TITLE: How do I take the limit as $n$ goes to $\infty$ of $\frac{\sqrt{n}}{\log(n)}$?
QUESTION [3 upvotes]: How do take this limit:
$$ \lim_{n\to\infty} \frac{\sqrt{n}}{\log(n)}$$
I have a feeling that it is infinity, but I'm not sure how to prove it. Should I use L'Hopitals Rule?

REPLY [3 votes]: Let $a(n) = \frac{\sqrt{n}}{\log(n)}$.
We want to show that $a(n)$ grows arbitrarily large.
$a(n^2) = \frac{n}{2\log(n)}$
so $\frac{a(n^2)}{a(n)} = \frac{\sqrt{n}}{2}$,
so, for $n >16$, $\frac{a(n^2)}{a(n)} > 2$.
Iterating or inducting or multiplying,
$\frac{a(n^{2^k})}{a(n)} > 2^k$,
so $a(n)$ gets arbitrarily large.


TITLE: what is this set
QUESTION [0 upvotes]: While trying previous year exam question paper of real analysis i got struck on this particular question and so I am posting here for help.

Let $f\colon\mathbb{R}^{n}\to\mathbb{R}$ be a linear map $f(0,0,\dots)=0$. Then the set $\{f(x_{1},...,x_{n} ):\sum_{j=1}^n x_{j}^2\leq 1\}$ equals

A. $[-a,a]$ for some $a\in\mathbb{R}$, $a\geq 0 $.
B. $[0,1]$
C. $[0,a]$ for some $a\in\mathbb{R}$, $a\geq 0 $.
D. $[a,b]$ for some $a\in\mathbb{R}$, $0\leq a<b $.
I have no idea on how this set will be and so i am sorry bt i have nothing which i can show as an attempt.
I request you to tell in which direction I should proceed .

REPLY [0 votes]: Hint: $f(-x)=-f(x).\,\,\,\,\,$


TITLE: What are some good sources to learn about real analytic manifolds?
QUESTION [7 upvotes]: Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade my knowledge to the real-analytic set up.
Also my target is not to go in the direction of complex algebraic geometry (like Griffiths-Harris). I am primarily interested in building "analytic diffeomorphisms" and "analytic flows" on abstract analytic manifolds. I am interested in some basic materials which will eventually lead me in that direction.
Another question I asked, that may be more precise: What are some general strategies to build measure preserving real-analytic diffeomorphisms?

REPLY [2 votes]: This is a group of references, so to say, for basic foundations on the field. Whenever one deals with real analytic functions, one gets involved with complex analytic functions and analytic sheaves. Even for differentiable matters this is true. Good examples are Malgrange's and Tougeron's classical books on Ideals of differentiable functions, or Bierstone-Milman solution of Glaeser's Problem (I mention this last as a top research example). Also, in my own experience, to do anything with real analytic functions one needs Complex Analytic Geometry. For this I would recommend Naramsimhan's Theory of Analytic Spaces,  Gunning-Rossi's Several Complex Variables, or a book by Whitney, I'm afraid a bit forgotten: Complex Analytic Varieties. To put a context to this comment, which may be far form your specific interests (and then I apologize), I refer to Is there a field of 'real analytic geometry'?
Another important matter is approximation: to what extent one can make analytic anything differential. The basic tool is Whitney's Approximaton Thm, and a very good reference for this is Narasimhan's Real and Complex Analysis.


TITLE: How to solve distributional equations?
QUESTION [0 upvotes]: I'm trying to solve this exercise
If $T \cdot x=1$ in the sense of distributions then $T=\textrm{p.v.}\left(\frac{1}{x}\right)+c \delta_{0}$
Where $\delta_{0}$ is the Dirac delta distribution centered at the origin and $$\left\langle\textrm{p.v.}\left(\frac{1}{x}\right), \varphi\right\rangle=\lim _{\varepsilon \rightarrow 0} \int_{\mathbb{R} \backslash(-\varepsilon, \varepsilon)} \frac{\varphi(x)}{x} d x$$
For any test function $\varphi \in C_c^\infty(\mathbb{R})$
Here's my attempt, equality in the distribution sense must mean equality for any test function, so
$$\int_{\mathbb{R}}T\cdot x \varphi dx = \int_\mathbb{R} \varphi dx \implies \int_\mathbb{R} \varphi (T \cdot x - 1)dx = 0 \implies \int_\mathbb{R} x\varphi (T - \frac{1}{x})dx = 0$$
Since $\varphi$ is any arbitrary test function $\psi = x \varphi$
$$\implies \int_{\mathbb{R} \setminus (-\varepsilon, \varepsilon)} \psi (T - \frac{1}{x})dx + \int_{-\varepsilon}^\varepsilon \psi (T - \frac{1}{x})dx = 0$$
Taking $\varepsilon \rightarrow 0$ gives
$$\int_{\mathbb{R}}T\psi = \lim _{\varepsilon \rightarrow 0} \int_{\mathbb{R} \backslash(-\varepsilon, \varepsilon)} \frac{\psi(x)}{x} d x$$
So $T$ is the Cauchy principal value. But where does the dirac delta appear? If I know nothing about $T$ is it justified to assume that $\langle T, \varphi \rangle = \int_\mathbb{R} T\varphi$ i.e. "integral of something" times the test function? The dirac delta does not seem to really follow example as its just defined as $\langle \delta, \varphi \rangle = \varphi(0)$, nor does the principal value.

REPLY [2 votes]: So $T$ is the Cauchy principal value. But where does the dirac delta appear?

The Dirac delta comes in as the solutions to the homogeneous equation $T \cdot x=0$. If $T_1$ and $T_2$ are two solutions to $T \cdot x=1$ then $T_1$ and $T_2$ differ by a solution to the homogeneous equation.

If I know nothing about $T$ is it justified to assume that $\langle T,
\varphi \rangle = \int_\mathbb{R} T\varphi$ i.e. "integral of
  something" times the test function?

No, it is not. The distribution doesn't have to be a function. For example, $\delta$ is not a function.
Because of this your solution is not good.
I'd like to suggest another approach:

Show that $x \cdot \textrm{p.v.}\left( \frac{1}{x} \right) = 1$ as a distribution.
Then add the solutions to the homogeneous equation.


TITLE: If $[K:k]=2$ then there is $a\in K$ s.t. $K=k(\alpha )$ and $\alpha ^2=a$.
QUESTION [4 upvotes]: 1) Let $K/k$ an extension field. If $[K:k]=2$ show that there is $a\in K$ s.t. $K=k(\alpha )$ and $\alpha ^2=a$.
2) Does it work for $[K:k]=p$ when $p$ is prime ? Or for other number ? (like non-prime)
My work
For 1), if $[K:k]=2$ there is an $a\in K$ that is not in $k$. Let $X^2-\alpha $ its minimal polynomial. Then $$a^2-\alpha =0\implies (a-\sqrt \alpha )(a+\sqrt \alpha )=0\implies a=\pm\sqrt \alpha $$
It looks to be very confusing... any idea ?

REPLY [2 votes]: You have to be careful in the logical order of your arguments here.
It is correct to say that $[K:k] = 2$ implies there is an $a \in K \setminus k$. If its minimal polynomial is $X^2 - \alpha$ for $\alpha \in k$ then you certainly have that $a^2 - \alpha = 0$. This implies that $a^2 = \alpha$. This means that $a$ is a square root of $\alpha$ and that $-a$ is another (indeed, the only other) square root of $\alpha$ in $K$. At this point, however, the symbol $\sqrt{\alpha}$ is meaningless, believe it or not. You have not chosen any embedding of $K$ into a field where $\sqrt{\alpha}$ has already been defined (for example, some algebraic closure of $k$) and so it is not yet actually defined. You could define $\sqrt{\alpha} := a$ in $K$ and then you would of course have $-\sqrt{\alpha} = -a$ but there is little to be gained from this.
Instead of all of that, since it is clear that there is some element in $K\setminus k$ (namely $a$), your only mission is to show that in fact $a$ generates the extension - i.e. $K = k(a)$. This will be true because $K$, being a finite field extension of $k$ of degree 2, is necessarily a $k$-vector space of dimension $2$. Since $a \not\in k$, it is linearly independent from $1 \in k$ and so $\{1,a\}$ is a maximally linearly independent set in the vector space $K$ and is thus a basis. This means every element $b \in K$ can be written as $b = c_1\cdot 1 + c_2\cdot a$ for $c_1, c_2 \in k$. Hence $K = k(a)$.
As for your second question, I think the answer to be given here is that for 'nice' extensions (those being finite, separable ones), the statement remains true. That is, an extension $K/k$ is generated by a single element (called a 'primitive element'). This result is known as the Primitive Element Theorem. The caution is that for extension degree higher than 2, a primitive element need not be a $d^{th}$ root of an element of the base field (for $d$ the degree of the extension).

