\label{T:1}
Suppose $n\geq 3$ is an odd integer and $p\geq 5$ is a prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$
that are Eisenstein at the prime $p$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}
\,.
$$
Here $\varepsilon(p)\to 0$ and is effectively computable.

\label{S:reallyFinal}

\begin{proof}[Proof of Theorem~\ref{T:2}]

Let $Y>0$ be a parameter to be chosen later.
Let $\mathcal{F}^*_{p,n}$ denote the subset of
$\mathcal{F}_{p,n}$ consisting of polynom

\begin{proof}
First note that
    $f(k) = {p\choose k}/p^k$
    is a decreasing function. Indeed, for $k\le p-1$
    $$\frac{f(k)}{f(k+1)} = \frac{(k+1)p}{p-k} > 0,$$
    so $f(k) > f(k+1)$ for all $k\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\ge 2$,
    $$\frac{{p\choose 2}}{p^2} - \frac{{p\choose 3}}{p^3} \le \frac{A_p(n)}{p^n} \le \frac{{p\choose 2}}{p^2}.$$
    \end{proof}

\section{Proof of the main result}\label{S:final}

\begin{proof}[Proof of Theorem~\ref{T:1}]
Adopt the notation from the statement of the theorem.
Further, write $N_{p,n}(X)$ to denote the number of
$f\in\mathcal{F}_{p,n}$ with $\Ht(f)\leq X$.
We have $N_{p,n}(X)\sim E(p)(2X)^n$ where $E(p)$
is given in (\ref{E:eisenstein}).
This follows from~\cite{MR1995564} or alternatively,
from Proposition~\ref{P:count} with $t=0$.

First we will establish the lower bound
$\liminf_{X\to\infty}\delta_{p,n}(X)\geq 2/27$.
Assume $p\neq 7,11,19$.
By Lemma~\ref{lem: Froby}, we can write $p=2u+3v$ with
$(2,u)=1$ and $(3,v)=1$ for $p\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.
Therefore, provided $f(x)$ has no roots in $\F_2$
or $\F_3$, Heilbronn's criterion applies.
(Recall Lemma~\ref{L:heilbronn} and the discussion in~\S\ref{S:EH}.)
Write $N^*_{p,n}(X)$ to denote the counting function for this collection
of polynomials.
Using Proposition~\ref{P:count}
with $q_1=2$, $q_2=3$, we have
$$
  N^*_{p,n}(X)\sim E_p(n)C_2(n)C_3(n)(2X)^n
  \,,
$$
and therefore,
via Lemma~\ref{lem: A_p formula} with $n\geq 3$,
we find
$$
\liminf_{X\to\infty}
\delta_{n,p}(X)\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27
\,.
$$

In the case where $p\in\{7,11,19\}$,
we can write $p=2u+5v$ with
$(2,u)=1$ and $(5,v)=1$.
The same approach proves
$$
\liminf_{X\to\infty}\delta_{n,p}(X)\geq C_2(n)C_5(n)\geq(1/4)(8/25)>2/27
\,.
$$
Indeed, by Lemma~\ref{L:Cbounds},
as $n\geq 3$ is odd,
one has $C_5(n)\geq 8/25$.

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
  D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1  \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
  &1-\liminf_{X\to\infty}\delta_{n,p}(X)
  \notag\\
  &\quad\leq
  \prod_{i=1}^t(1-C(q_i))+
  \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
  \notag\\
  &\quad\leq
  (1+tD)\prod_{i=1}^t(1-C(q_i))
  \notag\\
  &\quad\leq
  \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

We are interested in proving statistical results concerning
polynomials with integer coefficents.  Let $\mathcal{F}\subseteq\mathbb{Z}[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\mathbb{Z}[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\mathbb{Z}[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
  N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
  \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
  \delta(X)=\frac{N^*(X)}{N(X)}
  \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\mathbb{Z}[x]$ of degree $n$.
It is easy to show that
$$
   N(X)\sim (2X)^n
   \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}).  Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
  \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
  \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

On the other hand, one of the central objects of study in algebraic number theory
are number fields.  As an Eisenstein polynomial
$f\in\mathcal{F}^*_n$ is always irreducible,
it generates a number field $K$ of degree $n$.
That is, if we define $K=\mathbb{Q}[x]/(f)$ then one has $[K:\mathbb{Q}]=n$.
We can ask various questions about properties of
this number field as we vary the polynomial.
As is standard, we let $\O_K$ denote the ring of integers in $K$,
and let $N:K\to\mathbb{Q}$ denote the norm map.

One classical question is regarding when the Euclidean algorithm holds in $K$.
We call a number field $K$ norm-Euclidean if for every $\alpha,\beta\in\O_K$, $\beta\neq 0$,
there exists $\gamma,\rho\in\O_K$ such that $\alpha=\gamma\beta+\rho$
and $|N(\rho)|<|N(\gamma)|$.  This is the generalization of the
division property of $\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps.  The statement that $K$ is norm-Euclidean is equivalent to $\O_K$ being a Euclidean domain with respect to the function $\partial(\alpha)=|N(\alpha)|$.
The quadratic fields that are norm-Euclidean have been completely determined
(see~\cite{MR41883, MR41885, MR53162, MR1191702}),
but this has not been accomplished in full for any degree $n>2$.
The reader may be interested to consult the survey article~\cite{MR1362867}.

One knows that
all norm-Euclidean fields have class number one, but the converse is not
true in general. Recall that $K$ having class number one is equivalent to $\O_K$ being a unique factorization domain.
We expect the norm-Euclidean property to be rather rare in general, but proving this is quite difficult.
On the other hand, if $K$ is generated by an Eisenstein polynomial, we actually have a tool available to us, due to the existence of a totally ramified prime.  More specifically, if $f$ is Eisenstein at the prime $p$, then $p$ is totally ramified in $K$; that is, $(p)=\mathfrak{p}^n$ for some prime ideal $\mathfrak{p}$.  The tool we refer to is as follows
(see~\cite{MR1574525, MR35313, MR43134, MR752120}):

\begin{lemma}[Heilbronn's criterion]\label{L:heilbronn}
Let $K$ be a number field of degree $n$.
Let $p$ be a prime that is totally ramified in $K$.
If we can write $p=a+b$ where $\{a,-b\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.
\end{lemma}
For clarity, recall that we refer to $a\in \mathbb{Z}$ as a norm (from $\O_K$) when
there exists $\alpha\in \O_K$ such that $N(\alpha)=a$.
Note that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence
one can simply write $\{a,b\}$ where we have written $\{a,-b\}$
in the lemma above.  Additionally, note that the criterion never applies
when $p=2$ or $p=3$, so we should always be considering $p\geq 5$ in our applications
of this result.

