Spectral Algebraic Geometry

Job: develop algebraic geometry on \(\mathbb{E}_\infty\)-ring spectra — spectral schemes and Deligne–Mumford stacks — as in Lurie’s SAG, and locate it relative to derived AG (simplicial commutative rings).

Core

Spectral algebraic geometry replaces commutative rings by \(\mathbb{E}_\infty\)-ring spectra and builds geometry via functors on connective (or all) commutative ring spectra, étale morphisms, and spectrally ringed \(\infty\)-topoi. It is the natural home for topological modular forms, chromatic methods in geometry, and “brave new” algebraic geometry.

Layer Content
Affine \(\mathrm{Spec}^{\mathrm{Sp}} A\) for \(A\in\mathrm{CAlg}=\mathrm{CAlg}(\mathrm{Sp})\)
Schemes spectral schemes; étale/Zariski descent
Stacks spectral Deligne–Mumford stacks (Lurie SAG)
QCoh modules over structure sheaf; perfect complexes
vs DAG simplicial commutative rings (Toën–Vezzosi / Lurie DAG) vs \(\mathbb{E}_\infty\)

Reading map

Page Focus
Foundations \(\mathbb{E}_\infty\)-rings; \(\mathrm{Spec}^{\mathrm{Sp}}\); spectral schemes/DM stacks; SAG vs DAG
Elliptic Cohomology & TMF elliptic cohomology; Landweber exactness; Goerss–Hopkins–Miller \(\mathcal{O}^{\mathrm{top}}\); \(\mathrm{tmf}/\mathrm{TMF}/\mathrm{Tmf}\); Witten genus
Derived Moduli & Deformation cotangent complex; spectral \(\mathrm{Bun}_G\); formal groups / chromatic; derived Artin; DT / shifted symplectic
Spectral GAGA & Descent spectral GAGA; étale / \(\ell\)-adic descent; proper base change / coherent duality; \(p\)-adic / prismatic realization
Chromatic & Morava E-theory chromatic filtration / convergence; Morava \(E_n\) / Lubin–Tate; \(K(n)\)-local; Lubin–Tate tower / local Langlands

Dependency order

  1. Algebraic geometry (schemes, étale morphisms).
  2. Higher categories + \(\infty\)-operads (\(\mathbb{E}_\infty\), modules).
  3. Derived AG (parallel story with SCR).
  4. Model categories optional for spectra models; then SAG applications (TMF, …).

Interfaces

Area Contact
Derived AG sibling formalism; comparison functors
Higher categories \(\mathrm{CAlg}\), \(\mathrm{Mod}_A\) presentable stable
∞-topoi spectrally ringed topoi
K-theory algebraic \(K\) of \(\mathbb{E}_\infty\)-rings; trace methods
Algebraic topology spectra, chromatic homotopy
Homological algebra \(\pi_*A\) as graded ring; \(\mathrm{Tor}\) spectral sequences

Notation (paper-standard)

  • \(\mathrm{Sp}\), \(\mathrm{CAlg}\), \(\mathrm{CAlg}^{\mathrm{cn}}\); \(A\in\mathrm{CAlg}\); \(\pi_*A\)
  • \(\mathrm{Spec} A\) / \(\mathrm{Spec}^{\mathrm{Sp}} A\); \(\mathrm{Spét}\), spectral étale site
  • \(\mathrm{QCoh}(X)\), \(\mathrm{Perf}(X)\); \(\mathbb{L}_{B/A}\) cotangent complex (HA/SAG)
  • SAG \(=\) Lurie Spectral Algebraic Geometry; HA \(=\) Higher Algebra

See it

Elliptic cohomology and TMF. The Goerss–Hopkins–Miller theorem builds a sheaf of \(\mathbb{E}_\infty\)-rings on the moduli stack of elliptic curves \(\mathcal{M}_{\mathrm{ell}}\); global sections (after derived/spectral enhancement) yield tmf / TMF. This is geometry valued in spectra that has no purely discrete-ring avatar — a flagship application of spectral AG, not merely of simplicial commutative rings.

Check

When do connective \(\mathbb{E}_\infty\)-rings and simplicial commutative rings agree?

Over \(\mathbb{Q}\) (rationally), and more generally there are comparison functors; in char \(0\) for many deformation problems they present equivalent affine geometries. In char \(p\), power operations and Frobenius data distinguish \(\mathbb{E}_\infty\) from SCR — spectral and simplicial theories diverge.

What is \(\pi_0\) of a spectral scheme?

The underlying classical scheme/stack obtained from \(\pi_0\) of the structure sheaf (with care for nonconnective rings); spectral nilpotents and higher homotopy live in \(\pi_{>0}\mathcal{O}\).

Pitfalls

  • Connective vs nonconnective: SAG treats both; nonconnective rings break some “scheme-like” intuition (negative \(\pi_*\)).
  • Flatness / étaleness: definitions use cotangent complex and \(\pi_0\) conditions (HA/SAG) — not just \(\pi_0\)-étale.
  • DAG naming collision: “derived” sometimes means SCR, sometimes any homotopical rings; “spectral” specifically means \(\mathbb{E}_\infty\)-spectra based.
  • Model dependence of spectra: EKMM, symmetric spectra, orthogonal spectra present the same \(\infty\)-category \(\mathrm{Sp}\).

Refs

  • Lurie, Higher Algebra; Spectral Algebraic Geometry
  • Lurie, DAG series (comparisons); Toën–Vezzosi HAG (simplicial side)
  • Goerss–Hopkins, moduli of \(\mathbb{E}_\infty\)-rings; Behrens–Lawson TMF notes
  • Mathew, surveys on \(\mathbb{E}_\infty\)-rings and spectral AG

Next

Foundations · Elliptic Cohomology & TMF · Derived Moduli & Deformation · Spectral GAGA & Descent · Chromatic & Morava E-theory · Derived AG · Higher categories · ∞-operads · ∞-topoi · Algebraic geometry · K-theory