REPLY [1 votes]: Suppose the characteristic is not 2 and $[K:k]=2$. Let $x\in K$, $x$ is not in $k$, $1,x$ is a basis of the 2-dimensional space $K$, $x^2=ax+b, a,b\in K$, $x^2-ax-b=(x-a/2)^2-a^2/4-b=0$
thus $(x-a/2)^2=b+a^2/4$. We have $K=k(x-a/2)$ and $(x-a/2)^2\in k$.


TITLE: Expected number of times a random walk of n steps starting from origin and ending at x passes a point
QUESTION [0 upvotes]: I wonder what is the expected number of times a random walk of n steps starting from origin and ending at x passes a point m located between 0 and x or after x? I just found the occupancy equation and probability of return to the origin and no similar question.

REPLY [3 votes]: Let $W(x,n)$ be the number of walks of $n$ steps producing a net displacement of $x$ (equivalently, starting at $0$ and ending at $x$).
$$W(x,n)={n \choose \frac{x+n}{2}} \quad [x=n \pmod 2]\,[-n \le x \le n]$$
Let $Z_t$ be the indicator variable of the event "a random walk of $n$ steps starting at $0$ and ending at $x$ is found at position $m$ at step $t$".
Then $$ P(Z_t=1)=E(Z_t)=\frac{W(m,t)W(x-m,n-t)}{W(x,n)}$$
Then, by linearty of expectations, the desired result is
$$ E(\sum_{t=0}^{n-1} Z_t) =\sum_{t=0}^{n-1} \frac{W(m,t)W(x-m,n-t)}{W(x,n)} $$


TITLE: Calculate pairwise cosine distance only returning the lower triangular matrix
QUESTION [1 upvotes]: I have a matrix, where each row is a feature vector.
I would like to find out the pairwise cosine distance between all of these feature vectors.
The cosine value between all rows in a matrix could be calculated as
$\cos(M) = M' \cdot M'^T$
Where $M'$ is the row normalized matrix of $M$.
This creates a symmetric matrix of pairwise cosine distances, and it is pretty fast to calculate this using the sparse matrix library in scipy. I was wondering if there is an alternative method of only calculating the lower triangular matrix of this, rather than the full symmetric matrix? Hopefully this would be dividing the necessary computational time by two and saving a bit of space.
Thanks

REPLY [0 votes]: Y = pdist(X, 'cosine') does that directly -- see
scipy.spatial.distance.pdist .
(spatial.distance seems to be missing in the doc for scipy.spatial in v0.16.0, so hard to find :/ )


TITLE: Showing that $\frac{ 1- \sin\frac{5\pi}{18}}{\sqrt{3} \sin \frac{5\pi}{18}}= \tan\frac{\pi}{18} $
QUESTION [2 upvotes]: It's easy to verify on WolframAlpha, but I have difficulty deriving it.
It's easy to see
$$
\tan\left(\frac{\pi}{18}\right)=\frac{\sqrt{3}-\tan(5/18 \pi)}{1+\sqrt{3}\tan(5/18 \pi)}
$$

REPLY [1 votes]: Evaluate,
$$\frac1{\sin\frac{5\pi}{18}}-\sqrt3 \tan\frac{\pi}{18}
=\frac{\cos\frac{\pi}{18}-\sqrt3 \sin\frac{5\pi}{18}\sin\frac{\pi}{18}}{\sin\frac{5\pi}{18}\cos\frac{\pi}{18}}\tag 1$$
Examine the numerator,
$$\cos\frac{\pi}{18}-\frac{\sqrt3}2\cdot2 \sin\frac{5\pi}{18}\sin\frac{\pi}{18}$$
$$= \cos\frac{\pi}{18}-\cos\frac\pi6(\cos\frac{2\pi}{9}-\cos\frac{\pi}{3})$$
$$= \cos\frac{\pi}{18}-\frac12(\cos\frac{\pi}{18} + \cos\frac{7\pi}{18})+\cos\frac{\pi}{6}\cos\frac{\pi}{3}$$
$$= \frac12(\cos\frac{\pi}{18} - \cos\frac{7\pi}{18})+\frac12\cos\frac{\pi}{6}$$
$$= \sin\frac{\pi}{6} \sin\frac{2\pi}{9}+\frac12\cos\frac{\pi}{6}$$
$$= \frac12( \sin\frac{2\pi}{9}+\sin\frac{\pi}{3})
=\sin\frac{5\pi}{18}\cos\frac{\pi}{18}$$
Substitute the result for the numerator into (1) to have
$$\frac1{\sin\frac{5\pi}{18}}-\sqrt3 \tan\frac{\pi}{18}= 1$$
Then, rearrange to obtain,
$$\frac{ 1- \sin\frac{5\pi}{18}}{\sqrt{3} \sin\frac{5\pi}{18}}= \tan\frac{\pi}{18}$$


TITLE: A claiming series to be of rational values
QUESTION [0 upvotes]: $$P_m(q)=1+c_m\sum_{k=1}^{\infty}{n^{2m+1}{q^k\over 1-q^k}}\tag1$$

$m:=1,2,3,...$
$$P_1(q)=1+240\sum_{k=1}^{\infty}{n^{3}{q^k\over 1-q^k}}\tag a$$
$$P_2(q)=1-504\sum_{k=1}^{\infty}{n^{5}{q^k\over 1-q^k}}\tag b$$
$$P_3(q)=1+480\sum_{k=1}^{\infty}{n^{7}{q^k\over 1-q^k}}\tag c$$
See here for values of $c_m$
$m\ge1$ and $x\ge0$
Then we have $$P_m(q)=[-4(4x+3)]^{m+1}\tag2$$
where $$q=e^{-{\pi\over \sqrt{4x+3}}}$$

$(2)$ is apparently rational values but how do we show that $P_m(q)=[-4(4x+3)]^{m+1}?$

REPLY [1 votes]: (2) doesn't hold. These are Eisenstein series. We do have $P_3=P_1^2$
but not $P_3=P_1^{3/2}$. Indeed
$$P_1^3-P_2^2=1728q\prod_{n=1}^\infty(1-q^n)^{24}.$$


TITLE: If a random variable is a linear combination of another random variable does it have the same $\mu, \sigma^2$?
QUESTION [0 upvotes]: If a random variable is a linear combination of another random variable does it have the same $\mu, \sigma^2$ ?
Is the variance the same but not the expectation? Please explain me by which by principles this hold or not.

REPLY [4 votes]: If by 'linear combination' of $X$ you mean a random variable of the type $Y=aX+b$ then mean of $Y$ is $a\mu +b$ and variance is $a^{2}\sigma^{2}$ where $\mu$ is the mean of $X$ and $\sigma ^{2}$ is its variance. These follow immediately from the definitions.


TITLE: Motivation behind the definition of equivalence of categories.
QUESTION [1 upvotes]: The background:
The naive notion of isomorphism of categories is that: a functor $F:\mathscr{C}\rightarrow\mathscr{D}$ is an isomorphism if there exists $F^{-1}:\mathscr{D}\rightarrow\mathscr{C}$ such that $F^{-1}\circ F$ and $F\circ F^{-1}$ are equal to identity map.
I'm starting my studies about category theory from the sight of representations theory. The book that I'm following (Introduction to Representations Theory - Etingof et al) states that this naive notion is not useful and show the following example:

Let $\mathscr{C}_1$ the most simple category. $Ob(\mathscr{C}_1)=\{X\}$ and $\textrm{Hom}(X,X)=\{id_X\}$. Also let $\mathscr{C}_2$ the category that have the objects $X$ and $Y$, and the following four morphisms: $1_x,1_Y,a: X\rightarrow Y,b:Y\rightarrow X$. So we must have $a\circ b=1_Y$ and $b\circ a=1_X$. It easy to check that for any category $\mathscr{D}$ there is a natural bijection between the collections of isomorphism classes of functors $\mathscr{C}_1\rightarrow \mathscr{D}$ and $\mathscr{C}_2\rightarrow \mathscr{D}$.

This show us that for "practical purposes" the categories $\mathscr{C}_1$ and $\mathscr{C}_2$ are the same but using the naive notion they will be not isomorphic (they don't have the same number of objects).
My question:
My doubt maybe is quite simple and I would like if the answer could be detailed for a beginner on this subject. Why is it easy to check that for any category $\mathscr{D}$ there is a natural bijection between the collections of isomorphism classes of functors $\mathscr{C}_1\rightarrow \mathscr{D}$ and $\mathscr{C}_2\rightarrow \mathscr{D}$? How to check this step by step?

