What I’m up to…

$\textup{Spc}$ is the $\infty$-category of spaces.

Given an $\infty$-category $\mathscr{C}$, we let $\textup{Gr}(\mathscr{C}):=\textup{Fun}(\mathbb{Z},\mathscr{C})$ denote the $\infty$-category of graded objects in $\mathscr{C}$. Similarly, we let $\textup{Fil}(\mathcal{C}):=\textup{Fun}((\mathbb{Z},\leq),\mathscr{C})$ denote the $\infty$-category of decreasing filtered objects in $\mathscr{C}$.

$\mathscr{C}(-,-)$ encodes the mapping space, and $\textup{Map}_{X/}(-,-)$ and $\textup{Map}_{/X}(-,-)$ are alternative names for $\mathscr{C}_{X/}(-,-)$ and $\mathscr{C}_{/X}(-,-)$ (when clear from context we simply write $\textup{Map}_X(-,-)$)

$\textup{Mod}_R^{\heartsuit},\textup{Mod}_R,\textup{CAlg}_R^{\heartsuit},\textup{CAlg}_R$

$\otimes_R,\otimes_R^{\heartsuit}$ derived vs. classical tensor product over $R$

$M/^{\mathbb{L}}p$ derived quotient by $p$

$p$-nilpotence and $p$-completeness

Definition: A derived (etale) stack is a functor $\textup{CAlg}\to\textup{Spc}$ satisfying etale descent. These span an $\infty$-category $\textup{Stk}$. (We will have reason to consider other Grothendieck topologies later…)

To a derived stack $X$ we associate $\mathbf{APerf}(X)\subset\mathbf{Perf}(X)\subset\mathbf{QCoh}(X)$.

$\textup{Perf}(X)$ is the underlying $\infty$-groupoid

$\textup{Perf}$ is the derived moduli stack

$\textup{Tor}$-amplitude

Given derived stacks $Y$ and $Z$ over $X$, the associated derived mapping stack is

$$\underline{\textup{Map}}_X(Y,Z): \textup{CAlg}\to\textup{Spc},\qquad C\mapsto\textup{Map}_X(Y_C,Z).$$

If $X,Y,Z$ are all equipped with an action of a given derived group stack $G$ then we obtain the $G$-equivariant derived mapping stack

$$\underline{\textup{Map}}_X^G(Y,Z)\simeq\underline{\textup{Map}}_{[X/G]}([Y/G],[Z/G]).$$

Definition: A derived $p$-adic formal stack is a functor $\textup{CAlg}^{p\textup{-nilp}}\to\textup{Spc}$ satisfying etale descent. These span an $\infty$-category $\textup{FStk}_p$.

Each $X\in\textup{FStk}_p$ is equivalently viewed in terms of the restrictions $X_n: \textup{CAlg}_{\mathbb{Z}/p^n}\to\textup{Spc}$.

Let $\Theta$ denote the moduli stack of generalized Cartier divisors — i.e., its $C$-points identify with arrows $L\xrightarrow{t}C$ with $L$ a line bundle over $C$ and $t$ a $C$-linear cosection of $L$. We distinguish two points of $\Theta$: the open point $*\xrightarrow{\mathbb{1}}\Theta$ parametrizing trivialized line bundles $L\xrightarrow{\sim}C$ (i.e., the locus where $t$ is an isomorphism), and the closed point $B\mathbb{G}_m\xrightarrow{\mathbb{0}}\Theta$ parametrizing arrows $L\xrightarrow{0}C$ (i.e., the locus where $t$ vanishes). There is also an evident projection $\Theta\to B\mathbb{G}_m$, for which $\mathbb{0}$ is a section.

Remark: Our conventions associate $\Theta$ with decreasing filtrations. We can work with increasing filtrations by instead considering the moduli stack $\Theta_+$ parametrizing arrows $C\xrightarrow{u}L$ which describe line bundles equipped with a section. Analogous to above, we obtain the open point $*\xrightarrow{\mathbb{1}_+}\Theta_+$, the closed point $B\mathbb{G}_m\xrightarrow{\mathbb{0}_+}\Theta_+$, and the projection $\Theta_+\to B\mathbb{G}_m$. Note that $\Theta_{\pm}\simeq[\mathbb{A}_{\pm}^1/\mathbb{G}_m]$.

Over $\Theta$, we have a line bundle $\mathscr{O}(1)$ called the Tate twist given by $(L\xrightarrow{t}C)\mapsto L$. We denote the associated tensor powers by $\mathscr{O}(n)$ for $n\in\mathbb{Z}$, and use the same name and notation for the analogously defined line bundles over $B\mathbb{G}_m$.

Definition: Fix a derived stack $\mathcal{Y}$.

