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user196717

Theorem. Let $K$ be a perfectoid field and $A$ be a $K$-algebra local ring that is a localization of some finite type $K$-algebra. Then $A$ is regular iff there is a faithfully flat perfectoid covering $A\to A_{\infty}$. (It's just a special case of the main theorem in https://arxiv.org/abs/1803.03229)

Proof) ($\implies$) It's just proved by noether normalization theorem and adjoining $p^n$th root of unities. In more generally, we can use Cohen structure theorem and the correpondence theorem between perfectoid algebras and prisms.

($\Longleftarrow$) We can use noether normalization to reduce the problem to prove that if $\mathfrak{p}$ is the prime ideal and $\varphi:K[x_1,\cdots,x_n]_{\mathfrak{p}}\to A_{\infty}$ is a faithfully flat perfectoid covering, then the map Kähler differentlals $\Omega_{\varphi}$$\Omega_{\varphi}\to \Omega_{A_{\infty}/A}$ is zeroinjective, which can be proved by tilting equivalence and easy Frobenius technique. So the theorem concluded by the exact sequence of Kähler differentials and the theorem "étale=flat+unramified".

Remark. I might hope that my proof with mimicking of proof of Cohen structure theorem could give the full proof of main theorem of Bhatt-Iyengar-Ma but I don't know the detail.

Definition. Let $X$ be a scheme over a characteristic $p$ algebraically closed field $k$ and $p\ne \ell$. Then $X$ is $\ell$-cohomologically smooth if $f$ is of finite type and for $f:X\to \mathrm{Spec} k$ and any scheme $Y$ over $k$, $Rf^!_X:D^b_c(Y,\mathbb{Q}_{\ell})\to D^b_c(Y\times X,\mathbb{Q}_{\ell})$ is of form $Rf^!_X=\mathbb{Q}_{\ell}(2\dim X)[\dim X]\otimes f^*$. (The terminology and formal definition are from https://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)

Remark.Remark. Of course, there is a relative notion of smoothness by considering separably closed points.

Quasi-conjecture. Let $X$ be a separated finite type scheme over an algebraically closed char. $0$ perfectoid field $K$. Then the followings are equivalent.

(a) $X$ is smooth

(b) $X$ is $p$-cohomologically smooth

(c) There is a Riemann-Hilbert correspondense in sense of https://arxiv.org/abs/1602.06282

(d) There is somewhat smooth base change theorem

Remark. The equivalence of (a) and (b) might failfails because of the possibility of existence of imperfect $K$-algebra having zero cotangent complex as Bhatt pointed out (http://www-personal.umich.edu/~bhattb/math/trivial-cc.pdf).

Theorem. Let $K$ be a perfectoid field and $A$ be a $K$-algebra local ring that is a localization of some finite type $K$-algebra. Then $A$ is regular iff there is a faithfully flat perfectoid covering $A\to A_{\infty}$. (It's just a special case of the main theorem in https://arxiv.org/abs/1803.03229)

Proof) ($\implies$) It's just proved by noether normalization theorem and adjoining $p^n$th root of unities. In more generally, we can use Cohen structure theorem and the correpondence theorem between perfectoid algebras and prisms.

($\Longleftarrow$) We can use noether normalization to reduce the problem to prove that if $\mathfrak{p}$ is the prime ideal and $\varphi:K[x_1,\cdots,x_n]_{\mathfrak{p}}\to A_{\infty}$ is a faithfully flat perfectoid covering, then the Kähler differentlals $\Omega_{\varphi}$ is zero, which can be proved by tilting equivalence and easy Frobenius technique. So the theorem concluded by the exact sequence of Kähler differentials and the theorem "étale=flat+unramified".

Remark. I might hope that my proof with mimicking of proof of Cohen structure theorem could give the full proof of main theorem of Bhatt-Iyengar-Ma but I don't know the detail.

Definition. Let $X$ be a scheme over a characteristic $p$ algebraically closed field $k$ and $p\ne \ell$. Then $X$ is $\ell$-cohomologically smooth if $f$ is of finite type and for $f:X\to \mathrm{Spec} k$ and any scheme $Y$ over $k$, $Rf^!_X:D^b_c(Y,\mathbb{Q}_{\ell})\to D^b_c(Y\times X,\mathbb{Q}_{\ell})$ is of form $Rf^!_X=\mathbb{Q}_{\ell}(2\dim X)[\dim X]\otimes f^*$. (The terminology and formal definition are from https://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)

Remark. Of course, there is a relative notion of smoothness by considering separably closed points.

