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I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ a subcategory which is stable under cofibers. Let $L:D_0(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}Q=0$.

Then the map from $Ring_C\to Ring_A$ admits a left adjoint ($Ring_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. For the proof of this fact, it gives a reference to Proposition 12.26 in Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_{\ge0}(A)$ a subcategory which is stable under cofibers. Let $L:D_{\ge0}(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L(\phi_{p^m}^{*}Q)=0$, where $\phi_{p^m}\colon A\to A/p$ is the Frobenius $x\mapsto x^{p^m}$.

Then the map from $\operatorname{Ring}_C\to\operatorname{Ring}_A$ admits a left adjoint ($\operatorname{Ring}_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(\operatorname{Sym}(Q))=0$. For the proof of this fact, it gives a reference to Proposition 12.26 in Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ a subcategory which is stable under cofibers. Let $L:D_0(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}M=0$.

Then the map from $Ring_C\to Ring_A$ adimits a left adjoint ($Ring_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. For the proof of this fact, it gives a reference to Proposition 12.26 in Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ a subcategory which is stable under cofibers. Let $L:D_0(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}Q=0$.

Then the map from $Ring_C\to Ring_A$ admits a left adjoint ($Ring_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. For the proof of this fact, it gives a reference to Proposition 12.26 in Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ a subcategory which is stable under cofibers. Let $L:D_0(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}M=0$.

Then the map from $Ring_C\to Ring_A$ adimits a left adjoint ($Ring_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. For the proof of this fact, it gives a reference to Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?

I am trying to understand the proof of lemma 2.2.18 in Lucas Mann's thesis. Its statement is surprising for me, because it talks about general rings which are not necessarily characteristic $p$, and out of nowhere Frobenius appears in the statement. the statement is this:

Let $A$ be a ring and $C\subset D_0(A)$ a subcategory which is stable under cofibers. Let $L:D_0(A)\to C$ be left adjoint to the forgetful functor, and satisfying these conditions:

1. if $L(Q)=0$ then $L(Q\otimes M)=0$ for every $M$.
2. if $L(Q)=0$, then for every prime $m$ and every integer $m$, $L\phi_{p^m}^{*}M=0$.

Then the map from $Ring_C\to Ring_A$ adimits a left adjoint ($Ring_C$ is the category of $A$-algebras whose underlying "module" lies in $C$).

The proof is based on the fact that if $L(Q)=0$ then $L(Sym(Q))=0$. For the proof of this fact, it gives a reference to Proposition 12.26 in Scholze's Lectures on Analytic Geometry, where they only give a sketch of the proof and I can't understand it. Can someone explain the proof of this fact in more detail or give a reference?