survey Derived algebraic geometry
  • Lurie, Spectral Algebraic Geometry; Kerodon excerpts

Next

DAG index · Ind-Coherent Sheaves · Derived Moduli & Perfect · Shifted Symplectic & DT · D-Modules & Crystals · AG schemes · AG cohomology · Higher categories · \(\infty\)-operads · Homological algebra · K-theory · Prismatic cohomology · Condensed mathematics — condensed/analytic rings extend animated rings analytically · Geometric Langlands (derived) — IndCoh on derived \(\mathrm{LocSys}\) is the spectral side · SAG / derived moduli — the derived moduli / the cotangent complex in the spectral setting