REPLY [4 votes]: Functors $\mathscr{C}_1\rightarrow \mathscr{D}$ are completely determined by what object $X$ gets sent to. Two such functors $F,G$ are isomorphic iff the objects $F(X)$ and $G(X)$ are isomorphic, indeed the data of an isomorphism $F(X)\cong G(X)$ is all you need to define a natural isomorphism from $F$ to $G$. Hence isomorphism classes of functors $\mathscr{C}_1\rightarrow \mathscr{D}$ are indexed by isomorphism classes of objects in $\mathscr{D}$.
On the other hand, a functor $F:\mathscr{C}_2\rightarrow \mathscr{D}$ is given by a choice of two isomorphic objects $F(X)$ and $F(Y)$ and a choice of an isomorphism between them. This a priori seems like it is keeping track of more information, and it is, however we are looking at isomorphism classes of functors not the functors themselves. The claim now is that two such functors $F, G:\mathscr{C}_2\rightarrow \mathscr{D}$ are isomorphic iff $F(X)$ is isomorphic to $G(X)$.
The "only if" part I think should be clear, so to show the "if" part we want to take an isomorphism $\phi$ (of objects) from $F(X)$ to $G(X)$ and construct a natural isomorphism between $F$ and $G$. Well all that amounts to is constructing an isomorphism from $F(Y)$ to $G(Y)$, which makes the one commutative diagram you can possibly write down commute (I'll leave it to you to actually write it down).  Anyway $G(a)\circ \phi \circ F(b)$ should be exactly the isomorphism needed. Hence isomorphism classes of functors $\mathscr{C}_2\rightarrow \mathscr{D}$ are also indexed by isomorphism classes of objects in $\mathscr{D}$.


TITLE: Solving particle in vertical motion with air resistance using conservation of energy
QUESTION [0 upvotes]: A particle of unit mass is projected vertically upwards with speed u. At any height x, while the particle is moving upwards, it is found to experience a total force F, due to gravity and air resistance, given by $F = \alpha e^{-\beta x}$, where $\alpha$ and $\beta$ are positive constants. Calculate the energy expended in reaching a given height z. Show that
$F=\frac 1 2 \beta v^2 + \alpha - \frac 1 2 \beta u^2$
where $v$ is the speed of the particle, and explain why $\alpha = \frac 1 2 \beta u^2 + g$
  where $g$ is the acceleration due to gravity.
Determine an expression, in terms of $y$, $g$ and $\beta$, for the air resistance experienced by the particle on its downward journey when it is at a distance $y$ below its highest point.

Fairly long question but I can solve it for the most part.
The problem I have is the last part.
The model solution requires I use
$v\frac {dv} {dy}= g - \frac 1 2 \beta v^2$ (since the air res. is going the other way)
and it leads to $\frac 1 2 \beta v^2 = g(1-e^{-\beta y})$.
When I was solving it I tried to use conservation of energy, $mgy = \frac 1 2 v^2$, where $m$ = 1. Which lead to $\frac 1 2 \beta v^2 = \beta g y$. Why is this wrong?
Many thanks in advance,
Chris

REPLY [0 votes]: Air resistance (also any sort of friction) renders the use of conservation of energy useless for the problem, because these constitute the "loss" of energy to some ignored sink.  (For example, air resistance heats up the air, but you never carry that into your calculations, it would be impossible.)
This is why they call these "nonconservative" forces.
Of course, energy is conserved in the real world, but in just the problem you are considering, energy is magically "lost" because of the nonconservative force of friction or resistance.


text:
\section{Witt vector and $p$-adic cohomology}
\label{sec:witt_cohomology_and_wo_rational_sings}

In this section we recall the basic properties of Witt vector and $p$-adic cohomology groups and direct images.  For the results related to Witt vector cohomology we refer to \cite{Berthelot_Bloch_Esnault_On_Witt_vector_cohomology_for_singular_varieties} and \cite{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational} for original accounts, and to
\cite[Section 2.5]{Gongyo_Nakamura_Tanaka_Rational_points_on_log_Fano_threefolds_over_a_finite_field} for an accessible summary.  The necessary results concerning $p$-adic cohomology can be found in the classical reference \cite[Exposé V, VI]{SGA5}.  For the sake of clarity, we recall most of the definitions and some useful properties.

\subsection{Basics of W$\sO$-cohomology}

First, we recall that for every scheme defined over $\FF_p$ there exists a sequence of schemes $W_n(X)$, for $n \geq 1$, underlying the same topological space $X$ and with varying sheaves of rings $W_n\sO_X$ given by Witt vectors of length $n$, see \cite[II, Section 6]{Serre_Local_fields} or \cite[Section 2.5]{Gongyo_Nakamura_Tanaka_Rational_points_on_log_Fano_threefolds_over_a_finite_field},.  We define the sheaf $W\sO_X $ on $X$ as the following inverse limit taken in the category $\Ab(X)$ of sheaves of abelian groups
\[
W\sO_X \eqdef \varprojlim_n W_n\sO_X.
\]
We note that unlike $W_n(X) = (X,W_n\sO_X)$ the pair $(X,W\sO_X)$ is not a scheme.

As will be stated in \autoref{def:wo_rational_morphism}, one of the defining properties of $W\sO$-rational singularities is that $R^i f_* W\sO_{Y,\QQ}=0$ for every integer $i>0$ and every quasi-resolution of the singularities (see \autoref{defin:quasi_resolution}). The question is however how one defines $R^i f_* W\sO_{Y,\QQ}$. The naive idea is to let $R^i f_* W\sO_{Y,\QQ}$ be defined as $(R^i f_* W\sO_{Y}) \otimes_{\bZ} \bQ$. According to the understanding of the authors, it is not known whether this definition is the correct one.  The main reason is that it is not known whether the $p^{\infty}$-torsion in $R^i f_* W \sO_{Y}$ is annihilated by $p^M$ for a single integer $M>0$ or not.  If the inverse system $\{ R^i f_* W_n \sO_X \}_{n \in \NN}$ satisfied the Mittag-Leffler condition, then there would not be $p^\infty$-torsion of infinite height by the arguments of \cite[Prop 2.10]{Berthelot_Bloch_Esnault_On_Witt_vector_cohomology_for_singular_varieties}.
However, to the best of our knowledge, it is again not known whether this Mittag-Leffler property always holds true. Hence, to avoid these problems, the present definitions define $R^i f_* W\sO_{Y,\QQ}$ so that its $p^{\infty}$-torsion is annihilated by $p^M$ for a single integer $M>0$. In particular, the definition becomes slightly cumbersome, in a sense that it uses the notion of localization of a category in a Serre subcategory.

\begin{remark}
\label{remark:localization}
The notion of localization of a category in a full subcategory is well-explained in \stacksproj{02MN}.  For the original account we refer to \cite[Chapitre III]{Gabriel_Des_Categories_abeliennes}.
\end{remark}

In our case, this is applied to the full subcategory $\Ab(X)_{\btor}$ of the category $\Ab(X)$ of abelian sheaves on a scheme $X$, defined by the formula:
\[
A \in {\rm Ob}\left(\Ab(X)_{\btor}\right) \iff \exists N \in \bZ \setminus 0  \ : \  N \cdot \id_A = 0.
\]
The category obtained by localizing  $\Ab(X)$ in $\Ab(X)_{\btor}$ is denoted  $\Ab(X)_{\QQ}$. The category  $\Ab(X)_{\QQ}$  is called the category of $\QQ$-localized abelian sheaves, and it is an abelian category. One intuitive description of this localization is by formal inverting of all arrows in $\Ab(X)$ whose kernel and cokernel are in $\Ab(X)_{\btor}$.  We denote the natural projection functor by $q \colon \Ab(X) \to \Ab(X)_{\QQ}$ or simply $(-)_{\QQ}$, and we note that:
\begin{itemize}
\item  $q(A) = 0$ if and only if $A \in \Ab(X)_{\btor}$ \cite[3.7.2, line 6]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational},
\item $\Hom_{\Ab(X)_{\QQ}} (q(\sF) , q(\sG))= \Hom_{\Ab(X)}(q(\sF), q(\sG)) \otimes_{\bZ} \bQ$ for any $\sF, \sG \in \Ab(X)$ \cite[Prop 3.7.4]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}, and
 \item the functor $q$ is exact \stacksproj{02MN} .
 \end{itemize}
 In particular, the above setting is used to define $W\sO_{X,\QQ}$:
 \begin{equation*}
  W\sO_{X,\QQ} \eqdef q( W\sO_X) \in\Ab(X)_{\QQ}.
 \end{equation*}
The next goal is to define the derived pushforwards of $W\sO_{X,\QQ}$.
  By \cite[Cor 3.7.5]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}, if $f \colon X \to Y$ is a morphism of varieties over $k$, the natural functor $f_* \colon \Ab(X) \to \Ab(Y)$ descends to localized categories yielding $f_* \colon \Ab(X)_{\QQ} \to \Ab(Y)_{\QQ}$.  Moreover, by \cite[Prop 3.7.7]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}, both functors admit total right derived functors $Rf_*$ which satisfy compatibility condition expressed by commutativity of the diagram:
\[
\xymatrix{
D^+(\Ab(X)) \ar[d]_{D^+(q)} \ar[r]^{Rf_*} & D^+(\Ab(Y)) \ar[d]^{D^+(q)} \\
D^+(\Ab(X)_{\QQ}) \ar[r]^{Rf_*} & D^+(\Ab(Y)_{\QQ}).
}
\]
As a consequence, the cohomology groups satisfy the equation $(R^if_*A)_{\QQ} \isom R^if_*(A_{\QQ})$, and hence this setup allows ups to define $R f_* W \sO_{X,\bQ}$:

\begin{definition}
Let $f \colon X \to Y$ be a morphism of varieties over $k$.  We define the $i$-th derived pushforward of $W\sO_{X,\bQ}$ by $R^if_* \left( W\sO_{X,\QQ} \right)  \in \Ab(Y)_{\QQ}$.  In particular, the Witt vector cohomology $H^i(X,W\sO_{X,\QQ}) \in {\rm Mod(W(k))}_{\QQ}$ of a scheme $X$ defined over a perfect field $k$ is given by the formula:
\[
H^i(X,W\sO_{X,\QQ}) = R^i\Gamma  \left( W\sO_{X,\QQ} \right),
\]
where $\Gamma \colon X \to \Spec(k)$ is the structure map.
\end{definition}

\begin{remark}
\label{rem:Q_Witt_coho_proper_case}
According to \cite[Remark 2.17.(5)]{Gongyo_Nakamura_Tanaka_Rational_points_on_log_Fano_threefolds_over_a_finite_field},
if $X$ is proper over $k$, then $H^i(X,W\sO_{X,\QQ})=0$ if and only if $H^i(X,W\sO_{X}) \otimes_{\bZ} \bQ=0$. As explained in the first paragraph of the section, it is not known if the corresponding statement for a proper birational morphism holds true.
\end{remark}

Witt vector cohomology relates to finite length Witt vector cohomology via the following results. These results are deduced from Grothendieck spectral sequence for the composition of derived functor $R\lim \circ Rf_* = Rf_* \circ R\lim$ applied for the system $n \mapsto W_n\sO_X$ which satisfies Mittag-Leffler condition.

\begin{proposition}\label{prop:witt_vectors_and_lim}
\cite[Lem 2.18]{Gongyo_Nakamura_Tanaka_Rational_points_on_log_Fano_threefolds_over_a_finite_field}
Let $f \colon X \to Y$ be a morphism of varieties over $k$.  Then, for every $i \geq 0$, there exists a short exact sequence:
\[
0 \ra R^1  \varprojlim_{n} \left(R^{i-1}f_*W_n\sO_X \right) \ra R^if_* W\sO_X \ra \left( \varprojlim_n R^if_* W_n\sO_X \right) \ra 0.
\]
In particular, if $X$ is a proper scheme over  $k$, then for all $i$ the system $n \mapsto H^i(X,W_n\sO_X)$ satisfies the Mittag-Leffler condition, and hence
\[
H^i(X,W\sO_{X,\QQ}) = q\left( \varprojlim_{n} H^i(X,W_n\sO_X)\right) \expl{=}{\autoref{rem:Q_Witt_coho_proper_case}} \left( \varprojlim_{n} H^i(X,W_n\sO_X)\right) \otimes_{\bZ} \bQ.
\]
\end{proposition}

We will also use the following result relating Witt vector cohomology with the rigid cohomology, that is, the substitute of crystalline cohomology in the singular setting.  Since the main application of the result is actually the comparison between Witt vector and $p$-adic cohomology (see \autoref{thm:clr_rigid_zero_padic} and \autoref{cor:p_adic_slope_zero_of_witt_vector}), and rigid cohomology can be treated as a black box for this purpose, we do not include any details, referring to the standard literature on the topic such as \cite{Berthelot_cohomologie_rigide}.

\begin{theorem}[{\cite[Theorem 1.1]{Berthelot_Bloch_Esnault_On_Witt_vector_cohomology_for_singular_varieties}}]
\label{thm:bbe_rigid_witt_vector}
Let $k$ be a perfect field of characteristic $p > 0$ and let $X$ be a $k$-scheme of finite type. There exists a functorial isomorphism:
\[
H^{\bullet}_{\rm rig}(X/ K)^{<1} \isomto H^\bullet(X, W\sO_{X,\QQ}),
\]
where $K:=\Frac(W(k))$.
\end{theorem}

\subsection{W$\sO$-singularities}

Since singular schemes naturally show up in our approach to the main theorem, and since we use a Witt-vector cohomology based invariant to show non-uniruledness (see \autoref{lem:uniruled_witt_vector}), we need a characteristic $p>0$ notion of singularities allowing for the control of Witt vector cohomology.  We will use the notion of W$\sO$-rational singularities introduced in the papers \cite{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational,Blickle_Esnault_Rational_singularities_and_rational_points}.

The definition uses the notion of \emph{quasi-resolution} (see \cite[Definition 4.3.1]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}).  For the sake of clarity we recall some of its details.  We say that a normal equidimensional scheme $X$ defined over a field $k$ is a \emph{finite quotient} if there exists a finite surjective morphism $Y \to X$ from a scheme $Y$ smooth over $k$.  A scheme is a \emph{topological finite quotient} if there exists a universal homeomorphism from (equiv. to) a finite quotient as defined above.

\begin{definition}[{\cite[Definition 4.3.1]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}}]
\label{defin:quasi_resolution}
We say that a morphism between two integral varieties $f \colon X \to Y$ is a \emph{quasi-resolution} if the following conditions are satisfied:
\begin{enumerate}
    \item $X$ is a topological finite quotient,
    \item $f$ is projective, surjective, and generically finite,
    \item the extension of the function fields $k(Y) \subset k(X)$ is purely inseparable.
\end{enumerate}
\end{definition}
\noindent We note that if $Y$ is normal the last condition is actually equivalent to the Stein factorization $X' \to Y$ being a universal homeomorphism.  By the work of de Jong \cite{de_Jong_Smoothness_semi_stability_and_alterations} for every variety $X$ over a perfect field $k$ there exists a quasi-resolution with $X$ as a target (see \cite[Remark 4.3.2]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}).  More precisely, the alterations of de Jong could be chosen to be compositions of quotients of smooth varieties by finite group actions and quasi-resolutions.

\begin{lemma}
\label{lem:etale_base_change_for_quasi_resolutions}
An \'etale base change of a quasi-resolution is a quasi-resolution.
\end{lemma}

\begin{proof}
Let $X \to Y$ be a quasi-resolution and let $Y' \to Y$ be an \'etale morphism.  Take $U \to X$ to be a morphism furnishing the topological finite quotient structure on $X$.  We consider the following diagram of cartesian squares:
\[
\xymatrix{
    U' \ar[r]\ar[d]_{\text{\'etale}} & X' \ar[r]\ar[d]_{\text{\'etale}} & Y' \ar[d]_{\text{\'etale}} \\
    U \ar[r] & X \ar[r] & Y
}
\]
The morphism $U' \to U$ is \'etale and therefore $U'$ is smooth.  Consequently $U' \to X'$ yields a topological finite quotient structure on $X'$.  Using standard base change properties the map $X' \to Y'$ satisfies the conditions from \autoref{defin:quasi_resolution}, and hence $X' \to Y'$ is a quasi-resolution as desired.
\end{proof}

\begin{definition}[{\cite[Definition 4.4.1 and Proposition 4.4.6]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}}]\label{def:wo_rational_morphism}
Let $X$ be a variety over $k$.  We say that $X$ has \emph{$W\sO$-rational singularities} if and only if for every (equiv.\,some) quasi-resolution $f \colon Y \to X$ the following two conditions are satisfied:
\begin{enumerate}
\item the natural map induces an isomorphism $W\sO_{X,\QQ} \isom f_*W\sO_{Y,\QQ}$,
\item $R^if_*W\sO_{Y,\QQ} = 0$, for every $i > 0$.
\end{enumerate}
\end{definition}

\begin{remark}\label{remark:some_remarks_on_WO_rationality}
We note that by compatibility of derived functor of $f_*$ and localization the following statements concerning the definition hold true:
\begin{enumerate}[(a)]
\item if $X$ is normal then the first condition is automatically satisfied,
\item the conditions are equivalent to the fact that the natural map in the derived category of $\QQ$-localized abelian sheaves $W\sO_{X,\QQ} \to Rf_*W\sO_{Y,\QQ}$ is an isomorphism.
\end{enumerate}
In what follows we say that a morphism satisfying the above conditions is \emph{W$\sO$-rational}.  Using this notion we may rephrase W$\sO$-rationality of a scheme $X$ by postulating the existence of a W$\sO$-rational quasi-resolution $Y \to X$.
\end{remark}