• A grading (respectively, filtration) on $\mathcal{Y}$ is the data of a map $\mathcal{Y}\to B\mathbb{G}_m$ (respectively, $\mathcal{Y}\to\Theta$). When $\mathcal{Y}$ is equipped with such a map, we say it is a graded (respectively, filtered) derived stack. If $\mathcal{Y}$ is graded (respectively, filtered), it is $C$-pointed if it is equipped with a map $B\mathbb{G}_{m,C}\to\mathcal{Y}$ (respectively, $\Theta_C\to\mathcal{Y}$) of graded (respectively, filtered) derived stacks.
• For $\mathcal{Y}$ filtered, the associated graded derived stack is the pullback $\mathcal{Y}_{\textup{gr}}:=\mathcal{Y}_{\mathbb{0}}$ and the underlying derived stack is the pullback $\mathcal{Y}_{\textup{und}}:=\mathcal{Y}_{\mathbb{1}}$.
• For $\mathcal{Y}$ graded, the fixed point locus of $\mathcal{Y}$ is the derived stack $$Y^0:=\underline{\textup{Map}}_{B\mathbb{G}_m}(B\mathbb{G}_m,\mathcal{Y}).$$
• For $\mathcal{Y}$ filtered, the attracting locus or attractor is the derived stack $$Y^-:=\underline{\textup{Map}}_{\Theta}(\Theta,\mathcal{Y}).$$
• For $\mathcal{Y}$ graded, define $Y^-$ as the attractor of the pullback of $\mathcal{Y}$ along the projection $\Theta\to B\mathbb{G}_m$.
• For $\mathcal{Y}$ filtered, define $Y^0$ as the fixed point locus of $\mathcal{Y}_{\textup{gr}}$.

Note that a grading on $\mathcal{Y}$ is equivalent to the data of a line bundle over $\mathcal{Y}$.

Rees functor

$$\textup{Rees}: \textup{FilMod}_R\to\textup{GrMod}_R,\qquad \textup{Fil}^{\bullet}M\mapsto\bigoplus_{i\in\mathbb{Z}}\textup{Fil}^{-i}M.$$

Transition maps in filtration induce extra structure on this functor (endomorphism up to shifting)

Example: $\textup{Rees}(\textup{Fil}_p^{\bullet}\mathbb{Z}_p)\simeq\mathbb{Z}_p[u,t]/(ut-p)$

Theorem (Rees equivalence): $\textup{FilMod}_R\simeq\mathbf{QCoh}(\Theta_R)$ upgrading

$\textup{GrMod}_R\simeq\mathbf{QCoh}^{\mathbb{G}_m}(\textup{Spec}(R))\simeq\mathbf{QCoh}(B\mathbb{G}_{m,R})$

(shifting corresponds to Tate twisting; other structure…)

Rees equivalence globalizes (we will make extensive use of this)

Via the Rees equivalence, $\mathbf{Vect}(\Theta_R)$ identifies with the $\infty$-category of finitely filtered vector bundles over $R$: filtered vector bundles $\textup{Fil}^{\bullet}E$ over $R$ such that $\textup{gr}^{\bullet}E$ is a graded vector bundle over $R$ (thus supported in finitely many graded degrees).

Definition:

• Let $A_{\bullet}$ be an animated graded $R$-algebra, with underlying animated $R$-algebra $A$. The graded spectrum $\textup{GSpec}(A_{\bullet})$ of $A_{\bullet}$ is the quotient $[\textup{Spec}(A)/\mathbb{G}_m]$ of the induced right action of $\mathbb{G}_m$ on $\textup{Spec}(A)$, which is naturally an $A$-pointed graded derived stack.
• Let $\textup{Fil}^{\bullet}A$ be an animated filtered $R$-algebra, with underlying animated $R$-algebra $A$. The filtered spectrum of $\textup{Fil}^{\bullet}A$ is $\textup{FSpec}(\textup{Fil}^{\bullet}A):=\textup{GSpec}(\textup{Rees}(\textup{Fil}^{\bullet}A))$, which is naturally an $A$-pointed filtered derived stack.

Formal variants $\textup{GSpf}(-),\textup{FSpf}(-)$

We have $\textup{GrSpec}(A_{\bullet})^0\simeq\textup{Spec}(A_0)$ and $\textup{FSpec}(\textup{Fil}^{\bullet}A)^-\simeq\textup{Spec}(\textup{gr}^0A)$.

$\textup{FSpec}(\textup{Fil}^{\bullet}A)_{\textup{und}}\simeq\textup{Spec}(A)$ and $\textup{FSpec}(\textup{Fil}^{\bullet}A)_{\textup{gr}}\simeq\textup{GSpec}(\textup{gr}^{\bullet}A)$

Suppose that $\mathcal{Y}\to\Theta_R$ is a filtered derived $R$-stack which is relative locally almost finitely presented with almost perfect cotangent complex $\mathbb{L}_{\mathcal{Y}/\Theta_R}\in\mathbf{APerf}(\mathcal{Y})$. We can naturally extract from this crucial filtered and graded information that will be useful later to probe the structure of $\mathcal{Y}$. In more detail, we obtain $\textup{Fil}^{\bullet}\mathbb{L}_{\mathcal{Y}}^-\in\textup{Fil}(\mathbf{APerf}(Y^-))$ via

$$Y^-(C)\to\mathbf{APerf}(\Theta_C),\qquad (y: \Theta_C\to\mathcal{Y})\mapsto\textup{Fil}^{\bullet}\mathbb{L}_{\mathcal{Y}}^-(y):=y^*\mathbb{L}_{\mathcal{Y}/\Theta_R}.$$

Similarly, we obtain $\mathbb{L}_{\mathcal{Y},\bullet}^0\in\textup{Gr}(\mathbf{APerf}(Y^0))$ via

$$Y^0(C)\to\mathbf{APerf}(B\mathbb{G}_{m,C}),\qquad (y: B\mathbb{G}_{m,C}\to\mathcal{Y}_{\textup{gr}})\mapsto\mathbb{L}_{\mathcal{Y},\bullet}^0(y):=y^*\mathbb{L}_{\mathcal{Y}_{\textup{gr}}/B\mathbb{G}_{m,R}}.$$

These satisfy $\mathbb{L}_{Y^-}\simeq\textup{gr}^0\mathbb{L}_{\mathcal{Y}}^-$ and $\mathbb{L}_{Y^0}\simeq\mathbb{L}_{\mathcal{Y},0}^0$.