A natural question to ask is:  How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said.  For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\mathbb{Z}[x]$ if the criterion applies to the number field
$K=\mathbb{Q}[x]/(f)$.  We establish the following result:

We are interested in proving statistical results concerning
polynomials with integer coefficents. Let $\mathcal{F}\subseteq\mathbb{Z}[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\mathbb{Z}[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\mathbb{Z}[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
 N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
 \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
 \delta(X)=\frac{N^*(X)}{N(X)}
 \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\mathbb{Z}[x]$ of degree $n$.
It is easy to show that
$$
 N(X)\sim (2X)^n
 \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}). Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
 \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
 \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

On the other hand, one of the central objects of study in algebraic number theory
are number fields. As an Eisenstein polynomial
$f\in\mathcal{F}^*_n$ is always irreducible,
it generates a number field $K$ of degree $n$.
That is, if we define $K=\mathbb{Q}[x]/(f)$ then one has $[K:\mathbb{Q}]=n$.
We can ask various questions about properties of
this number field as we vary the polynomial.
As is standard, we let $\O_K$ denote the ring of integers in $K$,
and let $N:K\to\mathbb{Q}$ denote the norm map.

One classical question is regarding when the Euclidean algorithm holds in $K$.
We call a number field $K$ norm-Euclidean if for every $\alpha,\beta\in\O_K$, $\beta\neq 0$,
there exists $\gamma,\rho\in\O_K$ such that $\alpha=\gamma\beta+\rho$
and $|N(\rho)|<|N(\gamma)|$. This is the generalization of the
division property of $\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\O_K$ being a Euclidean domain with respect to the function $\partial(\alpha)=|N(\alpha)|$.
The quadratic fields that are norm-Euclidean have been completely determined
(see~\cite{MR41883, MR41885, MR53162, MR1191702}),
but this has not been accomplished in full for any degree $n>2$.
The reader may be interested to consult the survey article~\cite{MR1362867}.

\begin{lemma}[Heilbronn's criterion]\label{L:heilbronn}
Let $K$ be a number field of degree $n$.
Let $p$ be a prime that is totally ramified in $K$.
If we can write $p=a+b$ where $\{a,-b\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.
\end{lemma}
For clarity, recall that we refer to $a\in \mathbb{Z}$ as a norm (from $\O_K$) when
there exists $\alpha\in \O_K$ such that $N(\alpha)=a$.
Note that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence
one can simply write $\{a,b\}$ where we have written $\{a,-b\}$
in the lemma above. Additionally, note that the criterion never applies
when $p=2$ or $p=3$, so we should always be considering $p\geq 5$ in our applications
of this result.

A natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\mathbb{Z}[x]$ if the criterion applies to the number field
$K=\mathbb{Q}[x]/(f)$. We establish the following result:

Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
We order our polynomials by the natural height $\Ht(f)$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\eps(p)\right\}
\,,
$$
where $\eps(p)\to 0$ and is effectively computable.
In particular, the lower density tends to $1$ as $p\to\infty$
uniformly in $n$.
We also give a version of this result where we weaken the
condition on $\gcd(p-1,n)$.

We are interested in proving statistical results concerning
polynomials with integer coefficents. Let $\mathcal{F}\subseteq\Z[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\Z[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\Z[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
 N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
 \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
 \delta(X)=\frac{N^*(X)}{N(X)}
 \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\Z[x]$ of degree $n$.
It is easy to show that
$$
 N(X)\sim (2X)^n
 \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}). Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
 \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
 \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

A natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\Z[x]$ if the criterion applies to the number field
$K=\Q[x]/(f)$. We establish the following result:

When $p$ is large, there are many ways to write $p=a+b$ and hence
determining the exact density of polynomials that satisfy Heilbronn's
criterion is likely very difficult. However, the previous
theorem says that the lower density tends to $1$ as $p\to\infty$,
uniformly in the variable $n$.

\begin{theorem}\label{T:2}
Suppose $n\geq 2$ and $p\geq 5$ is a prime with
$$
\gcd(p-1,n)<\frac{p^{1/2}}{(\log p)^2}
\,.
$$
Define $\delta_{p,n}(X)$ as in Theorem~\ref{T:1}.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq 1-\hat{\eps}(p)
\,,
$$
where $\hat{\eps}(p)\to 0$ and is effectively computable.
\end{theorem}

Before proceeding, we require a couple definitions.
First we define
\begin{equation}\label{E:eisenstein}
E_p(n)\colonequals
\frac{1}{p^n}-\frac{1}{p^{n+1}}
\,,
\end{equation}
which is the $p$-Eisenstein density that contributes
to (\ref{E:dub}).
Secondly, we write $C_q(n)$ to denote --- the number of
$(a_0,\dots,a_{n-1})\in(\Z/q\Z)^n$ such that the corresponding polynomial
in $\F_q[x]$ has no roots in $\F_q$ ---
divided by $q^n$.
We are now ready to prove the following result, which
will be used in the proof of Theorem~\ref{T:1}.
\begin{prop}\label{P:count}
Let $n\in\Z^+$.
Suppose $q_1<\dots<q_t<p$ are primes.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
The number of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which
$f$ has no roots in $\F_q$ for all $q\in\{q_1,\dots,q_t\}$
equals
$$
 E_p(n)\left(\prod_{i=1}^t C_{q_i}(n)\right)(2X)^n+O\left(n 2^n q_1\dots q_t p^2 (2X)^{n-1}\right)
 \,.
$$
\end{prop}

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
 D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1 \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
 &1-\liminf_{X\to\infty}\delta_{n,p}(X)
 \notag\\
 &\quad\leq
 \prod_{i=1}^t(1-C(q_i))+
 \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
 \notag\\
 &\quad\leq
 (1+tD)\prod_{i=1}^t(1-C(q_i))
 \notag\\
 &\quad\leq
 \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

\begin{theorem}\label{T:1}
Suppose $n\geq 3$ is an odd integer and $p\geq 5$ is a prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\eps(p)\right\}
\,.
$$
Here $\eps(p)\to 0$ and is effectively computable.
\end{theorem}

When $p$ is large, there are many ways to write $p=a+b$ and hence
determining the exact density of polynomials that satisfy Heilbronn's
criterion is likely very difficult.  However, the previous
theorem says that the lower density tends to $1$ as $p\to\infty$,
uniformly in the variable $n$.