We thank Kay R\"ulling for the following result and the subsequent proof of \'etale invariance of $W\sO$-rational singularities.

\begin{proposition}\label{prop:wo-rationality_finite_level}
Let $f \colon Y \to X$ be a proper, surjective and generically finite morphism of varieties over $k$.  Then the following conditions are equivalent:
\begin{enumerate}
\item\label{prop:wo-rationality_finite_level_it1} the morphism $f$ is W$\sO$-rational, that is, $W\sO_{X,\QQ} \to f_*W\sO_{Y,\QQ}$ is an isomorphism and $R^if_*W\sO_{Y,\QQ} = 0$, for every $i>0$,
\item\label{prop:wo-rationality_finite_level_it2} there exists an integer $M>0$ such that for every $n \geq 1$ the abelian sheaves $f_*W_n\sO_Y/W_n\sO_X$ and $R^if_*W_n\sO_X$, for $i \geq 1$, are annihilated by $p^M$.
\end{enumerate}
\end{proposition}

\begin{proof}
We first prove that \autoref{prop:wo-rationality_finite_level_it2} $\Rightarrow$ \autoref{prop:wo-rationality_finite_level_it1}.  We set $C_n = f_*W_n\sO_Y/W_n\sO_X$ and consider the following exact sequence of systems of abelian sheaves on $X$:
\[
0 \ra \{ W_n\sO_X \}_{n \in \NN} \ra \{ f_*W_n\sO_Y \}_{n \in \NN} \ra \{ C_n \}_{n \in \NN} \ra 0.
\]
Using the long exact sequence of $R^\bullet\varprojlim$ groups, and recalling that $W_n\sO_X$ and $f_*W_n\sO_X$ satisfy Mittag-Leffler condition, we obtain:

\begin{center}
\begin{tikzpicture}[descr/.style={fill=white,inner sep=1.0pt}]
        \matrix (m) [
            matrix of math nodes,
            row sep=2em,
            column sep=2.5em,
            text height=1.5ex, text depth=0.25ex
        ]
        { 0 & \overbrace{W\sO_X}^{\varprojlim W_n\sO_X} & \overbrace{f_*W\sO_Y}^{\varprojlim f_*W_n\sO_Y} & \varprojlim C_n \\
            & \underbrace{R^1\varprojlim W_n\sO_X}_{\isom 0} & \underbrace{R^1\varprojlim f_*W_n \sO_Y}_{\isom 0} & R^1\varprojlim C_n & 0. \\ };

        \path[overlay,->, font=\scriptsize,>=latex]
        (m-1-1) edge (m-1-2)
        (m-1-2) edge (m-1-3)
        (m-1-3) edge (m-1-4)
        (m-1-4) edge[out=355,in=175] (m-2-2)
        (m-2-2) edge (m-2-3)
        (m-2-3) edge (m-2-4)
        (m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{center}

\noindent Taking images in the localized category, and hence annihilating the elements $\varprojlim C_n$ and $R^1\varprojlim C_n$, which are $p^M$-torsion, we obtain the desired statement for $i = 0$.  Then using \autoref{prop:witt_vectors_and_lim}, we have an exact sequence
\[
0 \ra R^1\varprojlim R^{i-1}f_*W_n\sO_Y \ra R^if_*W\sO_Y \ra \varprojlim R^if_*W_n\sO_Y \ra 0.
\]
Again all the elements in the relevant inverse systems are annihilated by $p^M$, for some fixed $M$, and therefore both the right and left elements in the above exact sequence are annihilated by $p^M$ as well.  Consequently, the middle term is $p^{2M}$-torsion and hence the implication is proven.

We now approach \autoref{prop:wo-rationality_finite_level_it1} $\Rightarrow$ \autoref{prop:wo-rationality_finite_level_it2}.  For this purpose we apply \autoref{remark:some_remarks_on_WO_rationality} and the definition of $\Ab(X)_{\bQ}$ to see that there is an integer $N>0$ such that

$\sQ_i$ is $p^N$-torsion, where
\begin{equation}
\label{eq:wo-rationality_finite_level:def_Q}
\sQ_i:=
\left\{
\begin{matrix}
\factor{f_* W \sO_Y}{W\sO_X} & \textrm{if }i=0 \\
R^i f_* W \sO_Y & \textrm{if }i\neq0
\end{matrix}
\right.
\end{equation}
Consider the exact sequence
\begin{equation}
\label{eq:wo-rationality_finite_level:short_exact_sequence}
\xymatrix{
0 \ar[r] & W\sO_Y \ar[r]^{V^n} & W \sO_Y \ar[r] & W_n \sO_Y \ar[r] & 0
}
\end{equation}
The long exact sequence obtained from \autoref{eq:wo-rationality_finite_level:short_exact_sequence} by applying derived pushforwards sandwiches $R^i f_* W_n \sO_Y$ between $\sQ_i$ and $\sQ_{i+1}$. As the latter are $p^N$-torsion, we obtain that $R^i f_* W_n \sO_Y$ is $p^{2N}$ torsion for all $i \geq 0$ and all $n \geq 1$.

We have to show the same statement for $\sQ_0$ too. For that, consider the following diagram where all rows and columns are exact. To see this, start with the two leftmost non-zero columns that we already know are exact. Apply then the snake lemma first two the first three non-zero entries of these columns, and then to the second and third non-zero rows.
\begin{equation*}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
0 \ar[r] & W\sO_X \ar[r]\ar[d]^{V^n} & f_*W\sO_Y \ar[r]\ar[d]^{V^n} & \sQ_0 \ar[d] \ar[r] & 0 \\
0 \ar[r] & W\sO_X \ar[r]\ar[d] & f_*W\sO_Y \ar[r]\ar[d] & \sQ_0 \ar[d]  \ar[r] & 0 \\
0 \ar[r] & W_n\sO_X \ar[r] \ar[d] & f_*W_n\sO_Y \ar[r] \ar[d] & \sQ_n \ar[r] \ar[d] & 0. \\
&  0  \ar[r] & \Ker(V^n : R^1 f_* W \sO_Y \to R^1 f_* W \sO_Y ) \ar[d] \ar[r] & \ar[d] \sE \ar[r] & 0 \\
&  & 0 & 0
}
\end{equation*}
As, we already know that $\sE$ and $\sQ_0$ are $p^N$ torsion, we obtain that $\sQ_n$ is indeed $p^{2N}$ torsion for every integer $n \geq 1$.

That is, we obtain the statement of \autoref{prop:wo-rationality_finite_level_it2}, for $M = 2N$.

\end{proof}

\noindent As a corollary of \autoref{prop:wo-rationality_finite_level} and \cite[Proposition 4.4.9]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}, we obtain:

\begin{proposition}\label{prop:properties_witt_rational}
Let $X$ be a variety over $k$.  Then the following assertions hold:
\begin{enumerate}
\item\label{it1:properties_witt_rational} the notion of $W\sO$-rationality is local in the \'etale topology,
\item\label{it2:properties_witt_rational} if $X' \to X$ is a universal homeomorphism then $X$ is W$\sO$-rational if and only if $X'$ is.
\end{enumerate}
\end{proposition}

\begin{proof}
The result \autoref{it2:properties_witt_rational} is proven \cite[Proposition 4.4.9]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}.  For \autoref{it1:properties_witt_rational}, we observe that \cite[I, Proposition 1.5.8]{Illusie_Complexe_de_de_Rham_Witt_et_cohomologie_cristalline} implies that for every surjective \'etale morphism $U \to X$ the associated map $W_n(U) \to W_n(X)$ is also surjective \'etale, and consequently for a quasi-resolution $f \colon Y \to X$ we obtain a cartesian diagram with surjective \'etale rows:
\[
\xymatrix{
    W_n Y \ar[d] & W_n(Y \times_X U) \ar[d]\ar[l] \\
    W_n X & W_n(U), \ar[l]^{W_n(\pi)}
}
\]
and $f_U \colon Y \times_X U \to U$ is a quasi-resolution according to \autoref{lem:etale_base_change_for_quasi_resolutions}.  Consequently, using flat base change (note that as above $W_n(\pi)$ is \'etale and hence flat), we see that
\[
R^if_{U,*} W_n\sO_{Y \times_X U} = W_n(\pi)^* R^if_* W_n\sO_Y.
\]
Since $W_n(\pi)$ is in fact faithfully flat, this implies that for every $M>0$ the integer $p^M$ annihilates $R^if_* W_n\sO_Y$ if and only if it annihilates $R^if_{U,*} W_n\sO_{Y \times_X U}$, which finishes the proof by \autoref{prop:wo-rationality_finite_level}.
\end{proof}