Definition: The $1$-bounded fixed point locus is the open derived substack $Y_{1\textup{-bdd}}^0\hookrightarrow Y^0$ parametrizing points $y$ such that $\mathbb{L}_{\mathcal{Y},i}^0=0$ for $i<-1$.

If $\mathcal{Y}\to\Theta_R$ is relative locally finitely presented then $\mathbb{L}_{\mathcal{Y}/R}$ is perfect hence dualizable and so we may consider the duals $\mathbb{T}_{\mathcal{Y}/\Theta_R}$, $\textup{Fil}^{\bullet}\mathbb{T}_{\mathcal{Y}}^-$, and $\mathbb{T}_{\mathcal{Y},\bullet}^0$.

Definition: A frame is the data of $\underline{A}=(I\xrightarrow{d}A,\textup{Fil}^{\bullet}A,\Phi,A\{1\},\phi_{\textup{BK}})$ with

• $A$ an animated ring;
• $I\xrightarrow{d}A$ a generalized Cartier divisor such that $A$ is derived $(p,I)$-complete;
• $\textup{Fil}^{\bullet}A$ a non-negatively filtered animated ring that is derived $(p,I)$-complete;
• $\Phi: \textup{Fil}^{\bullet}A\to\textup{Fil}_I^{\bullet}A$ a map of filtered animated rings, with underlying $\phi: A\to A$ an animated Frobenius lift;
• $A\{1\}$ an invertible $A$-module;
• $\phi_{\textup{BK}}: \phi^{\bigstar}A\{1\}\otimes_AI\xrightarrow{\sim}A\{1\}$ an $A$-linear isomorphism.

Morphisms of such objects are defined in the expected manner. We say that $\underline{A}$ is

• prismatic if $I\xrightarrow{d}A$ underlies an animated prism, whose associated animated Frobenius lift and Breuil-Kisin structure respectively identify with $\phi$ and $(A\{1\},\phi_{\textup{BK}})$;
• orientable if $I\simeq A$, in which case a choice of isomorphism constitutes an orientation of $\underline{A}$ (note that Breuil-Kisin structure should be trivializable in this case; orientation always exists locally);
• crystalline if $I\xrightarrow{d}A$ identifies with $A\xrightarrow{p}A$ (hence is orientable).

Taking quasi-ideal quotients yields the animated $A$-algebras $R_A:=A/\!\!/\textup{Fil}^1A$ and $\overline{A}:=A/\!\!/I$ (which is just $A/^{\mathbb{L}}p$ in the crystalline case). We think of $\underline{A}$ as lying over $R_A$, which the following examples clarify.

Example: Let $R$ be a semiperfectoid ring. The Nygaard frame is the prismatic frame over $R$ given by

$$\underline{\mathbf{\Delta}}_R:=(I\to\mathbf{\Delta}_R,\textup{Fil}_{\textup{Nyg}}^{\bullet}\mathbf{\Delta}_R,\Phi_{\textup{Nyg}}),$$

where $\mathbf{\Delta}_R$ is the (absolute) prismatic complex of $R$, $\textup{Fil}_{\textup{Nyg}}^{\bullet}\mathbf{\Delta}_R$ is the (absolute) Nygaard filtration, and $\Phi_{\textup{Nyg}}$ is the associated (absolute) divided Frobenius. If $R$ has characteristic $p$ (hence $R$ is semiperfect) then this frame is crystalline.

Example: Let $R$ be a $p$-adically complete ring. The Lau-Witt frame is the crystalline prismatic frame $\underline{W}(R):=(\textup{Fil}_{\textup{Lau}}^{\bullet}W(R),\Phi_{\textup{Lau}})$ over $R$ with $\textup{Fil}_{\textup{Lau}}^{\bullet}W(R)$ given by

$$\cdots\xrightarrow{p}F_{\bigstar}W(R)\xrightarrow{p}F_{\bigstar}W(R)\xrightarrow{V}W(R)$$

and $\Phi_{\textup{Lau}}$ given by $F$ in filtered degree $0$ and the identity in higher filtered degrees, where $F$ denotes the Witt vector Frobenius and $V$ the Verschiebung. If $\underline{A}$ is a crystalline prismatic frame then the natural map $\lambda_A: A\to W(R_A)$ arising via adjunction from $A\twoheadrightarrow R_A$ canonically lifts to $\underline{A}\to\underline{W}(R_A)$.