As a result, we immediately obtain the following consequence:
When $n\geq 3$ is odd,
a positive proportion of Eisenstein polynomials of degree $n$
fail to generate norm-Euclidean fields.

The condition from Theorem~\ref{T:1} that $\mathrm{gcd}(p-1,n) = 1$
is not a severe restriction as there are infinitely
many primes $p$ that satisfy this condition for each fixed $n$.
Nonetheless, we show that we can
relax this condition at the expense of considering larger primes.
This will allow us to deal with the case where $n$ is even.

\begin{theorem}\label{T:2}
Suppose $n\geq 2$ and $p\geq 5$ is a prime with
$$
\gcd(p-1,n)<\frac{p^{1/2}}{(\log p)^2}
\,.
$$
Define $\delta_{p,n}(X)$ as in Theorem~\ref{T:1}.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq 1-\hat{\varepsilon}(p)
\,,
$$
where $\hat{\varepsilon}(p)\to 0$ and is effectively computable.
\end{theorem}

\begin{corollary}\label{C:1}
Fix an integer $n\geq 2$.
A positive proportion of Eisenstein polynomials of degree $n$
fail to generate norm-Euclidean fields.
\end{corollary}

In \S\ref{S:EH} we discuss Eisenstein polynomials and Heilbronn's criterion,
and outline the strategy of our proof.  In addition, we establish some preliminary
lemmas.  In the case where the proofs are very short, we err on the side of giving
proofs of known results for the sake of completeness.
In \S\ref{S:gotime} we prove our main counting result for Eisenstein polynomials
satisfying certain other local conditions.  In \S\ref{S:local} we prove a formula
for the relevant local densities, which is then used to establish the necessary bounds.
In \S\ref{S:final}
we put everything together and prove Theorem~\ref{T:1}. Finally, in \S\ref{S:reallyFinal}, we prove Theorem \ref{T:2}.

In \,\S\ref{S:EH} the authors “discuss Eisenstein polynomials and Heilbronn's criterion, and outline the strategy of our proof,” and “establish some preliminary lemmas.” Then “in \,\S\ref{S:gotime} we prove our main counting result for Eisenstein polynomials satisfying certain other local conditions.” Next, “in \,\S\ref{S:local} we prove a formula for the relevant local densities, which is then used to establish the necessary bounds.” Finally, “in \,\S\ref{S:final} we put everything together and prove Theorem~\ref{T:1}.”

\label{S:EH}

Let $f\in\Z[x]$ and write
$$
  f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0
  \,.
$$
We say that $f$ is Eisenstein at the prime $p$
(or $p$-Eisenstein for short) if
$p \mid a_i$
for $i=0,\,

\begin{proof}[Proof of Lemma~\ref{L:heilbronn}]
Suppose $p$ is totally ramified and we can write $p=a+b$ as in
the statement of the result.
By way of contradiction, suppose $K$ is norm-Euclidean.
By hypothesis, $(p)=\mathfrak{p}^n$, where $N(\mathfrak{p})=p$.
It follows that $p=u\pi^n$ in $K$,
for some $u\in\mathcal{O}_K^\times$ and some
$\pi\in\mathcal{O}_K$ irreducible with $N(\pi)=p$.
Since $a$ is an $n$-th power residue modulo $n$, there exists $x\in\Z$ such that $x^n\equiv a\pmod{p}$.
Using the norm-Euclidean condition, we find
$x=\gamma\pi+\rho$ for some $\gamma,\rho\in\mathcal{O}_K$
with $|N(\rho)|<|N(\pi)|=p$.  We have $x\equiv \rho\pmod{\pi}$
and hence $x\equiv \rho^\sigma\pmod{\pi}$ for any
embedding $\sigma:K\to\C$.  This leads to
$a\equiv x^n\equiv N(\rho)\pmod{p}$.
We have $0<a<p$ and $-p<N(\rho)<p$.
Thus either $N(\rho)=a$ or $N(\rho)+p=a$.
This implies one of $\{a,-b\}$ is a norm,
a contradiction.
\end{proof}

\section{Counting $p$-Eisenstein polynomials with local specifications}\label{S:gotime}

Given $a=(a_0,\dots,a_{n-1})\in\Z^n$, we define
$$
\Ht(a)=\max\{|a_0|,\dots,|a_{n-1}|\}
$$
so that it agrees  with our polynomial height.
The next result is what allows us to count polynomials
of bounded height with local specifications.
This type of result is certainly well-known,
but the proof is short and so we include it for the
sake of completeness.
\begin{lemma}\label{L:sieve}
Let $S$ be a subset of $(\Z/m\Z)^n$.  We have
$$
\#\{a\in\Z^n\mid a\hspace{-1ex}\mod{m}\in S\,,\;\Ht(a)\leq X\}
=
\frac{|S|}{m^n}(2X)^n+O\left(n2^n m(2X)^{n-1}\right)
\,.
$$
\end{lemma}

\begin{proof}
Let $\mathcal{A}\colonequals [-X,X]^n\cap\Z^n$.
Choose the maximal $k\in\Z^+$ such that
$$
  \mathcal{B}\colonequals (-km,km]^n\cap\Z^n\subseteq \mathcal{A}
  \,.
$$
The set $\mathcal{B}$ is chosen so that
$$
\#\{a\in\mathcal{B}\mid a\hspace{-1ex}\mod{m}\in S\}
=
|S|(2k)^n
$$
Using the Binomial Theorem,
one has
$$
\left(\frac{X}{m}\right)^n \ge k^n \ge \left(\frac{X}{m}-1\right)^n
=
\left(\frac{X}{m}\right)^n+O\left(n2^n\left(\frac{X}{m}\right)^{n-1}\right)
\,,
$$
and therefore
$$
\#\{a\in\mathcal{B}\mid a\hspace{-1ex}\mod{m}\in S\}=
\frac{|S|}{m^n}(2X)^n+O\left(n 2^n m(2X)^{n-1}\right)
\,.
$$
Next one bounds
$$
  \#(\mathcal{A}\setminus\mathcal{B})
  \leq
  (2X+1)^n-(2km)^n
  \ll
  n2^n m(2X)^{n-1}
  \,.
$$
The result follows.
\end{proof}