Quasi-conjecture. Let $X$ be a separated finite type scheme over an algebraically closed char. $0$ perfectoid field $K$. Then the followings are equivalent.

(a) $X$ is smooth

(b) $X$ is $p$-cohomologically smooth

(c) There is a Riemann-Hilbert correspondense in sense of https://arxiv.org/abs/1602.06282

(d) There is somewhat smooth base change theorem

Remark. The equivalence of (a) and (b) might fail because of the possibility of existence of imperfect $K$-algebra having zero cotangent complex as Bhatt pointed out (http://www-personal.umich.edu/~bhattb/math/trivial-cc.pdf).

Theorem. Let $K$ be a perfectoid field and $A$ be a $K$-algebra local ring that is a localization of some finite type $K$-algebra. Then $A$ is regular iff there is a faithfully flat perfectoid covering $A\to A_{\infty}$. (It's just a special case of the main theorem in https://arxiv.org/abs/1803.03229)

Proof) ($\implies$) It's just proved by noether normalization theorem and adjoining $p^n$th root of unities. In more generally, we can use Cohen structure theorem and the correpondence theorem between perfectoid algebras and prisms.

($\Longleftarrow$) We can use noether normalization to reduce the problem to prove that if $\mathfrak{p}$ is the prime ideal and $\varphi:K[x_1,\cdots,x_n]_{\mathfrak{p}}\to A_{\infty}$ is a faithfully flat perfectoid covering, then the map Kähler differentlals $\Omega_{\varphi}\to \Omega_{A_{\infty}/A}$ is injective, which can be proved by tilting equivalence and easy Frobenius technique. So the theorem concluded by the exact sequence of Kähler differentials and the theorem "étale=flat+unramified".

Remark. I might hope that my proof with mimicking of proof of Cohen structure theorem could give the full proof of main theorem of Bhatt-Iyengar-Ma but I don't know the detail.

Definition. Let $X$ be a scheme over a characteristic $p$ algebraically closed field $k$ and $p\ne \ell$. Then $X$ is $\ell$-cohomologically smooth if $f$ is of finite type and for $f:X\to \mathrm{Spec} k$ and any scheme $Y$ over $k$, $Rf^!_X:D^b_c(Y,\mathbb{Q}_{\ell})\to D^b_c(Y\times X,\mathbb{Q}_{\ell})$ is of form $Rf^!_X=\mathbb{Q}_{\ell}(2\dim X)[\dim X]\otimes f^*$. (The terminology and formal definition are from https://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)

Remark. Of course, there is a relative notion of smoothness by considering separably closed points.

Quasi-conjecture. Let $X$ be a separated finite type scheme over an algebraically closed char. $0$ perfectoid field $K$. Then the followings are equivalent.

(a) $X$ is smooth

(b) $X$ is $p$-cohomologically smooth

(c) There is a Riemann-Hilbert correspondense in sense of https://arxiv.org/abs/1602.06282

(d) There is somewhat smooth base change theorem

Remark. The equivalence of (a) and (b) might fails because of the possibility of existence of imperfect $K$-algebra having zero cotangent complex as Bhatt pointed out (http://www-personal.umich.edu/~bhattb/math/trivial-cc.pdf).

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user196717
user196717

What is the definition of smoothness in scheme theory?

I want to ask if there is a somewhat desirable definition of "smoothness".

Definition. Let $k$ be a field and $X$ be a separated finite type scheme over $k$. Then $X$ is smooth if the Kähler differentials $\Omega_{X/k}$ is locally free of rank equal to $\dim X$.

Smoothness is a crucial condition to compute various cohomologies. For example, to compute étale cohomology in curve case, the standard method computes algebraic curve case first and can take normalization with Leray spectral sequence.

When I was an undergraduate student, I was fascinated in the following theorem theorem I proved.

Theorem. Let $K$ be a perfectoid field and $A$ be a $K$-algebra local ring that is a localization of some finite type $K$-algebra. Then $A$ is regular iff there is a faithfully flat perfectoid covering $A\to A_{\infty}$. (It's just a special case of the main theorem in https://arxiv.org/abs/1803.03229)

Proof) ($\implies$) It's just proved by noether normalization theorem and adjoining $p^n$th root of unities. In more generally, we can use Cohen structure theorem and the correpondence theorem between perfectoid algebras and prisms.