Example: Let $R$ be an $\mathbb{F}_p$-algebra and $n\geq1$ a level parameter. The $n$-truncated Lau-Witt frame is the unique crystalline frame $\underline{W}_n(R)$ over $R$ such that $W(R)\twoheadrightarrow W_n(R)$ lifts to $\underline{W}(R)\to\underline{W}_n(R)$; the assumption that $R$ is an $\mathbb{F}_p$-algebra ensures that the truncated Witt vector Frobenius $F$ actually defines a ring endomorphism of $W_n(R)$. If $\underline{A}$ is a crystalline frame such that $R_A$ is an animated $\mathbb{F}_p$-algebra and $\phi$ lifts the Frobenius $\textup{Fr}$ on $R_A$ then the natural map $A\twoheadrightarrow R_A$ canonically lifts to $\underline{A}\to\underline{W}_1(R_A)$. This follows from the identification

$$\textup{Rees}(\textup{Fil}_{\textup{Lau}}^{\bullet}W_1(R))\simeq R[t]\times_{\textup{Fr}_{\bigstar}R}\textup{Fr}_{\bigstar}R[u],$$

where $|t|=1$ and $|u|=-1$.

Definition: A lamination on a prismatic frame $\underline{A}$ is the data of $\underline{\lambda}: \underline{A}\to\underline{W}(R_A)$ lifting $\lambda_A: A\to W(R_A)$. Equivalently, it is the data of an isomorphism of generalized Cartier divisors

$$(I\otimes_AW(R_A)\to W(R_A))\xrightarrow{\sim}(W(R_A)\xrightarrow{p}W(R_A)).$$

We refer to the pair $(\underline{A},\underline{\lambda})$ as a laminated frame.

As observed above, every crystalline prismatic frame admits a canonical lamination.

Remark: Given any frame $\underline{A}$, $\Phi$ induces a map $R_A\to\phi_{\bigstar}\overline{A}$ of animated $A$-algebras. If $\underline{A}$ is prismatic equipped with a lamination then the composition $R_A\to\phi_{\bigstar}\overline{A}\to F_{\bigstar}(\overline{A}\otimes_AW(R_A))$ identifies with the canonical map $R_A\to F_{\bigstar}W(R_A)/^{\mathbb{L}}p$. This will be useful later on.

Construction: To $\underline{A}$ we associate the filtered derived $p$-adic formal stack $\mathcal{R}(\underline{A}):=\textup{FSpf}(\textup{Fil}^{\bullet}A)$ (using the $(p,I)$-adic topology), equipped with open immersions $\sigma,\tau: \textup{Spf}(A)\to\mathcal{R}(\underline{A})$ which are defined as follows.

• $\tau$ is the pullback of the canonical map $\mathcal{R}(\underline{A})\xrightarrow{r}\Theta$ along $\textup{Spf}(\mathbb{Z}_p)\xrightarrow{\mathbb{1}}\Theta$.
• $\sigma$ is the composition $\textup{Spf}(A)\xrightarrow{\sim}\textup{FSpf}(\textup{Fil}_{I,\pm}^{\bullet}A)\xrightarrow{\textup{FSpf}(\Phi_{\pm})}\mathcal{R}(\underline{A})$, where $\Phi_{\pm}: \textup{Fil}^{\bullet}A\to\textup{Fil}_{I,\pm}^{\bullet}A$ is the canonical extension of $\Phi$.

We define the (abstract) syntomification of $\underline{A}$ to be the derived $p$-adic formal stack $\mathcal{S}(\underline{A})$ given by the coequalizer of $\sigma$ and $\tau$. We will endow $\mathcal{S}(\underline{A})$ with a natural $R_A$-pointed graded structure.

Viewing $A\{1\}$ as a line bundle over $\textup{Spf}(A)$ and using the Rees equivalence, $\tau^*A\{1\}\otimes r^*\mathscr{O}(1)$ identifies with a lift of $A\{1\}$ to a filtered line bundle $\textup{Fil}^{\bullet}A\{1\}$. Moreover, $\phi_{\textup{BK}}$ identifies with an isomorphism $\sigma^*\textup{Fil}^{\bullet}A\{1\}\xrightarrow{\sim}\tau^*\textup{Fil}^{\bullet}A\{1\}$ and so $\textup{Fil}^{\bullet}A\{1\}$ descends to a line bundle over $\mathcal{S}(\underline{A})$ which then equips $\mathcal{S}(\underline{A})$ with a grading. The map $A\twoheadrightarrow R_A$ induces

$$\textup{Spf}(R_A)\to B\mathbb{G}_{m,R_A}\xrightarrow{\mathbb{0}}\Theta_{R_A}\to\mathcal{R}(\underline{A})$$

factoring through $\tau$. The composition

$$B\mathbb{G}_{m,R_A}\to\mathcal{R}(\underline{A})\to\mathcal{S}(\underline{A})\to B\mathbb{G}_m$$

is the projection and so equips $\mathcal{S}(\underline{A})$ with a natural $R_A$-pointed graded structure. We also obtain a natural $R_A$-pointed filtered structure on $\mathcal{R}(\underline{A})$.

Lemma: Let $R_A\to R_{A’}$ be a $p$-completely etale map of animated rings. Then, there exists unique $\underline{A}\to\underline{A}’$ lifting $R_A\to R_{A’}$ such that the underlying $A\to A’$ is $(p,I)$-completely etale. Moreover, if $\underline{A}$ is prismatic/orientable/crystalline then so is $\underline{A}’$.

Fix a $C$-pointed graded derived $R$-stack $(\mathcal{Y},y^0)$. Given a map of derived $R$-stacks $\mathcal{Z}\to\mathcal{Y}$, we define $Z^0\to\textup{Spec}(C)$ as the fixed point locus of the graded derived $C$-stack $\mathcal{Z}_{y^0}\to B\mathbb{G}_{m,C}$ obtained via base change along $y^0: B\mathbb{G}_{m,C}\to\mathcal{Y}$. In the same manner, we obtain the locus $Z_{1\textup{-bdd}}^0\hookrightarrow Z^0$.

