Spectral Algebraic Geometry: Foundations¶

Job: define $\mathbb{E}_\infty$-ring spectra and their spectra $\mathrm{Spec}^{\mathrm{Sp}}$, outline spectral schemes and Deligne–Mumford stacks, and relate SAG to derived AG (Lurie / Toën–Vezzosi).

Core¶

$\mathbb{E}_\infty$-ring spectra¶

An $\mathbb{E}_\infty$-ring is a commutative algebra object in the symmetric monoidal $\infty$-category $\mathrm{Sp}$ of spectra: $A\in\mathrm{CAlg}(\mathrm{Sp})$. Equivalently: algebra over an $\mathbb{E}_\infty$-operad in spectra ($\infty$-operads).

Structure	Meaning
Multiplication	$A\wedge A\to A$ associative/commutative up to coherent homotopy
Unit	$\mathbb{S}\to A$
$\pi_*A$	graded-commutative ring (signs); $\pi_0A$ commutative ring
Modules	$\mathrm{Mod}_A$ stable presentable $\infty$-category

Connective $\mathbb{E}_\infty$-rings: $\pi_{<0}A=0$. Eilenberg–MacLane $HR$ for ordinary commutative $R$ embeds discrete rings. Examples: $\mathbb{S}$, $KU$, $ko$, $MU$, $\mathrm{tmf}$, spherical group rings $\mathbb{S}[\pi]$, and $\mathbb{E}_\infty$-structures on $\mathrm{THH}$, etc.

Postnikov / connectivity. Maps of connective $\mathbb{E}_\infty$-rings analyzed by square-zero extensions classified by $\mathrm{Der}$ / cotangent complex in HA — parallel to DAG.

Affine spectral geometry: $\mathrm{Spec}^{\mathrm{Sp}}$¶

For $A\in\mathrm{CAlg}$, the spectrum $\mathrm{Spec} A$ (notation $\mathrm{Spec}^{\mathrm{Sp}} A$ when contrasting classical $\mathrm{Spec}$) is the spectral Deligne–Mumford stack / spectrally ringed $\infty$-topos representing $$ B\;\mapsto\;\mathrm{Map}_{\mathrm{CAlg}}(A,B) $$ on connective commutative rings (functor of points), equipped with a structure sheaf of $\mathbb{E}_\infty$-rings in the étale topology.

Étale morphisms. $A\to B$ is étale if $\mathbb{L}_{B/A}\simeq 0$ and $B$ is compact / finitely presented over $A$ in the appropriate sense (Lurie HA/SAG: equivalently $\pi_0A\to\pi_0B$ étale and $\pi_*B\cong\pi_*A\otimes_{\pi_0A}\pi_0B$). Zariski: localizations. These generate sites for descent.

Classical	Spectral
$\mathrm{Spec} R$, $R$ ring	$\mathrm{Spec} A$, $A\in\mathrm{CAlg}$
$\mathcal{O}$ sheaf of rings	structure sheaf valued in $\mathrm{CAlg}$
$R$-Mod	$\mathrm{Mod}_A$
Étale cover	spectral étale cover

Spectral schemes and DM stacks (pointer)¶

Spectral scheme (Lurie): spectrally ringed $\infty$-topos locally étale-equivalent to $\mathrm{Spec} A$ for connective $A$, with $0$-truncated / discrete underlying topos conditions matching classical schemes on $\pi_0$.

Spectral Deligne–Mumford stack: analogous to classical DM stacks — étale diagonal (representable by spectral algebraic spaces) and étale atlas by spectral schemes. Proper, smooth, flat morphisms defined via cotangent complex and $\pi_0$ properties.

Functor of points: geometric objects as accessible sheaves on $\mathrm{CAlg}^{\mathrm{cn}}$ for the étale topology, satisfying spectral analogues of Artin/DM conditions (SAG).

Notion	SAG reference role
Algebraic space	étale sheaf with schematic diagonal
DM stack	étale atlas; finite inertia flavour
Artin stack	smooth atlas (more general; spectral Artin in literature)
$\mathrm{QCoh}(X)$	limit of $\mathrm{Mod}_A$ over $\mathrm{Spec} A\to X$

Cotangent complex and deformations¶

For $A\to B$ in $\mathrm{CAlg}$, $\mathbb{L}_{B/A}\in\mathrm{Mod}_B$ as in HA: derived module of Kähler differentials. Controls:

étaleness / smoothness (perfect amplitude conditions);
square-zero extensions and obstruction classes in $\mathrm{Ext}^{1,2}$;
cotangent complex of spectral DM stacks for deformation theory of maps.

Same formal deformation package as DAG, with $\mathbb{E}_\infty$ cotangent.

