Spectral GAGA, Descent, and Base Change¶

Job: develop the spectral GAGA comparison (algebraic vs analytic $\mathrm{QCoh}$), the étale / $\ell$-adic descent, the spectral proper base change and coherent duality, and the $p$-adic / prismatic / cyclotomic realizations — with the spectral GAGA for $\mathrm{Spec}\,H\mathbb{Z}$ and $\mathrm{tmf}\to\mathrm{tmf}\otimes\mathbb{C}$ as the worked example.

Core¶

The spectral GAGA theorem¶

The classical GAGA (Serre) compares the algebraic coherent sheaves on a projective complex variety with the analytic coherent sheaves. The spectral GAGA (Lurie, the $\mathbb{E}_\infty$-version): for a proper / projective spectral scheme $X$ over $\mathbb{C}$, the comparison between the spectral algebraic $\mathrm{QCoh}(X)$ and the spectral analytic $\mathrm{QCoh}(X^{\mathrm{an}})$ (the analytic / complex-analytic enhancement via the derived / spectral analytic spaces).

GAGA comparison	Algebraic	Analytic
Coherent sheaves	$\mathrm{Coh}(X)$ (spectral algebraic)	$\mathrm{Coh}(X^{\mathrm{an}})$ (spectral analytic)
Pushforward	$Rf_*$ (algebraic)	$Rf_*^{\mathrm{an}}$ (analytic)
Proper base change	the algebraic version	the analytic version
The theorem (proper / projective over $\mathbb{C}$)	$\mathrm{QCoh}(X)\simeq\mathrm{QCoh}(X^{\mathrm{an}})$	the GAGA equivalence

The Betti / the analytic realization of the spectral schemes (the $\mathbb{C}$-points as a complex analytic space, the $\mathbb{E}_\infty$-structure sheaf): a spectral scheme $X$ over $\mathbb{C}$ gives $X(\mathbb{C})$ a complex analytic space with a sheaf of $\mathbb{E}_\infty$-rings (the analytification $X^{\mathrm{an}}$), and the GAGA compares the algebraic and the analytic $\mathrm{QCoh}$.

Étale descent and the $\ell$-adic completion¶

The étale descent in SAG (the descent of $\mathrm{QCoh}$ / $\mathrm{Perf}$ along étale covers, the spectral analog of the classical faithfully-flat descent): for an étale cover $\{U_i\to X\}$, $\mathrm{QCoh}(X)\simeq\lim(\prod\mathrm{QCoh}(U_i)\rightrightarrows\prod\mathrm{QCoh}(U_{ij})\rightrightarrows\cdots)$ (the descent / the cosimplicial limit). The $\ell$-adic completion of spectral schemes (the $\ell$-completion $X\to X^\wedge_\ell$ and the relation to the étale homotopy type, the $\ell$-adic homotopy groups via the $\ell$-adic realization of the $\mathbb{E}_\infty$-rings).

The $\mathbb{A}^1$-homotopy / the motivic realization as a descent / base change (the relation of SAG to the motivic homotopy via the $\mathbb{E}_\infty$-realization: a spectral scheme gives a motivic space, the $\mathbb{E}_\infty$-structure sheaf as the motivic spectrum).

Proper base change and the spectral proper base change theorem¶

Theorem (spectral proper base change). For a proper morphism $f:X\to Y$ of spectral schemes and a cartesian square $$ \begin{matrix}X'\ar[r]^{g'}\ar[d]_{f'}&Y'\ar[d]^f\X\ar[r]_g&Y\end{matrix} $$ (with $g$ Tor-independent / flat), the base change $g^*Rf_*\simeq Rf'_*g'^*$ holds (the spectral version of the classical proper base change).

Theorem (coherent duality, Grothendieck duality in SAG). For a proper morphism $f:X\to Y$ of spectral schemes, $Rf_*R\underline{\mathrm{Hom}}(E,f^!F)\simeq R\underline{\mathrm{Hom}}(Rf_*E,F)$ — the six-functor formalism on spectral schemes (the Grothendieck–Verdier duality in the $\mathbb{E}_\infty$-setting). The $\mathrm{Perf}$ / the perfect complexes are the dualizable objects in $\mathrm{QCoh}(X)$ (the relation to the $\mathbb{E}_\infty$-module categories and the orientation / the Thom isomorphism).