\label{S:local}

The following lemma gives a formula for our local density
$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity
defined immediately before Proposition~\ref{P:count}
and $A_p(n)$ is defi

\begin{proof}
First note that
    $f(k) = {p\choose k}/p^k$
    is a decreasing function. Indeed, for $k\le p-1$
    $$\frac{f(k)}{f(k+1)} = \frac{(k+1)p}{p-k} > 0,$$
    so $f(k) > f(k+1)$ for all $k\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\ge 2$,
    $$\frac{{p\choose 2}}{p^2} - \frac{{p\choose 3}}{p^3} \le \frac{A_p(n)}{p^n} \le \frac{{p\choose 2}}{p^2}.$$
    \end{proof}

\section{Proof of the main result}\label{S:final}

\begin{proof}[Proof of Theorem~\ref{T:1}]
Adopt the notation from the statement of the theorem.
Further, write $N_{p,n}(X)$ to denote the number of
$f\in\mathcal{F}_{p,n}$ with $\Ht(f)\leq X$.
We have $N_{p,n}(X)\sim E(p)(2X)^n$ where $E(p)$
is given in (\ref{E:eisenstein}).
This follows from~\cite{MR1995564} or alternatively,
from Proposition~\ref{P:count} with $t=0$.

First we will establish the lower bound
$\liminf_{X\to\infty}\delta_{p,n}(X)\geq 2/27$.
Assume $p\neq 7,11,19$.
By Lemma~\ref{lem: Froby}, we can write $p=2u+3v$ with
$(2,u)=1$ and $(3,v)=1$ for $p\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.
Therefore, provided $f(x)$ has no roots in $\F_2$
or $\F_3$, Heilbronn's criterion applies.
(Recall Lemma~\ref{L:heilbronn} and the discussion above.)
Write $N^*_{p,n}(X)$ to denote the counting function for this collection
of polynomials.
Using Proposition~\ref{P:count}
with $q_1=2$, $q_2=3$, we have
$$
  N^*_{p,n}(X)\sim E_p(n)C_2(n)C_3(n)(2X)^n
  \,,
$$
and therefore,
via Lemma~\ref{lem: A_p formula} with $n\geq 3$,
we find
$$
\liminf_{X\to\infty}
\delta_{n,p}(X)\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27
\,.
$$

In the case where $p\in\{7,11,19\}$,
we can write $p=2u+5v$ with
$(2,u)=1$ and $(5,v)=1$.
The same approach proves
$$
\liminf_{X\to\infty}\delta_{n,p}(X)\geq C_2(n)C_5(n)\geq(1/4)(8/25)>2/27
\,.
$$
Indeed, by Lemma~\ref{L:Cbounds},
as $n\geq 3$ is odd,
one has $C_5(n)\geq 8/25$.

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
  D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1  \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
  &1-\liminf_{X\to\infty}\delta_{n,p}(X)
  \notag\\
  &\quad\leq
  \prod_{i=1}^t(1-C(q_i))+
  \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
  \notag\\
  &\quad\leq
  (1+tD)\prod_{i=1}^t(1-C(q_i))
  \notag\\
  &\quad\leq
  \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

This completes the proof of the main theorem.

Question:
Let \(n\ge 3\) be an odd integer and \(p\ge 5\) a prime such that \(\gcd(p-1,n)=1\). Let \(\mathcal F_{p,n}\) be the set of monic polynomials
\[
f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb Z[x]
\]
of degree \(n\) that are Eisenstein at \(p\), i.e. \(p\mid a_i\) for every \(0\le i\le n-1\) and \(p^2\nmid a_0\). For such a polynomial, define its height by
\[
\operatorname{Ht}(f)=\max\{1,|a_{n-1}|,\dots,|a_0|\}.
\]
For each \(X>0\), let \(\delta_{p,n}(X)\) be the proportion of polynomials \(f\in \mathcal F_{p,n}\) with \(\operatorname{Ht}(f)\le X\) for which Heilbronn's criterion applies to the number field \(K=\mathbb Q[x]/(f)\), meaning that there exist integers \(a,b\) with \(p=a+b\), such that \(a\) is an \(n\)-th power residue modulo \(p\) and neither \(a\) nor \(-b\) is a norm from \(\mathcal O_K\). Under these assumptions, which quantitative estimate holds?

One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).

Choices:
B.
One has
\[
\lim_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).
C.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \frac{2}{27}.
\]
D.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge 1-\varepsilon(p),
\]
where \(\varepsilon(p)\to 0\) as \(p\to\infty\).
E.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(n)\right\},
\]
where \(\varepsilon(n)\) is effectively computable and satisfies \(\varepsilon(n)\to 0\) as \(n\to\infty\).


Question: Let \(n\ge 3\) be an odd integer and let \(p\ge 5\) be a prime such that \(\gcd(p-1,n)=1\). Define \(\mathcal F_{p,n}\) to be the set of monic polynomials \(f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in\mathbb Z[x]\) of degree \(n\) that are Eisenstein at \(p\). For a polynomial \(f\), let its height be \(\operatorname{Ht}(f)=\max\{|a_0|,|a_1|,\dots,|a_{n-1}|,1\}\). For \(X>0\), let \(\delta_{p,n}(X)\) be the proportion of polynomials \(f\in\mathcal F_{p,n}\) with \(\operatorname{Ht}(f)\le X\) for which Heilbronn's criterion applies, meaning that for the number field \(K=\mathbb Q[x]/(f)\), one can write \(p=a+b\) with \(\{a,-b\}\) not norms from \(\mathcal O_K\) and with \(a\) an \(n\)-th power residue modulo \(p\). As \(X\to\infty\), which limiting statement holds?

Choices:
B.
One has
\[
\lim_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).
C.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \frac{2}{27}.
\]
D.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge 1-\varepsilon(p),
\]
where \(\varepsilon(p)\to 0\) as \(p\to\infty\).
E.
One has
\[
\limsup_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).


\begin{lemma} \label{lem1}
For $k\geq 2$ we have that
$$f_k = (p_{k}-1) f_{k-1}.$$
\end{lemma}

\begin{theorem} \label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.
\end{theorem}

The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

\begin{abstract}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.
We relate this sequence to a concrete counting problem.
Choose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \leq i \leq k$ and set $P_k = \prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \emph{at most} one of the $k$ chosen residue classes.
Interestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.

\section{Introduction}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

We aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \ldots, p_k$ and arbitrary choice of residue classes $r_1, \ldots, r_k$ with $P_k = \prod_{i=1}^k p_i$.
In particular, we want to understand whether an integer $1\leq n \leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.