Relation to derived algebraic geometry¶

Aspect	Spectral AG (SAG)	Derived AG (SCR / HAG)
Affine input	$\mathbb{E}_\infty$-rings	simplicial commutative rings
Smash / tensor	$\wedge$ in spectra	$\otimes$ on simplicial modules
Natural for	TMF, chromatic, THH/TC	classical moduli, Illusie, char $0$ DT
Embedding of rings	$HR$	discrete constant simplicial
Char $p$	power ops, $\mathbb{E}_\infty$-Frobenius	SCR Frobenius / animated rings

Comparison. There are functors relating connective $\mathbb{E}_\infty$-rings under $H\mathbb{Z}$ to simplicial commutative rings / animated rings; over $\mathbb{Q}$ the theories are closely aligned. Lurie’s DAG series and SAG carefully track when geometric conclusions transfer. Animated rings (π-finite simplicial commutative / condensed variants) mediate modern comparisons.

Nonconnective geometry. Allowing nonconnective $A$ is largely a spectral phenomenon (e.g.\ $KU$); classical and SCR-based DAG usually emphasize connective objects.

Quasi-coherent modules¶

For a spectral DM stack $X$, $\mathrm{QCoh}(X)=\lim_{\mathrm{Spec} A\to X}\mathrm{Mod}_A$ in $\mathrm{Pr}^L_{\mathrm{st}}$. Perfect complexes $=$ dualizable objects. Push-pull, projection formula, and base change hold under SAG’s proper/flat hypotheses — designed to recover and extend classical AG theorems (proper base change, coherent duality, …) in the spectral setting.

See it¶

Spectra of $HR$. $\mathrm{Spec}\, HR$ recovers the classical scheme $\mathrm{Spec}\, R$ as the underlying scheme, with structure sheaf the Eilenberg–MacLane sheaf. Modules over $HR$ are chain complexes of $R$-modules (stable Dold–Kan / HA), so spectral geometry over $HR$ specializes to derived geometry of ordinary schemes with stable $\mathrm{QCoh}\simeq D_{\mathrm{qc}}$.

Goerss–Hopkins–Miller / $\mathcal{O}^{\mathrm{top}}$. On the étale site of the moduli stack of elliptic curves, there is a sheaf of even-periodic $\mathbb{E}_\infty$-rings whose stalk at an elliptic curve $E$ is the elliptic spectrum associated to $E$ (Landweber, Goerss–Hopkins–Miller). Spectral DM geometry packages this sheaf; global sections give $\mathrm{Tmf}$. This requires honest $\mathbb{E}_\infty$-multiplications — the defining example of spectral (not merely simplicial) geometry.

Check¶

State the étale criterion for connective $\mathbb{E}_\infty$-maps $A\to B$ in one line.

$\mathbb{L}_{B/A}\simeq 0$ and finite presentation, equivalently $\pi_0A\to\pi_0B$ étale of commutative rings and $\pi_nB\cong\pi_nA\otimes_{\pi_0A}\pi_0B$ for all $n$ (no exotic higher homotopy).

Why is $\mathrm{Mod}_{\mathbb{S}}\simeq\mathrm{Sp}$ the ‘absolute’ QCoh of the spectral point?

The sphere is the initial $\mathbb{E}_\infty$-ring; $\mathrm{Spec}\,\mathbb{S}$ is final among spectral schemes in the connective functor-of-points sense; modules over $\mathbb{S}$ are spectra. Absolute geometric objects pull back along $\mathbb{S}\to A$.

Pitfalls¶

$\pi_*$ graded commutativity includes signs; evenness hypotheses appear constantly in chromatic geometry.
Smashing vs tensor of $\pi_0$: $\pi_*(A\wedge_R B)$ is not $\pi_*A\otimes_{\pi_*R}\pi_*B$ without flatness / Künneth hypotheses — use spectral sequences.
DM vs Artin: spectral DM stacks have étale atlases; many moduli problems need Artin-type smooth atlases (still active packaging).
Hypercompleteness of topoi of sheaves: descent for spectra may require hypercomplete sheaves (∞-topoi).

Refs¶

Lurie, Higher Algebra (CAlg, cotangent, modules)
Lurie, Spectral Algebraic Geometry
Toën–Vezzosi, HAG II (comparison mindset)
Goerss, Topological modular forms (survey); Lurie, A survey of elliptic cohomology

Next¶

SAG index · Elliptic Cohomology & TMF · Derived Moduli & Deformation · Spectral GAGA & Descent · Chromatic & Morava E-theory · Derived AG · DAG index · Higher categories · ∞-operads · ∞-topoi · Algebraic geometry · K-theory · Condensed mathematics — condensed $\mathbb{E}_\infty$-rings; analytic ring spectra · Motivic homotopy — motivic spectra / $\mathrm{MGL}$ as $\mathbb{E}_\infty$-ring spectra over schemes

Classical	Spectral
\(\mathrm{Spec} R\), \(R\) ring	\(\mathrm{Spec} A\), \(A\in\mathrm{CAlg}\)
\(\mathcal{O}\) sheaf of rings	structure sheaf valued in \(\mathrm{CAlg}\)
\(R\)-Mod	\(\mathrm{Mod}_A\)
Étale cover	spectral étale cover