Warning: $Z^0\to\textup{Spec}(C)$ should not be confused with the fixed point locus of the composition $\mathcal{Z}\to\mathcal{Y}\to B\mathbb{G}_{m,R}$. The pointed structure on $\mathcal{Y}$ introduces a crucial twist.

Definition: A $1$-bounded derived stack over $(\mathcal{Y},y^0)$ (or simply $\mathcal{Y}$ if $y^0$ is clear from context) is the data of a pair $\mathcal{X}=(\mathcal{X}^{\diamondsuit},X^0)$ with

• $\mathcal{X}^{\diamondsuit}\to\mathcal{Y}$ a relative locally almost finitely presented derived Artin $R$-stack, which encodes the underlying derived $R$-stack of $\mathcal{X}$;
• $X^0\hookrightarrow\mathcal{X}_{1\textup{-bdd}}^{\diamondsuit,0}$ an open immersion, which encodes the fixed point locus of $\mathcal{X}$.

Such objects naturally pullback along maps of pointed graded derived $R$-stacks.

$X^{\pm}:=\mathcal{X}^{\diamondsuit,\pm}\times_{\mathcal{X}^{\diamondsuit,0}}X^0$

$\textup{Fil}^{\bullet}\mathbb{L}_{\mathcal{X}^{\diamondsuit}}^-$ and $\mathbb{L}_{\mathcal{X}^{\diamondsuit},\bullet}^0$ should admit variants that involve $X^0$ and $X^-$

Example: Let $M_{\bullet}$ be a graded $R$-module corresponding to an object of $\mathbf{APerf}(B\mathbb{G}_{m,R})$. To this we associate $\mathbb{V}(M_{\bullet})\to B\mathbb{G}_{m,R}$. Equipping $B\mathbb{G}_{m,R}$ with the tautological $R$-pointed graded derived $R$-stack structure yields $\mathbb{V}(M_{\bullet})^0\simeq\mathbb{V}(M_0)$. There is a natural graded map $\mathbb{V}(M_0)\times B\mathbb{G}_{m,R}\to\mathbb{V}(M_{\bullet})$, and we find that $\mathbb{L}_{\mathbb{V}(M_{\bullet}),\bullet}^0\in\textup{Gr}(\mathbf{APerf}(\mathbb{V}(M_0)))$ identifies via a globalization of the Rees equivalence with the pullback of $M_{\bullet}$ along the projection $\mathbb{V}(M_0)\times B\mathbb{G}_{m,R}\to B\mathbb{G}_{m,R}$. Letting $\textup{Van}_{i<-1}(M_i)$ denote the open vanishing locus in $\textup{Spec}(R)$ of $M_i$ for $i<-1$, we have $\mathbb{V}(M_{\bullet})_{1\textup{-bdd}}^0\simeq\mathbb{V}(M_0)\times_R\textup{Van}_{i<-1}(M_i)$. We obtain the $1$-bounded derived stack

$$\mathcal{P}(M_{\bullet}):=(\mathbb{V}(M_{\bullet}),\mathbb{V}(M_0)\times_R\textup{Van}_{i<-1}(M_i))$$

over $B\mathbb{G}_{m,R}$.

Example: Equip $\textup{Perf}$ with the trivial grading $\textup{Perf}\times B\mathbb{G}_m\to B\mathbb{G}_m$. Then, $(\textup{Perf}\times B\mathbb{G}_m)^0$ identifies with the derived moduli stack $\textup{Perf}^{\textup{gr}}$ of perfect graded complexes. Letting $M_{\textup{taut},\bullet}$ denote the tautological perfect graded complex, we have $\mathbb{L}_{\textup{Perf}\times B\mathbb{G}_m,\bullet}^0\simeq(M_{\textup{taut},\bullet}^{\vee}\otimes M_{\textup{taut},\bullet})[-1]$ and so

$$(\textup{Perf}\times B\mathbb{G}_m)_{1\textup{-bdd}}^0\simeq\textup{Van}_{i+j<-1}((M_{\textup{taut},-i})^{\vee}\otimes M_{\textup{taut},j})\hookrightarrow\textup{Perf}^{\textup{gr}}.$$

In particular, we can consider $\textup{Perf}^{[0,1]}:=\textup{Van}_{i\neq-1,0}(M_{\textup{taut},i})\hookrightarrow\textup{Perf}^{\textup{gr}}$. We obtain the $1$-bounded derived stack

$$\mathcal{P}^{[0,1]}:=(\textup{Perf}\times B\mathbb{G}_m,\textup{Perf}^{[0,1]})$$

over $B\mathbb{G}_m$.

Example: Restricting $\mathcal{P}^{[0,1]}$ to the setting of vector bundles yields the $1$-bounded derived stack

$$\mathcal{V}^{[0,1]}:=(\textup{Vect}\times B\mathbb{G}_m,\textup{Vect}^{[0,1]})$$

over $B\mathbb{G}_m$. Points of $\textup{Vect}^{[0,1]}$ identify with graded vector bundles $E_{\bullet}$ supported in graded degrees $-1,0$. Letting $\textup{Vect}^{h,d}$ denote the open and closed locus of points where $\textup{rank}(E_{-1})=d$ and $\textup{rank}(E_0)=h-d$, we obtain

$$\mathcal{V}^{h,d}:=(\textup{Vect}\times B\mathbb{G}_m,\textup{Vect}^{h,d}).$$

Such $E_{\bullet}$ is said to have type $(h,d)$. Note that there is a decomposition $\textup{Vect}^{[0,1]}\simeq\bigsqcup_{0\leq d\leq h}\textup{Vect}^{h,d}$ into connected components.