Assume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets
$s_i$, $1\leq i \leq k$.
For each integer $1\leq n \leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\gamma(n)$.
We denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.
\begin{align*}
av(k) &:= \# \{n : \gamma(n) \leq 1 \} \\
free(k) &:=\# \{n : \gamma(n) = 0 \} \\
occ(k) &:=\# \{n : \gamma(n) > 1 \}
\end{align*}
To illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.
It is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.

\begin{theorem} \label{thm2}
Let $p_{j_1}, \ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\prod_{i=1}^k p_{j_i}$ be their product.
For $k\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \ldots, p_{j_k})$.
In other words, for every choice of residue classes, there are $A_k(p_{j_1}, \ldots, p_{j_k})$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $F_k(p_{j_1}, \ldots, p_{j_k})$ integers with $\gamma(n)=0$.
\end{theorem}

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

\begin{remark}
Lemma \ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \cite{oeis2} in terms of determinants.
\end{remark}

\begin{remark}
As a side product, the proof of Theorem \ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.
\end{remark}

The paper is organized as follows. In Section \ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \ref{thm:main} is proven in Section \ref{sec4} and generalised in Section \ref{sec5}.

The post-theorem introduction does not give an explicit proof outline for Theorem~\ref{thm:main}, but it indicates the structure used to prove it: Section~\ref{sec2} "derive[s] structural results about $a_k$ and $f_k$"; Section~\ref{sec3} "analyse[s] the counting problem" and "obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions"; then "Theorem~\ref{thm:main} is proven in Section~\ref{sec4}" (and "generalised in Section~\ref{sec5}").

The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).

\label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.,

Question:
Fix a positive integer $k$. Let $p_1,\dots,p_k$ be the first $k$ prime numbers and let $P_k=\prod_{i=1}^k p_i$. For each $i$, choose one residue class $r_i \pmod{p_i}$, and for each integer $n$ with $1\le n\le P_k$ define
$$\gamma(n)=\#\{i\in\{1,\dots,k\}: n\equiv r_i \pmod{p_i}\}.$$
Call $n$ \emph{available} if $\gamma(n)\le 1$ and \emph{free} if $\gamma(n)=0$. Also define
$$a_k=\det\begin{bmatrix}
2 & 1 & \ldots & 1\\
1 & 3 & \ldots & 1\\
\vdots & \vdots & \ddots & \vdots\\
1 & 1 & \ldots & p_k
\end{bmatrix},$$
and
$$f_k=(-1)^k\det\begin{bmatrix}
1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1 & 1\\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_k & 1
\end{bmatrix}.$$
Which statement holds for every choice of the residue classes $r_1,\dots,r_k$?

The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=a_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=f_k.$$

Choices:
B.
The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$, provided the chosen residue classes $r_1,\dots,r_k$ are pairwise distinct as integers; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=a_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=f_k$$
for every such choice of pairwise distinct residues.
C.
The number of free integers in $[1,P_k]$ is exactly $f_k$ for every choice of the residue classes $r_1,\dots,r_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)=0\}=f_k.$$
D.
For every choice of the residue classes $r_1,\dots,r_k$, the number of available integers in $[1,P_k]$ is at least $a_k$, and equality holds if and only if no integer of $[1,P_k]$ lies in more than two of the chosen residue classes; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}\ge a_k.$$
E.
The number of available integers in $[1,P_k]$ is exactly $f_k$, and the number of free integers in $[1,P_k]$ is exactly $a_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=f_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=a_k.$$

dropped the available-count conclusion while keeping the free-count conclusion

exact recurrence-derived equality for available numbers

distinction between available and free counts


Question:
Let $p_1,\dots,p_k$ be the first $k$ prime numbers, and let $P_k=\prod_{i=1}^k p_i$. For each $i$, choose a residue class $r_i\pmod{p_i}$. For $1\le n\le P_k$, define
$$\gamma(n)=\#\{i: n\equiv r_i\pmod{p_i}\}.$$
Call $n$ available if $\gamma(n)\le 1$ and free if $\gamma(n)=0$. Also define
$$a_k=\det\begin{bmatrix}
2&1&\cdots&1\\
1&3&\cdots&1\\
\vdots&\vdots&\ddots&\vdots\\
1&1&\cdots&p_k
\end{bmatrix}$$
and
$$f_k=(-1)^k\det\begin{bmatrix}
1&1&\cdots&1&1\\
2&1&\cdots&1&1\\
1&3&\cdots&1&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
1&1&\cdots&p_k&1
\end{bmatrix}.$$
Which statement holds for the integers in $[1,P_k]$, uniformly over all choices of the residue classes $r_1,\dots,r_k$?

For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $f_k$.

Choices:
B. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $a_k-f_k$.
C. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)=0$ is exactly $f_k$.
D. There exists a choice of residue classes $r_i\pmod{p_i}$ such that the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $f_k$.
E. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=1$ is exactly $f_k$.

relation between available and free counts

dropped the available-count conclusion involving $a_k$

uniformity over all residue-class choices

confusion between free integers and singly covered integers


\begin{proposition}
    \label{prop:CM_or_Gorensteinnes_morphisn_derived_invariance}
    Let $f_i \colon Y_i \to S$ be projective flat morphisms to a separated Noetherian scheme with geometrically integral fibers. Assume that $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$. Then $f_1$ is Cohen--Macaulay (resp.\ Gorenstein) if, and only if, $f_2$ satisfies the same condition.
\end{proposition}

\begin{remark}
    \label{rmk:equivalences_induced}
    Let $\Phi\colon D_{\operatorname{qc}}(Y_1) \to D_{\operatorname{qc}}(Y_2)$ be an exact functor where each $Y_i$ is Noetherian. Then the following are equivalent:
    \begin{enumerate}
        \item \label{rmk:equivalences_induced1} $\Phi$ is an equivalence
        \item \label{rmk:equivalences_induced2} $\Phi$ restricts to an equivalence on $D^-_{\operatorname{coh}}$
        \item \label{rmk:equivalences_induced3} $\Phi$ restricts to an equivalence on $D^b_{\operatorname{coh}}$
        \item \label{rmk:equivalences_induced4} $\Phi$ restricts to an equivalence on $\operatorname{Perf}$.
    \end{enumerate}
    Indeed, by \cite[Theorem 3.2(iv)]{Neeman:2022} and \cite[Theorem B]{Canonaco/Haesemeyer/Neeman/Stellari:2024}, $\eqref{rmk:equivalences_induced1} \implies \eqref{rmk:equivalences_induced2} \implies \eqref{rmk:equivalences_induced3} \implies \eqref{rmk:equivalences_induced4}$. Additionally, if $\eqref{rmk:equivalences_induced4}$ holds, then $\Phi$ admits a right adjoint that preserves small coproducts \cite[Theorem 5.1]{Neeman:1996}. So, \Cref{lem:equivalence_or_fully_faithful_via_compacts} tells us $\eqref{rmk:equivalences_induced4} \implies \eqref{rmk:equivalences_induced1}$.
\end{remark}