\begin{proposition}
\label{prop:separable_base_change_for_WO_rationality}
If $X$ is a W$\sO$-rational variety over $k$, and $K/k$ is a  separable (possibly infinite) algebraic field extenstion, then the variety $X_K$ is W$\sO$-rational.
\end{proposition}

\begin{proof}
Let $f \colon Y \to X$ be a W$\sO$-rational quasi-resolution, and
let $\{K_\lambda\}_{\lambda \in I}$ be an ascending system of finite separable extension $k \subset K_\lambda \subset K$ such that $\bigcup_\lambda K_\lambda = K$. Set $\pi \colon X_K \to X$, $\psi \colon Y_K \to Y$, $\pi_{K_\lambda} \colon X_{K_\lambda} \to X$ and $\psi_{K_\lambda} \colon Y_{K_\lambda} \to Y$ to be the natural maps.   We shall prove that the base change $f_K \colon Y_K \to X_K$ is a W$\sO$-rational quasi-resolution.  It is a quasi-resolution by the arguments of  \autoref{lem:etale_base_change_for_quasi_resolutions}.  In order to see that it is also W$\sO$-rational we reason as follows.  Using \autoref{prop:wo-rationality_finite_level}, we choose an $M$ such that $p^M \cdot R^if_*W_n\sO_Y = 0$.  We claim it is also true for $f_K$ which concludes the proof using \autoref{prop:wo-rationality_finite_level} once again.  We consider the following diagrams, where by abuse of notation we identify every morphism $g$ with $W_n(g)$:
\[
\xymatrix{
    W_nY \ar[dd]_{f} & & W_n Y_K \ar[ll]_{\psi_K}\ar[dd]^{f_K} \ar[ld] \\
    & W_n Y_{K_\lambda} \ar[lu]^{\psi_{K_\lambda}} \ar[dd]|\hole \\
    W_nX & & W_n X_K \ar[ll]_(0.4){\pi_K} \ar[ld] \\
    &  W_n X_{K_\lambda} \ar[lu]^{\pi_{K_\lambda}} .\\
}
\]
Since the map $\pi_K$ is affine, we see that $p^M \cdot R^i f_{K,*} W_n\sO_{X_K} = 0$ if and only if $p^M \cdot \pi_{K,*} R^i f_{K,*} W_n\sO_{X_K} = 0$.  By the commutativity of the diagram above and the fact that $\psi_K$ is also affine, this is equivalent to the vanishing $p^M \cdot R^i f_*(\psi_{K,*} W_n \sO_{Y_K})$.  Since the forgetful functor from sheaves of rings to sheaves of sets preserves direct limits and $W_n$ is the $n$-fold self-product set theoretically, the pushforward $\psi_{K,*} W_n \sO_{Y_K}$ is in fact isomorphic to \[
\varinjlim_{\lambda} \psi_{K_\lambda,*}W_n \sO_{Y_{K_\lambda}}.
\]
Now, since direct images commute with direct limits for quasi-compact quasi-separated morphisms (see \stacksproj{07U6}) we see that
\begin{align*}
p^M \cdot R^i f_*(\psi_{K,*} W_n \sO_{Y_K}) & \isom p^M \cdot R^i f_*(\varinjlim_{\lambda} \psi_{K_\lambda,*} W_n \sO_{Y_{K_\lambda}})  \\
& \isom p^M \cdot \varinjlim_{\lambda} \pi_{K_\lambda,*} R^i f_{K_\lambda,*} W_n\sO_{Y_{K_\lambda}} = \varinjlim_{\lambda} p^M \cdot \pi_{K_\lambda,*} R^i f_{K_\lambda,*} W_n\sO_{Y_{K_\lambda}}.
\end{align*}
However, since the morphisms $X_{K_\lambda} \to X$ are in fact \'etale, by the argument in \autoref{prop:properties_witt_rational} all the terms in the last direct limit vanish which finishes the proof.
\end{proof}

\begin{remark}
We note that one would hope that a similar proof actually works for an arbitrary extension.  Unfortunately, a purely inseparable extension of height $k$ leads to the substitution of the exponent $M$ by $M+k$.  The limiting procedure cannot be therefore performed.
\end{remark}

\subsection{Primer on $p$-adic cohomology}

For the sake of completeness, in this section we present the basic results concerning $p$-adic cohomology and direct images which are going to be necessary in our argument.  Most the results are contained in \cite[Exposé V ``Systemes projectifs $J$-adiques'']{SGA5} and \cite[Exposé VI ``Cohomologie $\ell$-adique'']{SGA5} by Jean-Pierre Jouanolou where the formalism of $\ell$-adic sheaves is developed.  We emphasize that the standard assumption $\ell \neq p$ showing up in $\ell$-adic cohomology considerations is not necessary for neither of the results in \emph{loc. cit}.

We begin with a short description of the necessary notion of Artin-Rees category $\AR(\ZZ_p)$ of $p$-adic sheaves.  We refer to \cite[Exposé V, \S 2.2]{SGA5} for the original account.  All the sheaves considered here are in  \'etale topology. The objects of the category are projective systems
\[
\cF_\bullet = \left(\ \cF_n \ ,\ \phi^\cF_n \colon \cF_n \to \cF_{n-1}\ \right)_{n \in \ZZ}
\] of constructible (and hence torsion) sheaves of $\ZZ_p$-modules such that $\cF_n = 0$, for $n \ll 0$.  The morphisms are defined in a slightly subtle way, by the formula:
\[
\Hom_{\AR(\ZZ_p)}(\cF_\bullet,\cG_\bullet) = \varinjlim_{d \in \ZZ} \  \Hom_{\rm systems}(\cF_\bullet[d],\cG_\bullet)
\]
where $\cF[d]$ denotes the system defined by the association $\cF_\bullet[d]_n = \cF_{n+d}$, and the colimit is taken along the maps $\Hom_{\rm systems}(\cF_\bullet[d],\cG_\bullet) \to \Hom_{\rm systems}(\cF_\bullet[d+1],\cG_\bullet)$
induced by the diagram:
\[
\xymatrix{
    & & \cF_{n+d+1} \ar[r]^{\phi^{\cF}_{n+d+1}}\ar[dd]_{\phi^{\cF}_{n+d+1}} & \cF_{n+d} \ar[r]^{\psi_n}\ar[dd]_{\phi^{\cF}_{n+d}} & \cG_n \ar[dd]^{\phi^{\cG}_{n}} \\
    \psi = \Big(\psi_n \colon \cF_{n+d} \to \cG_n\Big)_{n \in \ZZ} \mapsto \Big(\psi_n \circ \phi^{\cF}_{n+d+1} \Big)_{n \in \ZZ} & & \\
    & & \cF_{n+d} \ar[r]_{\phi^{\cF}_{n+d}} & \cF_{n+d-1} \ar[r]_{\psi_{n-1}} & \cG_{n-1}.
}
\]
\noindent The right square in the diagram is in fact a morphism $\cF[d+1] \to \cF[d]$ of systems, and the above map is a composition therewith.