Example: We want to construct a $1$-bounded derived stack

$$\mathcal{B}(\mathcal{G},\mu):=(\mathcal{B}(\mathcal{G},\mu)^{\diamondsuit},BZ_{\mu})$$

over the pointed graded stack $B\mathbb{G}_{m,\mathcal{O}}\to B\mathbb{G}_m$, which serves as a torsor-theoretic generalization of $\mathcal{V}^{h,d}$ in the sense that $\mathcal{B}(\textup{GL}_h,\mu_{h,d})\simeq\mathcal{V}^{h,d}$.

Let’s start by looking at the general linear case. The classifying map $B\mu_{h,d}: B\mathbb{G}_m\to B\textup{GL}_h$ defines a $\textup{GL}_h$-torsor $\mathcal{Q}_{\textup{can}}^{h,d}\to B\mathbb{G}_m$ which identifies with $\mathscr{O}(-1)^{\oplus d}\oplus\mathscr{O}^{\oplus(h-d)}$. Our goal is to characterize $\textup{GL}_h$-torsors over $B\mathbb{G}_m$ which are locally isomorphic to $\mathcal{Q}_{\textup{can}}^{h,d}$.

TO DO: Look at Breen’s perspective on this… (automorphisms of potentially nontrivial torsors and role played by twisting)

The torsor automorphism group $\textup{GL}_h\{d\}:=\underline{\textup{Aut}}(\mathcal{Q}_{\textup{can}}^{h,d})$ defines a graded group stack.

Semidirect product $\textup{GL}_{h,d}:=\textup{GL}_h\rtimes_{\mu_{h,d}}\mathbb{G}_m$ linked with $\textup{GL}_h\{d\}$-torsors

$\textup{GL}_d\times\textup{GL}_{h-d}\simeq Z_{h,d}\to\textup{GL}_h\{d\}$

We begin by noting a handful of useful facts.

• The classifying map $B\mu: B\mathbb{G}_{m,\mathcal{O}}\to B\mathcal{G}_{\mathcal{O}}$ induces a canonical $\mathcal{G}$-torsor $\mathcal{Q}_{\textup{can}}^{\mu}\to B\mathbb{G}_{m,\mathcal{O}}$. This has automorphism group $\mathcal{G}\{\mu\}:=\underline{\textup{Aut}}(\mathcal{Q}_{\textup{can}}^{\mu})$.
• The graded stack $[\mathcal{G}_{\mathcal{O}}/\mathbb{G}_{m,\mathcal{O}}]\to B\mathbb{G}_{m,\mathcal{O}}$ induced by $\textup{ad}_{\mu}$ has fixed point locus $Z_{\mu}$ and attractor/repeller $P_{\mu}^{\pm}$ by earlier work (and carries residual action of $\mathcal{G}$).
• $\mathcal{G}_{\mu}$-equivariant $\mathcal{Q}\to\textup{Spec}(C)$ restricting to a $\mathcal{G}$-torsor over $C$

$H^1(\mathbb{G}_{m,\mathcal{O}},\mathcal{G}_{\mathcal{O}})$

There are two perspectives on $\mathcal{B}(\mathcal{G},\mu)^{\diamondsuit}$. We claim that every $\mathcal{O}$-linear representation $(V,\rho)\in\textup{Rep}_{\mathcal{O}}(\mathcal{G})$ naturally induces a graded vector bundle $E_{\bullet}(V,\rho)$ over $\mathcal{B}(\mathcal{G},\mu)^{\diamondsuit,0}$.

$\mathbb{L}_{(B\mathcal{G}_{\mathcal{O}}\times B\mathbb{G}_{m,\mathcal{O}})/B\mathbb{G}_{m,\mathcal{O}}}\simeq\textup{pr}_1^*\mathbb{L}_{B\mathcal{G}_{\mathcal{O}}}$

Let $e$ denote the identity section of $\mathcal{G}_{\mathcal{O}}$. Using the equivalence $\mathbf{QCoh}(B\mathcal{G}_{\mathcal{O}})\simeq\mathbf{QCoh}^{\mathcal{G}}(\textup{Spec}(\mathcal{O}))$, we have

$$\mathbb{L}_{B\mathcal{G}_{\mathcal{O}}}\simeq e^*\mathbb{L}_{\mathcal{G}_{\mathcal{O}}}[-1]\simeq e^*\Omega_{\mathcal{G}_{\mathcal{O}}}^1[-1]\simeq e^*\mathfrak{g}^{\vee}[-1].$$

$\mathbb{T}_{B\mathcal{G}_{\mathcal{O}}}\simeq e^*\mathfrak{g}[-1]$ and $E_{\bullet}(\mathfrak{g})[-1]$

$\mathcal{B}(\mathcal{G},\mu)^{\diamondsuit}\simeq B\mathcal{G}\{\mu\}\simeq B\mathcal{G}_{\mathcal{O}}\times B\mathbb{G}_{m,\mathcal{O}}$

$\mathcal{B}(\mathcal{G},\mu)^{\diamondsuit,0}$ described using $\mathcal{G}_{\mu}$

$\mathcal{B}(\mathcal{G},\mu)_{1\textup{-bdd}}^{\diamondsuit,0}$

$\mathcal{B}(\mathcal{G},\mu)^0\simeq BZ_{\mu}$

We now introduce a key construction which ties together the structure of $\mathcal{X}$.