\begin{proposition}
    \label{prop:invariance_under_unit_preserve_adjointness}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\prime$ for the kernel of the integral transform obtained in \Cref{prop:right_adjoint_pullback} which is right adjoint to $\Phi_K$ on $D_{\operatorname{qc}}$. If $\Phi_K$ is fully faithful on $D_{\operatorname{qc}}$, then $\mathbf{L} t^\ast_1$ preserves the unit $\eta$ of the adjoint pair $\Phi_K$ and $\Phi_{K^\prime}$ on $D_{\operatorname{qc}}$. That is, $\mathbf{L}t_1^\ast \eta_E$ is the unit of $\mathbf{L}t_1^\ast E$ for the adjoint pair $\Phi_{\mathbf{L}(t^\prime)^\ast K}$ and $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{proposition}

\begin{corollary}
    \label{cor:special_fibers_genii_invariance}
    Let $S$ be a separated Noetherian $\mathbb{Q}$-scheme. Suppose $f_i\colon Y_i \to S$ are proper flat morphisms whose special fibers are geometrically integral curves. Assume $f_1$ is locally projective. If $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$, then then genii of the special fibers of $Y_1$ coincide with that of $Y_2$.
\end{corollary}

\begin{proposition}
    \label{prop:smoothness_derived_invariance}
    Let $Y_1$ and $Y_2$ be proper schemes over a field $k$ (which need not be perfect). Suppose $Y_1$ and $Y_2$ are Fourier--Mukai partners. If $Y_1$ is geometrically regular, then so is $Y_2$. In other words, smoothness is a derived invariance.
\end{proposition}

\begin{corollary}
    \label{cor:smooth_morphisn_derived_invariance}
    Let $f_i \colon Y_i \to S$ be locally projective flat morphisms to a separated Noetherian scheme. Assume that $Y_1$ and $Y_2$ are Fourier--Mukai partners over $S$. Then $f_1$ is smooth if, and only if, $f_2$ is smooth.
\end{corollary}

\begin{theorem}
    [see \Cref{cor:ascending_faithfully}]
    \label{introthm:fully_faithfulness_equivalence}
    Consider affine morphism $t\colon T \to S$ locally of finite tor-dimension of Noetherian schemes whose target is semiseparated. Suppose $Y_1$ and $Y_2$ are proper and flat $S$-schemes. Denote by $\pi \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}\pi^\ast K}$.
\end{theorem}

\begin{example}
    \label{ex:curves_isomorphic_after_base_change}
    There is a pair of non-isomorphic elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$ that are Fourier--Mukai partners (see \cite[Example 2.3.11]{Ward:2014}). From \cite[Theorem 2.2]{Orlov:2002}, it follows that any triangulated equivalence $D^b_{\operatorname{coh}}(E_1) \to D^b_{\operatorname{coh}}(E_2)$ is given by an integral transform.
    Note that an elliptic curve is geometrically integral and projective.
    By \Cref{prop:curves_isomorphic_after_base_change}, there is a finite field extension $L/\mathbb{Q}$ such that $E_1 \times_{\mathbb{Q}} \operatorname{Spec}(L) \cong E_2 \times_{\mathbb{Q}} \operatorname{Spec}(L)$.
\end{example}

\begin{proposition}
    \label{prop:curves_isomorphic_after_base_change}
    Let $k$ be a field of characteristic zero. Suppose $C$ and $C^\prime$ are geometrically integral projective curves over $k$. If $C$ and $C^\prime$ are Fourier--Mukai partners over $k$, then $\operatorname{Spec}(L)\times_k C \cong \operatorname{Spec}(L)\times_k C^\prime$ for some finite field extension $L/k$ (i.e.\ are $k$-forms).
\end{proposition}

\label{sec:intro}

\subsection{Setting the stage}
\label{sec:intro_stage}

The context of our work starts with Mukai \cite{Mukai:1981}. Loc.\ cit.\ introduced the notion of integral transforms to study triangulated equivalences between derived categories associated with abelian varieties. Over time, integral transforms have played a central role in the interplay between derived categories and algebraic geometry (e.g.\ \cite{Bridgeland:2002, Kawamata:2002a}), with further applications in string theory \cite{Andres/HernandezRuiperez:2005, Andres/Scheidegger/Sharpe:2005}.

Much of this progress has focused on smooth varieties, where geometric results rely on categorical methods. However, extending these methods to singular varieties remains challenging, partly because bounded coherent complexes are typically not perfect. Thus, an important task is to develop new results or techniques that improve our understanding of integral transforms in such singular settings.

Recent work has begun to address these difficulties. Criteria determining when an integral transform is fully faithful or an equivalence have been established for varieties over algebraically closed fields, generalizing results from the smooth case \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009} and building upon foundational work of Bondal--Orlov \cite{Bondal/Orlov:1995}. Likewise, Bondal--Orlov's reconstruction theorem for smooth projective varieties has been extended to projective curves \cite{Spence:2023} and (anti-)Fano Gorenstein varieties \cite{Ballard:2011}.

There is also growing interest in studying integral transforms in the relative setting. Specifically, let $Y_1$ and $Y_2$ be proper $S$-schemes where $S$ is Noetherian. An \textit{integral transform} is an exact functor $\Phi_K := \mathbf{R}(p_2)_\ast(\mathbf{L}p_1^\ast(-)\otimes^{\mathbf{L}} K) \colon D_{\operatorname{qc}}(Y_1) \to D_{\operatorname{qc}}(Y_2)$ where $K \in D_{\operatorname{qc}}(Y_1\times_S Y_2)$ and $p_i \colon Y_1\times_S Y_2 \to Y_i$ are the projection morphisms. If $\Phi_K$ is an equivalence, then we say $Y_1$ and $Y_2$ are \textit{Fourier--Mukai partners (over $S$)}.