($\Longleftarrow$) We can use noether normalization to reduce the problem to prove that if $\mathfrak{p}$ is the prime ideal and $\varphi:K[x_1,\cdots,x_n]_{\mathfrak{p}}\to A_{\infty}$ is a faithfully flat perfectoid covering, then the Kähler differentlals $\Omega_{\varphi}$ is zero, which can be proved by tilting equivalence and easy Frobenius technique. So the theorem concluded by the exact sequence of Kähler differentials and the theorem "étale=flat+unramified".

Remark. I might hope that my proof with mimicking of proof of Cohen structure theorem could give the full proof of main theorem of Bhatt-Iyengar-Ma but I don't know the detail.

So we can regard perfectoid covering as "contractible covering"!

Definition. Let $M$ be a topological space. Then $M$ is a manifold iff there is a open covering $\{U_i\}$ which is of form $U_i\cong \mathbb{R}^n$

Theorem. Let $X$ be a finite type scheme over a perfectoid field $K$. Then $X$ is smooth iff there is a perfectoid covering $\{U_i\to X\}$ in fpqc topology.

I wanted to make some pseudo-mathematics that the idea of Plato's sense with the slogan "the smoothness is the property that is covered by somewhat good objects that have higher homotopy, cohomology,...etc $0$". But finding the formal definition was so hard. I was fascinated in the theory of cohesive topos and formulation of differential cohomology but soon I felt there is no possibility to apply cohesive topos theory to number theory, that cohesive topos theory requires so strong contractibility property!!! I don't know how to compute higher homotopy or higher K-theory of perfectoid spaces (The $p$-adic version of higher K-theory of perfectoid spaces can be computed as topological cyclic homology and it makes a link between BMS's proof of integral Hodge theory and Niziol's K-theoretic proof of p-adic Hodge theory.) so there is no possibility to apply perfectoid space theory to prove motivic theorems such as Beilinson conjecture. So I doubt even the notion of smoothness in scheme theory can be an analogue of the notion of smoothness in manifold theory. It might be happen because the notion of "point" is somewhat different in two theories. My undergraduate dream was scattered and I was desperated.

Of course, I hope somewhat connections between various smoothness in scheme theory based on perfectoid covering.

Definition. Let $X$ be a scheme over a characteristic $p$ algebraically closed field $k$ and $p\ne \ell$. Then $X$ is $\ell$-cohomologically smooth if $f$ is of finite type and for $f:X\to \mathrm{Spec} k$ and any scheme $Y$ over $k$, $Rf^!_X:D^b_c(Y,\mathbb{Q}_{\ell})\to D^b_c(Y\times X,\mathbb{Q}_{\ell})$ is of form $Rf^!_X=\mathbb{Q}_{\ell}(2\dim X)[\dim X]\otimes f^*$. (The terminology and formal definition are from https://www.math.uni-bonn.de/people/scholze/EtCohDiamonds.pdf)

Remark. Of course, there is a relative notion of smoothness by considering separably closed points.

Quasi-conjecture. Let $X$ be a separated finite type scheme over an algebraically closed char. $0$ perfectoid field $K$. Then the followings are equivalent.

(a) $X$ is smooth

(b) $X$ is $p$-cohomologically smooth

(c) There is a Riemann-Hilbert correspondense in sense of https://arxiv.org/abs/1602.06282

(d) There is somewhat smooth base change theorem

Remark. The equivalence of (a) and (b) might fail because of the possibility of existence of imperfect $K$-algebra having zero cotangent complex as Bhatt pointed out (http://www-personal.umich.edu/~bhattb/math/trivial-cc.pdf).

and I also want to conjecture to identify "being perfectoid space" and "vanishing of cohomology on somewhat holonomic D-modules(of course this is not yet defined)" but it is not the main point of this post. My main question is

My Question. Is there real definition of smoothness on schemes/adic spaces analogue to the theory of manifolds?

As above, the notion of smoothenss on scheme/adic space/complex manifold theory is so weak than the notion of smoothness on manifold theory! Even when there is no really smooth scheme over a field $k$, I would not be surprised. Am I right? Sorry for ambiguous question!