Construction: Let $(B\mathbb{G}_{m,C’}\xrightarrow{w^0}\mathcal{W})\to(B\mathbb{G}_{m,C}\xrightarrow{y^0}\mathcal{Y})$ be a map of pointed graded derived $R$-stacks. To this we associate the (total) mapping space

$\textup{Map}_{\mathcal{Y}}(\mathcal{W},\mathcal{X})=\textup{Map}_{(\mathcal{Y},y^0)}((\mathcal{W},w^0),\mathcal{X}):=\textup{Map}_{\mathcal{Y}}(\mathcal{W},\mathcal{X}^{\diamondsuit})\times_{\mathcal{X}^{\diamondsuit,0}(C’)}X^0(C’),$

where $\textup{Map}_{\mathcal{Y}}(\mathcal{W},\mathcal{X}^{\diamondsuit})\to\mathcal{X}^{\diamondsuit,0}(C’)\simeq\textup{Map}_{\mathcal{Y}}(B\mathbb{G}_{m,C’},\mathcal{X}^{\diamondsuit})$ is induced via restriction along $w^0$.

TO DO: properties of this construction as $(\mathcal{W},w^0)$ varies “in families”

Let $\mathcal{X}$ be a $1$-bounded derived stack over $\mathcal{S}_n(\underline{A})$.

Definition: The sheaf of $\underline{A}$-sections of $\mathcal{X}$ is the etale sheaf $\Gamma_{\underline{A}}(\mathcal{X})$ over $R_A$ given by $R_{A’}\mapsto\textup{Map}_{\mathcal{S}_n(\underline{A})}(\mathcal{S}_n(\underline{A}’),\mathcal{X})$.

$\mathbf{\Delta}(R),\textup{Nyg}(R),\textup{Syn}(R),\textup{HT}(R)$

$\textup{Nyg}(R)_{\textup{dR}},\textup{Nyg}(R)_{\textup{HT}},\textup{Nyg}(R)_{\textup{Hdg}}$ components of $\textup{Nyg}(R)_{(p=0)}$

Prismatization: $j_{\textup{dR}},j_{\textup{HT}},\rho_{\textup{dR}},\rho_{\textup{HT}},\rho_{(A,I)},F$

Filtered prismatization: $\rho_{\textup{dR}}^{\mathcal{N}},\rho_{\textup{HT}}^{\mathcal{N}},r_{\mathcal{N}},\pi_{\mathbf{\Delta}}$

Lemma: Let $(R’\twoheadrightarrow R,\gamma)$ be an animated PD-thickening. Then, there is an induced map $\rho_{\textup{dR},\gamma}: \textup{Spf}(R’)\to\mathbf{\Delta}(R)$ compatible with $\rho_{\textup{dR},R}$ and $\rho_{\textup{dR},R’}$.

We equip $\textup{Syn}(R)$ with the grading induced by the Breuil-Kisin twist $\mathscr{O}_{\textup{syn}}\{1\}$.

Proposition: Let $(\underline{A},\underline{\lambda})$ be a laminated frame. Then, there is an induced map $\mathcal{R}(\underline{A})\to\textup{Nyg}(R_A)$.

Theorem: Let $R$ be differentially semiperfectoid and equip $\underline{\mathbf{\Delta}}_R$ with the $\textup{Tot}$-lamination. Then, the induced map $\mathcal{R}(\underline{\mathbf{\Delta}}_R)\to\textup{Nyg}(R)$ is an isomorphism.

Let $\mathcal{X}$ be a $1$-bounded derived stack over $\textup{Syn}_n(R)$.

$\Gamma_{\textup{syn}}^{(n)}(\mathcal{X}): C\mapsto\textup{Map}_{\textup{Syn}_n(R)}(\textup{Syn}_n(C),\mathcal{X})$

$\Gamma_{\mathcal{N}}^{(n)}(\mathcal{X}): C\mapsto\textup{Map}_{\textup{Syn}_n(R)}(\textup{Nyg}_n(C),\mathcal{X})$

$\Gamma_{\mathbf{\Delta}}^{(n)}(\mathcal{X}): C\mapsto\textup{Map}_{\textup{Syn}_n(R)}(\mathbf{\Delta}_n(C),\mathcal{X}^{\diamondsuit})$

$\widetilde{\mathbb{0}}_+=\mathbb{0}_{+,\textup{Fr}_{\bigstar}R}: B\mathbb{G}_{m,\textup{Fr}_{\bigstar}R}\to\Theta_{+,\textup{Fr}_{\bigstar}R}$

$\widetilde{\mathbb{0}}_-=\mathbb{0}_{-,R}\circ\textup{Fr}: B\mathbb{G}_{m,\textup{Fr}_{\bigstar}R}\to\Theta_{-,R}$