Such transforms naturally arise in the study of \textit{fibrations}; that is, in situations involving proper and flat morphisms. See e.g.\ \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, which dealt with families whose special fibers have mild singularities. In particular, loc.\ cit.\ investigated when an integral transform between locally projective fibrations $Y_i \to S$ is an equivalence for $S$ of finite type over an algebraically closed field. It was shown that the induced integral transform on special fibers (i.e.\ closed points) over $S$ controls the behavior of equivalences and full faithfulness.

In our work, we reveal additional depth to this perspective of exploiting special fibers. Specifically, we study the behavior of integral transforms under changes of the base scheme in more general settings that includes local algebra. Moreover, our approach naturally extends several results in the literature to arbitrary fields, and at the same time yields new insight for singularities of fibrations.

\subsection{Preservation of adjointness}
\label{sec:intro_preservation_adjoint}

Neeman has developed a new framework called approximable triangulated categories (see e.g.\ \cite{Canonaco/Neeman/Stellari:2025,Neeman:2023}), which has resolved open conjectures in algebraic geometry \cite{Neeman:2021a,Neeman:2022}. As a consequence of this machinery, any integral transform $\Phi_K$ inducing an equivalence on $D_{\operatorname{qc}}$ restricts to equivalences on the subcategories of bounded complexes with coherent cohomology $D^b_{\operatorname{coh}}$ and perfect complexes $\operatorname{Perf}$. Moreover, an equivalence on these `smaller' categories implies an equivalence on the `larger' category. See \Cref{rmk:equivalences_induced} for details.

For those concerned with equivalences, this allows the study of integral transforms to be reduced to their behavior on these subcategories.  Whether an integral transform restricts to these smaller categories is now well-understood \cite{Rizzardo:2017,Ballard:2009,Dutta/Lank/ManaliRahul:2025}. These results shed light to the existence of left or right adjoints to integral transforms and the conditions under which such adjoints also restrict to these subcategories.

However, the behavior of adjointness under base change is not yet fully understood. Our focus is on base changes induced by affine morphisms of finite tor-dimension. In particular, this leads us to the following, which summarizes \Cref{sec:preserve_adjoints}.

\begin{proposition}
    \label{prop:intro_preservation_adjoint}
    Suppose $t\colon T \to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\pi \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Consider $K\in D^-_{\operatorname{coh}}(Y_1\times_S Y_2)$.
    \begin{enumerate}
        \item \label{prop:intro_preservation_adjoint1} If $t$ is affine and $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$ (resp.\ $\operatorname{Perf}$), then so does $\Phi_{\mathbf{L}\pi^\ast K}$ (see \Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).
        \item \label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension.
        \begin{enumerate}
            \item \label{prop:intro_preservation_adjoint2a} If $\Phi_K$ restricts to $\operatorname{Perf}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its right adjoint (see  \Cref{prop:right_adjoint_pullback}).
            \item \label{prop:intro_preservation_adjoint2b} If $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its left adjoint (see \Cref{prop:left_adjoint_pullback}).
        \end{enumerate}
    \end{enumerate}
\end{proposition}

In short, \Cref{prop:intro_preservation_adjoint} describes the behavior of integral transforms under base change when restricted to the small categories, specifically addressing whether the preservation of adjoints occurs. Key instances where it applies are morphisms $t$ arising from field extensions or finite surjective morphisms to a Dedekind scheme (i.e.\ smooth projective curves).

\subsection{Fully faithfulness \& equivalences}
\label{sec:intro_fully_faithfulness_equivalence}

Now, we have a clearer understanding of how integral transforms behave under base change. Particularly, regarding the preservation of adjointness and the restriction to smaller categories. Next, we shift gears to studying fully faithfulness and equivalences under base change; specifically, the ascent and descent of these properties. Before doing so, we review the relevant literature a bit more.

As mentioned earlier, the work of \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009} studied these questions for fibrations whose base is of finite type over an algebraically closed field. A key ingredient in loc.\ cit.\ was to look at the behavior on special fibers. More recently, such questions have also been studied for smooth proper morphisms over a Noetherian base scheme (see \cite[Proposition/Definition 2.3]{Kurama:2024}). However, smoothness constraints can restrict the examples which one might be interested in, e.g.\ fibrations admitting singular fibers.

A result of Orlov addresses similar questions in the case of smooth varieties over a field. Specifically, for smooth projective varieties over a field, an integral transform is fully faithful or an equivalence if, and only if, the induced integral transform satisfies the same property after any base field extension \cite[Lemma 2.12]{Orlov:2002}. The strategy of loc.\ cit.\ leverages `canonical morphisms' between the kernels of integral transforms that describe the (co)units of adjunction. However, it remains unclear whether such canonical morphisms exist in the singular setting. This uncertainty is another source of motivation for our work, as we aimed to bypass the lack of these tools in singular cases using different techniques.

The brings attention to the following, which summarizes \Cref{sec:fully_faithful_equivalence}.

There is also growing interest in studying integral transforms in the relative setting. Specifically, let $Y_1$ and $Y_2$ be proper $S$-schemes where $S$ is Noetherian. An \textit{integral transform} is an exact functor $\Phi_K := \mathbf{R}(p_2)_\ast(\mathbf{L}p_1^\ast(-)\otimes^{\mathbf{L}} K) \colon D_{\operatorname{qc}}(Y_1) \to D_{\operatorname{qc}}(Y_2)$ where $K \in D_{\operatorname{qc}}(Y_1\times_S Y_2)$ and $p_i \colon Y_1\times_S Y_2 \to Y_i$ are the projection morphisms. If $\Phi_K$ is an equivalence, then we say $Y_1$ and $Y_2$ are \textit{Fourier--Mukai partners (over $S$)}.

Such transforms naturally arise in the study of \textit{fibrations}; that is, in situations involving proper and flat morphisms. See e.g.\ \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007, Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, which dealt with families whose special fibers have mild singularities. In particular, loc.\ cit.\ investigated when an integral transform between locally projective fibrations $Y_i \to S$ is an equivalence for $S$ of finite type over an algebraically closed field. It was shown that the induced integral transform on special fibers (i.e.\ closed points) over $S$ controls the behavior of equivalences and full faithfulness.