The $p$-adic / prismatic realization and the cyclotomic descent¶

The $p$-adic realization of spectral schemes (the $\mathbb{F}_p$ / the $p$-complete $\mathbb{E}_\infty$-rings, the relation to the prismatic cohomology via the $\mathrm{TC}$ / the cyclotomic structure). The cyclotomic descent (the descent along the cyclotomic extension / the $\mathbb{E}_\infty$-Frobenius, the Bhatt–Morrow–Scholze / the BMS spectral sequence).

Realization	Target	Comparison
Betti / analytic (over $\mathbb{C}$)	$\mathrm{Sp}$ / spectral analytic	the GAGA (proper / projective over $\mathbb{C}$)
$\ell$-adic / étale	$\ell$-complete $\mathbb{E}_\infty$-rings	the $\ell$-adic realization, the étale homotopy type
$p$-adic / prismatic / cyclotomic	$p$-complete $\mathbb{E}_\infty$-rings / $\mathrm{TC}$	the BMS / the cyclotomic descent, the prismatic comparison
Motivic ($\mathbb{A}^1$-homotopy)	$\mathrm{SH}(S)$	the $\mathbb{E}_\infty$-realization, the motivic descent

The realization functors commute with the six functors (the proper base change, the coherent duality) in the appropriate setting — the comparison theorems of the spectral AG.

See it¶

Spectral GAGA for $\mathrm{Spec}\,H\mathbb{Z}$ and the Betti realization. For the spectral scheme $X=\mathrm{Spec}\,H\mathbb{Z}$ (the Eilenberg–MacLane of $\mathbb{Z}$, the "constant" spectral scheme), the Betti realization over $\mathbb{C}$ gives $X(\mathbb{C})=\mathrm{pt}$ (a point, the $\mathbb{C}$-points of $\mathrm{Spec}\,\mathbb{Z}$), and $$ \mathrm{QCoh}(X)=\mathrm{Mod}_{H\mathbb{Z}}\simeq D(\mathbb{Z}) $$ (the derived category of $\mathbb{Z}$-modules) compares to the analytic / Betti $\mathrm{QCoh}(\mathrm{pt})=\mathrm{Mod}_{H\mathbb{Z}}$ (the same — the GAGA isomorphism is the identity for the constant scheme: the algebraic and the analytic $\mathrm{QCoh}$ coincide).

Contrast: the $\mathrm{tmf}$ case. The spectral scheme $\mathrm{Spec}\,\mathrm{tmf}$ (the global sections of the $\mathcal{O}^{\mathrm{top}}$ sheaf), the Betti realization sends $\mathrm{tmf}\to\mathrm{tmf}\otimes\mathbb{C}$ (the complex modular forms, the analytification), and the GAGA comparison $$ \mathrm{QCoh}(\mathrm{Spec}\,\mathrm{tmf})\to\mathrm{QCoh}((\mathrm{Spec}\,\mathrm{tmf})^{\mathrm{an}}) $$ is the relation of the integral and the analytic modular forms (the $\mathrm{tmf}$ as the integral / algebraic, the $\mathrm{tmf}\otimes\mathbb{C}$ as the analytic / over-$\mathbb{C}$ — the spectral GAGA is the integrality statement: the algebraic $\pi_*\mathrm{tmf}$ is the integral modular forms, the analytic $\pi_*(\mathrm{tmf}\otimes\mathbb{C})=\pi_*\mathrm{tmf}\otimes\mathbb{C}$ is the analytic modular forms, and the GAGA identifies the categories of coherent sheaves, with the integrality (the $\mathbb{Z}$- vs $\mathbb{C}$-valued modular forms) preserved by the GAGA equivalence).

Check¶

Why is the spectral GAGA for $\mathrm{Spec}\,H\mathbb{Z}$ the identity, and what changes for a non-constant spectral scheme?

$\mathrm{Spec}\,H\mathbb{Z}$ is the constant spectral scheme (the Eilenberg–MacLane of $\mathbb{Z}$); its $\mathbb{C}$-points are a single point (the $\mathbb{C}$-points of $\mathrm{Spec}\,\mathbb{Z}$, the trivial scheme), and $\mathrm{QCoh}(\mathrm{Spec}\,H\mathbb{Z})=\mathrm{Mod}_{H\mathbb{Z}}=D(\mathbb{Z})$ is the derived category of $\mathbb{Z}$-modules. The analytification $(\mathrm{Spec}\,H\mathbb{Z})^{\mathrm{an}}$ is the point with the $H\mathbb{Z}$-structure sheaf, so $\mathrm{QCoh}(\mathrm{pt})=\mathrm{Mod}_{H\mathbb{Z}}$ — the same. The GAGA is the identity (the algebraic and the analytic coincide for the constant scheme). For a non-constant spectral scheme (e.g. $\mathrm{Spec}\,\mathrm{tmf}$, $\mathrm{Spec}\,KU$), the analytification introduces the analytic / $\mathbb{C}$-valued structure (the complex modular forms, the complex $K$-theory), and the GAGA is the integrality statement (the algebraic / $\mathbb{Z}$-valued vs the analytic / $\mathbb{C}$-valued).