• $\textup{Fil}_{\textup{Hdg}}^{\bullet}M^-$
• $\textup{Fil}_{\bullet}^{\textup{conj}}M^+$
• $M^+\xrightarrow{\sim}M^-$
• $\textup{gr}_{\bullet}^{\textup{conj}}M^+\xrightarrow{\sim}\textup{Fr}^{\bigstar}\textup{gr}_{\textup{Hdg}}^{\bullet}M^-$

TO DO: highlight where the unramified hypothesis will be used

$\textup{Gauge}(\underline{A})$ and $\textup{F-Gauge}(\underline{A})$ (and level variants)

$\textup{Wind}_{\underline{A},n}^{\mathcal{G},\mu}(R_A)\simeq\textup{Map}_{B\mathbb{G}_m}(\mathcal{S}_n(\underline{A}),\mathcal{B}(\mathcal{G},\mu))$

$\mathcal{B}_n^{\textup{BK}}(\mathcal{G},\mu)$ the level $n$ Breuil-Kisin modification

Given $n\geq1$, let $p\textup{-div}_n$ denote the $p$-adic formal moduli stack of $n$-truncated $p$-divisible groups.

Let $\textup{F-Zip}^{\mathcal{G},\mu}$ denote the $k$-stack of $(\mathcal{G},\mu)$-zips (which are equivalent to $F$-zips with $\mathcal{G}$-structure of type $\mu$), which only depends on the special fiber of $(\mathcal{G},\mu)$.

$\textup{BT}_n^{\mathcal{G},\mu}(-)\simeq\textup{Map}_{\textup{Syn}_n(\mathcal{O})}(\mathcal{B}(\mathcal{G},\mu),\textup{Syn}_n(-))$

$\textup{Disp}_n^{\mathcal{G},\mu}(-)\simeq\textup{Map}_{\mathcal{S}(\underline{W}_n(\mathcal{O}))}(\mathcal{B}(\mathcal{G},\mu),\mathcal{S}(\underline{W}_n(-)))$

$\textup{Disp}_1^{\mathcal{G},\mu}\simeq\textup{F-Zip}^{\mathcal{G},\mu}$

$\mathcal{S}_n(\underline{\mathbf{\Delta}}_R)$ vs. $\mathcal{S}_n(\underline{W}(R))$ vs. $\mathcal{S}(\underline{W}_n(R))$

$\textup{BT}_n^{G,\mu}$ satisfies an analogue of Grothendieck-Messing theory: certain commutative squares associated to nilpotent animated PD-thickenings are Cartesian.

$\textup{BT}_n^{\mathcal{G},\mu}$ is a represented by a $0$-dimensional quasicompact smooth derived Artin formal stack over $\textup{Spf}(\mathcal{O})$ with affine diagonal. Moreover, the natural maps $\textup{BT}_{n+1}^{\mathcal{G},\mu}\to\textup{BT}_n^{\mathcal{G},\mu}$ are smooth.

There is a natural map $\textup{BT}_1^{\mathcal{G},\mu}\otimes\mathbb{F}_p\to\textup{F-Zip}^{\mathcal{G},\mu}$ that is relatively representable by a smooth $0$-dimensional derived Artin stack over $\textup{Spec}(k)$ with relatively affine diagonal. In fact, it is a gerbe banded by the Lau group scheme $\textup{Lau}_1^{\mathcal{G},\mu}$, which is a specific finite flat commutative $p$-group scheme of height $1$. In particular, $\textup{BT}_1^{\mathcal{G},\mu}\otimes\mathbb{F}_p$ is a smooth $0$-dimensional derived Artin stack over $\textup{Spec}(k)$ with affine diagonal.

The classical truncation of $\textup{BT}_n^{h,d}$ is isomorphic to $p\textup{-div}_n^{h,d}$.

There are several ways we can hope to generalize the theory of prismatic $(\mathcal{G},\mu)$-apertures, by individually loosening constraints on $\mathcal{G}$ and $\mu$. We pose these as questions, starting with the hardest one.

• What if $E_{\mathcal{O}}/\mathbb{Q}_p$ is ramified?

“decoupling” $\textup{BT}_{n,m}^{\mathcal{G},\mu}$ (and general problems with assuming integrality; working “directly” with $B\mu$ closer to rational setting)

• What if $\mathcal{G}$ is not smooth?

Description of $\mathbb{L}_{B\mathcal{G}}$ would be more complicated (and weight condition on $\mu$ would need to be reformulated)

• What if $\mu$ is not $1$-bounded?

(strongly) Fontaine-Laffaille, quasi-minuscule

• What if $\mathcal{G}$ is instead defined over the ring of integers $\mathcal{O}_L$ of a finite extension $L/\mathbb{Q}_p$?

(ramified) $\mathcal{O}_L$-typical Witt vectors and (animated) $\mathcal{O}_L$-typical divided powers

Related work (S. Bartling-M. Hoff)

General linear groups

General case (S. Y. Lee-K. Madapusi)

Uniformization

Serre-Tate ordinary lifting (K. Madapusi)

Hecke orbits

Galois representations

Extracting more information from the link with $\textup{F-Zip}^{\mathcal{G},\mu}$, $\textup{Disp}_n^{\mathcal{G},\mu}$, and $\textup{Lau}_n^{\mathcal{G},\mu}$

A. Frederick thesis
Y. Sulyma arXiv preprint

Also want to generalize known connections to the $\mathcal{O}$-typical setting

• Derived integral and local models for Shimura varieties
• Derived arithmetic intersection theory (click here for more information)