We remark that the above definition is in fact equivalent to the localization of the category of systems of sheaves of $\ZZ_p$-modules satisfying the condition $\cF_n = 0$, for $n \ll 0$, by the subcategory of \emph{null systems}, that is, systems $\cN_{\bullet}$ for which the natural map $\cN[d] \to \cN$, induced by composition of $d$ transition maps, is zero for some $d \in \NN$ (see \cite[Exposé V, Proposition 2.4.3]{SGA5}).
\noindent A system $\cF_\bullet = (\cF_n)$ is \emph{strict} if $\cF_n = 0$, for $n<0$, and for $n \geq 0$ the sheaf $\cF_n$ is annihilated by $p^{n+1}$ and satisfies $\cF_n \otimes_{\ZZ/p^{n+1}} \ZZ/p^n \isom \cF_{n-1}$.  A strict system $\cF_\bullet$ is \emph{constructible} if all the sheaves $\cF_n$ are moreover constructible.

\begin{definition}[{$p$-adic sheaves}]
\label{defin:padic_sheaves}
We define the categories of $\ZZ_p$ and $\QQ_p$ sheaves as follows:
\begin{enumerate}
    \item the category of constructible $\ZZ_p$-sheaves is defined as the full subcategory of the category $\AR(\ZZ_p)$ of systems isomorphic to a strict constructible system.
    \item the category of constructible \emph{$\QQ_p$-sheaves} (see \cite[Exposé VI, Définition 1.4.3]{SGA5}) is defined as the localization of the category of $\ZZ_p$-sheaves by the Serre subcategory of sheaves annihilated by a fixed power of $p$ (see \autoref{remark:localization} for a brief description of the localization procedure).
\end{enumerate}
Let $f \colon X \to Y$ be a proper finite type morphism of locally noetherian schemes.  The most important reason for the introduction of the above admittedly complex definition is that it allows for the direct images of a $\ZZ_p$-sheaf $\cF = (\cF_n)_{n \in \NN}$ to be well-defined using the formula:
\[
R^if_*\cF = (R^if_*\cF_n)_{n \in \NN}.
\]
We remark that the above higher direct image systems are often not strict constructible themselves.  However, by the proper base change for $\ZZ/p^n$-constructible sheaves and the main result of \cite[Exposé VI, \S 2.2]{SGA5}, they are Artin--Rees isomorphic to a strict constructible system.  In fact, the only condition which is likely not satisfied is strictness, which however holds after an Artin--Rees isomorphism.  Moreover, if $f \colon \overline{X} \to \overline{k}$ is a structure morphism of a base change over the algebraic closure of a variety $X/k$, we recover the usual definition of $p$-adic cohomology using \cite[Exposé VI, 1.2 ``$\ZZ_\ell$ faisceaux constants tordus constructible'']{SGA5}.  We note that in this case all sheaves $R^if_*\ZZ/p^n$ are in fact torsion constant and hence clearly satisfy requirements of \emph{loc. cit}.

\end{definition}

We shall need the following standard results concerning behaviour of constructible $\ZZ_p$-sheaves.  We remark that although the following results are stated in \cite[Exposé VI]{SGA5} only for $\ZZ_p$-sheaves, the author explains in \cite[Exposé VI, 2.2.5]{SGA5} that all the results can be carried through to the $\QQ_p$ setting using straightforward localization.

\begin{proposition}[{\cite[Exposé VI, Proposition 2.2.4]{SGA5}}]
\label{prop:leray_spectral_seq_QQ_p}
Let $f \colon X \to Y$ and $g \colon Y \to Z$ be proper finite type morphisms of locally noetherian schemes.  Then for every constructible $p$-adic sheaf $\cF$ there exists a convergent Leray spectral sequence:
\[
E^{pq}_2 = R^pg_*(R^qf_*\cF)\Rightarrow R^{p+q}(g \circ f)_* \cF.
\]
In particular, if $Y$ is a proper variety defined over an algebraically closed field and $f \colon X \to Y$ is proper finite type morphism then there exists a convergent Leray spectral sequence:
\[
E^{pq}_2 = H^p_{\et}(Y,R^qf_*\QQ_p) \Rightarrow H^{p+q}_{\et}(Y,\QQ_p).
\]
\end{proposition}

\noindent In fact, the result in \emph{loc. cit} works for higher direct images with compact support but we will only need it in the stated generality.

\begin{proposition}[{\cite[Exposé VI, 2.2.3 B)]{SGA5}}]
\label{prop:QQ_p_proper_base_change}
Suppose that the diagram
\[
\xymatrix{
    X \ar[d]_f & X' \ar[l]_v \ar[d]^g \\
    Y & \ar[l] Y' \ar[l]^u
}
\]
is a cartesian square of locally noetherian schemes.  Assume that the morphism $f$ is proper and of finite type.  Then for every constructible $\ZZ_p$-sheaf $\cF$ (resp. $\QQ_p$-sheaf) the natural base change map:
\[
u^*R^if_*\cF \to R^ig_*(v^*\cF)
\]
is an isomorphism.
\end{proposition}

\begin{theorem}[{\cite[Théorème 5.1]{Chambert_Loir_Points_Rationnels_et_Groupes_Fondamentaux}}]
\label{thm:clr_rigid_zero_padic}
For every proper finite type scheme $X$ defined over an algebraically closed field the following identification holds true:
\[
H^\bullet_{\et}(X,\QQ_p) \otimes K \isom H^\bullet_{\rm rig}(X)^{=0},
\]
where $K:=\Frac(W(k))$.
\end{theorem}

\noindent As a direct combination of \autoref{thm:bbe_rigid_witt_vector} and \autoref{thm:clr_rigid_zero_padic} we obtain the following:

\begin{corollary}
\label{cor:p_adic_slope_zero_of_witt_vector}
For every finite type scheme $X$ defined over an algebraically closed field the following identification holds true:
\[
H^\bullet_{\et}(X,\QQ_p) \otimes K \isom H^\bullet(X,W\sO_{X,\QQ})^{=0},
\]
where $K:=\Frac(W(k))$.
\end{corollary}

\subsection{$p$-adic cohomology and W$\sO$-rational morphisms}

In \autoref{cor:p_adic_slope_zero_of_witt_vector} we saw that the absolute $p$-adic cohomology is closely related to the Witt vector cohomology.  The subsequent lemma provides a similar relation in the relative setting.  We only present this simplistic version of the theorem, because a fully-fledged comparison will most likely require application of rigid cohomology and $F$-isocrystals (as do the full versions of \autoref{thm:bbe_rigid_witt_vector} and \autoref{thm:clr_rigid_zero_padic}).

The main tool that we utilize is the system of Witt Artin--Schreier sequences exact in \'etale topology:
\begin{align}
\label{eq:witt_artin_schreier}
0 \ra \ZZ/p^n \ra W_n\sO_X \xrightarrow{1 - F} W_n\sO_X \ra 0.
\end{align}

\begin{lemma}
\label{lemma:wo_rational_implies_QQp_rational}
Let $f \colon X \to Y$ be a proper W$\sO$-rational morphism (see \autoref{remark:some_remarks_on_WO_rationality}).  Then $f_*\QQ_p = \QQ_p$ and $R^if_*\QQ_p = 0$ for every $i>0$.
\end{lemma}

\begin{proof}
Looking at degree $i>1$ terms of the long exact sequences of cohomology associated to \eqref{eq:witt_artin_schreier} we obtain exact sequences:
\begin{equation}
\label{eq:wo_rational_implies_QQp_rational:LES}
\cdots \ra R^{i-1}f_*W_n\sO_X \ra R^if_*\ZZ/p^n \ra R^if_*W_n\sO_X \ra \cdots.
\end{equation}
Since $f$ is a W$\sO$-rational, according to \autoref{prop:wo-rationality_finite_level}, the left and right terms of \autoref{eq:wo_rational_implies_QQp_rational:LES} are annihilated by a fixed number $p^M$ and hence for every $n>0$ the middle term is killed by $p^{2M}$.  Using \autoref{defin:padic_sheaves}, this directly implies that $R^if_*\QQ_p = 0$, for $i>1$. For low degrees, for every $n>0$ we consider the diagram of sheaves on the \'etale site of $Y$:
\begin{equation}
\label{eq:wo_rational_implies_QQp_rational:big}
\xymatrix{
    & 0 \ar[d] & 0 \ar[d] & & 0 \ar[d] \\
    0 \ar[r] & (\ZZ/p^n)_Y \ar[r]\ar[d] & W_n\sO_Y \ar[rr]^{1 - F_Y}\ar[d] & & W_n\sO_Y \ar[r]\ar[d] & 0 \ar[d] \\
    0 \ar[r] & f_*(\ZZ/p^n)_X \ar[r]\ar[d] & f_*W_n\sO_X \ar[rr]^{f_*(1 - F_X)}\ar[d] & & f_*W_n\sO_X \ar[r]\ar[d] & \coker\left(f_*(1 - F_X)\right) \ar[d] \ar[r] & 0  \\
    0 \ar[r] & K_n \ar[d] \ar[r] & C_n \ar[rr]\ar[d] & & C_n \ar[d]\ar[r] & C_n/f_*(1-F_X) C_n \ar[r] & 0 \\
    &  0 & 0 & & 0
}
\end{equation}
A priori, we only know that the leftmost three non-zero columns, as well as the middle row and the row above that are exact in \autoref{eq:wo_rational_implies_QQp_rational:big}. Then, Snake lemma applied to the first three non-zero entries of these two rows yields, that the row below the middle row is also exact. Finally, Snake lemma applied to the second and the third non-zero column yields that also the rightmost column is exact. So, eventually we obtain that all columns and rows of \autoref{eq:wo_rational_implies_QQp_rational:big} are exact.