Show that the spectral proper base change $g^*Rf_*\simeq Rf'_*g'^*$ for a proper $f$ and a flat $g$ follows from the coherence of $Rf_*$ and the flat base change.

For $f$ proper, $Rf_*$ preserves coherence (the properness / the coherence theorem in SAG), and for $g$ flat (Tor-independent), the derived base change $g^*Rf_*E\simeq Rf'_*g'^*E$ holds for $E\in\mathrm{QCoh}(X)$ (the flat base change in the derived / spectral setting, the spectral analog of the classical flat base change). The proof: reduce to the affine case (the $f$ proper, the $g$ flat), use the projection formula $Rf_*(E\otimes f^*F)\simeq Rf_*E\otimes F$ (for $F\in\mathrm{QCoh}(Y)$) and the coherence / the perfectness to extend from the perfect to the coherent case. The properness ensures the coherence of $Rf_*E$ (so the base change preserves the coherent category), and the flatness ensures the Tor-independence (no derived Tor corrections to the base change).

Pitfalls¶

The spectral GAGA holds for proper / projective spectral schemes over $\mathbb{C}$ (the analytification, the comparison of $\mathrm{QCoh}$), not for arbitrary spectral schemes — the properness / projectivity is essential (the same as the classical GAGA).
The $\ell$-adic / étale realization requires the $\ell$-completion (the $\ell$-adic realization is the $\ell$-complete $\mathbb{E}_\infty$-ring, not the integral); the étale homotopy type is the $\ell$-adic homotopy type, not the integral one.
The $p$-adic / prismatic / cyclotomic realization is the BMS / the cyclotomic descent (the $\mathrm{TC}$ / the cyclotomic structure on the $\mathbb{E}_\infty$-rings), not a direct geometric realization — the prismatic comparison is a deep theorem (the Bhatt–Morrow–Scholze), not a formal consequence of the GAGA.
The proper base change requires the Tor-independence / flatness of $g$ (the square must be Tor-independent); without it the derived base change has the Tor corrections (the spectral base change with the derived tensor).
The coherent duality (the Grothendieck duality in SAG) requires $f$ proper (the $f^!$ is the dualizing, the $Rf_*R\underline{\mathrm{Hom}}(E,f^!F)\simeq R\underline{\mathrm{Hom}}(Rf_*E,F)$); for non-proper $f$ the duality is more subtle (the Borel–Moore / the exceptional pullback $f^!$).
The realization functors commute with the six functors only in the appropriate setting (the proper base change, the coherent duality); the comparison theorems are not formal — they require the properness / flatness / coherence hypotheses.

Refs¶

Lurie, Spectral Algebraic Geometry (the spectral GAGA, the proper base change, the coherent duality)
Serre, "Géométrie algébrique et géométrie analytique" (Ann. Inst. Fourier 1956); the spectral analog (Lurie SAG)
Bhatt–Morrow–Scholze, "Integral $p$-adic Hodge theory" (Publ. Math. IHÉS 2019); the cyclotomic descent / the prismatic comparison
Toën–Vezzosi, HAG II (the spectral base change, the coherent duality); Lurie, Higher Algebra (the six functors)

Next¶

SAG index · Foundations · Elliptic Cohomology & TMF · Derived Moduli & Deformation · Chromatic & Morava E-theory · Prismatic cohomology · Motivic homotopy

Realization	Target	Comparison
Betti / analytic (over \(\mathbb{C}\))	\(\mathrm{Sp}\) / spectral analytic	the GAGA (proper / projective over \(\mathbb{C}\))
\(\ell\)-adic / étale	\(\ell\)-complete \(\mathbb{E}_\infty\)-rings	the \(\ell\)-adic realization, the étale homotopy type
\(p\)-adic / prismatic / cyclotomic	\(p\)-complete \(\mathbb{E}_\infty\)-rings / \(\mathrm{TC}\)	the BMS / the cyclotomic descent, the prismatic comparison
Motivic (\(\mathbb{A}^1\)-homotopy)	\(\mathrm{SH}(S)\)	the \(\mathbb{E}_\infty\)-realization, the motivic descent