By the middle columns of \autoref{eq:wo_rational_implies_QQp_rational:big} we obtain that $C_n$ are annihilated by $p^{M}$ for every $n>0$. Then, the lowest non-zero row of \autoref{eq:wo_rational_implies_QQp_rational:big} implies that $K_n$ are annihilated by $p^{2M}$ for every $n>0$. Again, using \autoref{defin:padic_sheaves} and the fact that localization by Serre subcategories is an exact functor, this implies that $f_* \QQ_p = \QQ_p$.  Moreover, for $i=1$, we observe that \autoref{eq:wo_rational_implies_QQp_rational:LES} and  the middle row of \autoref{eq:wo_rational_implies_QQp_rational:big}  lead to the exact sequence
\[
0 \ra \coker\left(f_*(1 - F_X)\right) \ra R^1f_*(\ZZ/p^n)_X \ra R^1f_*W_n\sO_X.
\]
Since both $\coker\left(f_*(1 - F_X)\right)$, which injects into $C_n/f_*(1-F_X) C_n$, and $R^1f_*W_n\sO_X$ are $p^{M}$-torsion, the sheaf $R^1f_*(\ZZ/p^n)_X$ is $p^{2M}$-torsion for every $n>0$ and hence $R^1f_*\QQ_p = 0$.  This finishes our proof.
\end{proof}

\noindent In the spirit of \autoref{remark:some_remarks_on_WO_rationality}, the morphism satisfying the properties in stated in \autoref{lemma:wo_rational_implies_QQp_rational} is called \emph{$\QQ_p$-rational}. That is a proper $f : X \to Y$ is $\QQ_p$-rational if $f_*\QQ_p = \QQ_p$ and $R^if_*\QQ_p = 0$ for every $i>0$.

\begin{proposition}
\label{prop:geom_generic_fibres_QQp_rational}
Let $f \colon X \to T$ be a flat morphism of varieties defined over $k$.  Assume that $X$ is W$\sO$-rational.  Then the geometric generic fibre $X_{\overline{\eta}}$ of $f$ admits a $\QQ_p$-rational quasi-resolution.
\end{proposition}

\begin{proof}
Let $\overline{\eta}$ be a geometric generic point of $T$.  Let $\xi \colon Y \to X_{\overline{\eta}}$ be a quasi-resolution.  As $Y$ and $\xi$ are of finite type over $k\left(\overline{\eta} \right)$, both are defined over a finite extension $L$ of $k(\eta)$. Taking the normalization $U_{\pre}$ of $T$ in $L$, and restricting to the open set where the separable part of $U_{\pre} \to T$ is \'etale we obtain:
\begin{itemize}
\item a morphism $U \to T$, which is a composition of an \'etale map and a universal homeomorphism, and
\item  a quasi-resolution $\pi \colon Z \to X_U$ such that $\xi = \pi_{\overline{\eta}}$.
\end{itemize}
Using \autoref{prop:properties_witt_rational}, the scheme $X_U$ is W$\sO$-rational and hence $\pi$ is a W$\sO$-rational morphism.  By \autoref{lemma:wo_rational_implies_QQp_rational}, the morphism $\pi$ is also $\QQ_p$-rational.  Applying the proper base change (see \autoref{prop:QQ_p_proper_base_change}) this property is inherited by $\xi = \pi_{\overline{\eta}}$ which finishes the proof.
\end{proof}

\begin{remark}
Although \autoref{prop:geom_generic_fibres_QQp_rational} is  sufficient for our purpose,  it is  admittedly a workaround. A clearer claim would be that the geometric generic fibre is in fact W$\sO$-rational itself.  We were unfortunately unable to prove it.  The main obstacle is the lack of a proper base change type theorem, at the geometric generic point, for Witt vector higher direct images.
\end{remark}

config: arxiv


TITLE: Reference request: the dual Coleman family
QUESTION [2 upvotes]: Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an eigenform of weight k, level N (not necessary new) with nebentype $\chi$ and f be (one of the) p-stabilization of $f^{\circ}$,   we can construct several modular forms from f:
assume the $q$-expansion of f is $\sum_{n\geq1}a_nq^n$,
$(i)$ The conjugate form $f^*$ defined by $f^*(\tau)=\overline{f(-\bar{\tau})}$, the q-expansion of $f^*$ is $\sum_{n\geq1}\bar{a}_nq^n$;
$(ii)$ The twisted form $f_{\chi^{-1}}$ by $\chi^{-1}$ whose q-expansion is $\sum_{n\geq1}\chi^{-1}(n)a_nq^n$;
$(iii)$ and moreover as in Def.2.4 of the paper a note on p-adic Rankin-Selberg L-functions, the form $f^c$ which is the unique form which has level $Np^r$ for some r and whose Hecke eigenvalue away from N is same as $f_{\chi^{-1}}$.
What is the relation of the three forms?  Would anyone provide some reference? Thanks
If F is a Coleman family of tame level N and nebentype $\chi$, could we define the corresponding family $F^*$, $F_{\chi^{-1}}$ or $F^c$ satisfying similar description on $q$-expansion? The paper  A p-adic Gross-Zagier formula for diagoanl cycles(page 41) uses $F^*$ (for Hida family), but no details provided there and Lem.3.4 of 1 claims the existence of $F^c$ (at least for F is new at N) but I could not spell out the details. Would anyone please provide me some reference containing more details?
Thanks

REPLY [2 votes]: The definition of $f^c$ that I give in my Canad Math Bulletin paper is only well-defined when $f$ is a p-stabilised newform (i.e. f has level $Np^r$ for some $r \ge 1$ and $p \nmid N$, and either f is new of level $Np^r$, or $r = 1$ and f is the p-stabilisation of a newform of level N). If f is not locally new at the primes dividing N, then I didn't attempt to define $f^c$, and I suspect there is no good way of doing so.
In the "crystalline" case, when f is a p-stabilisation of a level N newform $f^\circ$, then $f^c$ and $f^*$ are both well-defined, and they are both p-stabilised newforms, but they are not generally the same. The relation between the two is the following. There is a level N newform $(f^\circ)^*$, and it has two p-stabilisations at level Np; one of these is $f^*$ and the other one is $f^c$. One computes that $a_p(f^c) = \chi(p)^{-1} a_p(f)$, but $a_p(f^*) = p^{k-1} / a_p(f)$. Hence the observation in my Remark 2.5 that if f is ordinary, then so is $f^c$, but $f^*$ is not ordinary unless k = 1.
(You can see immediately from those two formulae that $a_p(f^*)$ does not interpolate as f varies over specialisations of a Coleman family F, because $p^{k-1}$ is not a p-adically continuous function of k in weight space. I'm not 100% sure what Darmon and Rotger mean by $F^*$ when F is a Hida family, but I suspect they mean what I'm calling $F^c$.)
EDIT. You asked how $F^c$ is constructed. A Coleman family of tame level N has a "tame nebentype" $\chi$, which is a Dirichlet character mod $N$. If you twist $F$ by $\chi^{-1}$, you get another Coleman family whose tame nebentype is $\chi^{-1}$. A priori this family might have tame level larger than $N$, maybe up to $N^2$; but you know that it has many classical specialisations of tame level $N$ (in fact it suffices for it to have a single non-critical-slope specialisation of tame level $N$), so in fact this family has tame level $N$ and you are done.


TITLE: Geometry Problem about Circles and Tangents
QUESTION [1 upvotes]: It is the second problem from my maths notebooks, which is still unsolved. I translated it from Russian, so their may be some discrepancies in translation. So, I also added image.
First problem was Geometry Problem about Cyclic Quadrilateral.

The circle $S_1$ with centre $C_1 (a_1 , b_1)$ and radius $r_1$ touches (or tangent to) externally the circle $S_2$ with centre $C_2 (a_2 , b_2)$ and radius $r_2$. Their common tangent passes through the origin.
Show that, $(a_1)^2 - (a_2)^2 + (b_1)^2 - (b_2)^2 = (r_1)^2 - (r_2)^2 \; \; \; $  (i)
If, also other tangents from origin are perpendicular, prove that $|a_2b_1 -a_1b_2| = |a_1a_2 + b_1b_2| \; \; \; $  (ii)

I have solved (i), so I need solution only for (ii).

REPLY [1 votes]: Continuing from the hint given by $tomi, we have:-
$\frac {b_1}{a_1}$ is the slope of $OC_1$; and similarly, $\frac {b_2}{a_2}$ is the slope of $OC_2$.
Finding the angle between 2 lines (passing through the origin) with slopes known is given by a standard formula. It can be found in like:-
http://www.vitutor.com/geometry/line/angle_lines.html