Neeman has developed a new framework called approximable triangulated categories (see e.g.\ \cite{Canonaco/Neeman/Stellari:2025,Neeman:2023}), which has resolved open conjectures in algebraic geometry \cite{Neeman:2021a,Neeman:2022}. As a consequence of this machinery, any integral transform $\Phi_K$ inducing an equivalence on $D_{\operatorname{qc}}$ restricts to equivalences on the subcategories of bounded complexes with coherent cohomology $D^b_{\operatorname{coh}}$ and perfect complexes $\operatorname{Perf}$. Moreover, an equivalence on these `smaller' categories implies an equivalence on the `larger' category. See \Cref{rmk:equivalences_induced} for details.

\begin{proposition}
 \label{prop:intro_preservation_adjoint}
 Suppose $t\colon T \to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\pi \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Consider $K\in D^-_{\operatorname{coh}}(Y_1\times_S Y_2)$.
 \begin{enumerate}
 \item \label{prop:intro_preservation_adjoint1} If $t$ is affine and $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$ (resp.\ $\operatorname{Perf}$), then so does $\Phi_{\mathbf{L}\pi^\ast K}$ (see \Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).
 \item \label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension.
 \begin{enumerate}
 \item \label{prop:intro_preservation_adjoint2a} If $\Phi_K$ restricts to $\operatorname{Perf}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its right adjoint (see \Cref{prop:right_adjoint_pullback}).
 \item \label{prop:intro_preservation_adjoint2b} If $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its left adjoint (see \Cref{prop:left_adjoint_pullback}).
 \end{enumerate}
 \end{enumerate}
\end{proposition}

A result of Orlov addresses similar questions in the case of smooth varieties over a field. Specifically, for smooth projective varieties over a field, an integral transform is fully faithful or an equivalence if, and only if, the induced integral transform satisfies the same property after any base field extension \cite[Lemma 2.12]{Orlov:2002}. The strategy of loc.\ cit.\ leverages `canonical morphisms' between the kernels of integral transforms that describe the (co)units of adjunction. However, it remains unclear whether such canonical morphisms exist in the singular setting. This uncertainty is another source of motivation for our work, as we aimed to bypass the lack of these tools in singular cases using different techniques.

The brings attention to the following, which summarizes \Cref{sec:fully_faithful_equivalence}.

\begin{proposition}
 \label{prop:intro_preservation_adjoint}
 Suppose $t\colon T \to S$ is a morphism of Noetherian schemes. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Denote by $\pi \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Consider $K\in D^-_{\operatorname{coh}}(Y_1\times_S Y_2)$.
 \begin{enumerate}
 \item \label{prop:intro_preservation_adjoint1} If $t$ is affine and $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$ (resp.\ $\operatorname{Perf}$), then so does $\Phi_{\mathbf{L}\pi^\ast K}$ (see \Cref{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}). Additionally, the converse holds if $t$ is faithfully flat (see \Cref{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}).
 \item \label{prop:intro_preservation_adjoint2} Assume $t$ is affine and locally of finite tor-dimension.
 \begin{enumerate}
 \item \label{prop:intro_preservation_adjoint2a} If $\Phi_K$ restricts to $\operatorname{Perf}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its right adjoint (see \Cref{prop:right_adjoint_pullback}).
 \item \label{prop:intro_preservation_adjoint2b} If $\Phi_K$ restricts to $D^b_{\operatorname{coh}}$, then $\mathbf{L}\pi^\ast$ preserves the adjunction of $\Phi_K$ with its left adjoint (see \Cref{prop:left_adjoint_pullback}).
 \end{enumerate}
 \end{enumerate}
\end{proposition}

The brings attention to the following, which summarizes \Cref{sec:fully_faithful_equivalence}.

In practice, many schemes we may consider are separated, and hence, semiseparated. As a pleasant consequence, this allows us to partially generalize Orlov's result to the singular setting.

\begin{theorem}
 [see \Cref{cor:ascending_faithfully_affine}]
 \label{introthm:ascending_faithfully_affine}
 Consider affine morphism $t\colon T \to S$ to a semiseparated Noetherian scheme. Suppose $Y_1\to S$ is flat, locally projective and $Y_2\to S$ is flat, proper. Denote by $t^\prime \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{theorem}

\begin{theorem}
 \label{thm:ascending}
 Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. If $\Phi_K$ is fully faithful (resp.\ an equivalence) on $D_{\operatorname{qc}}$, then so is $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{theorem}

\begin{corollary}
 \label{cor:ascending_faithfully}
 Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{corollary}

\begin{proposition}
 \label{prop:adjoints_generalized_affine}
 Consider \Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\prime$ the kernel of the integral transform associated to the right adjoint of $\Phi_K \colon D_{\operatorname{qc}}(Y_1) \to D_{\operatorname{qc}}(Y_2)$ obtained in \Cref{prop:right_adjoint_pullback}. If $\Phi_K$ is fully faithful on $D_{\operatorname{qc}}$, then $\mathbf{L} t^\ast_1$ preserves the unit $\eta$ of the adjoint pair $\Phi_K$ and $\Phi_{K^\prime}$ on $D_{\operatorname{qc}}$. That is, $\mathbf{L}t_1^\ast \eta_E$ is the unit of $\mathbf{L}t_1^\ast E$ for the adjoint pair $\Phi_{\mathbf{L}(t^\prime)^\ast K}$ and $\Phi_{\mathbf{L}(t^\prime)^\ast K^\prime}$.
\end{proposition}

\begin{corollary}
 \label{cor:ascending_faithfully_affine}
 Consider \Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{corollary}

\begin{corollary}
    \label{cor:ascending_faithfully}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{corollary}

\begin{corollary}
    \label{cor:ascending_faithfully_affine}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is affine, $f_1$ is locally projective, and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{corollary}

\begin{corollary}
    \label{cor:bounded_pseudocoherence_perfectness_faithfully_flat_affine}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine faithfully flat morphism. Let $K\in D^-_{\operatorname{coh}}(Y_1\times_S Y_2)$. Then $\mathbf{L}(t^\prime)^\ast K$ is relatively perfect over $Y_1\times_S T$ (resp.\ $Y_2\times_S T$) if, and only if, $K$ is relatively perfect over $Y_1$ (resp.\ $Y_2$).
\end{corollary}

\begin{lemma}
    \label{lem:induced_dbcoh_perf_preservation_upon_pullback_with_t_affine}
    Consider \Cref{setup:fm_cube_pullback} with $t$ an affine morphism. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\Phi_{\mathbf{L} (t^\prime)^\ast K}$ restricts to an exact functor on $D^b_{\operatorname{coh}}$. Additionally, a similar statement holds for $\operatorname{Perf}$ if $K$ is relatively perfect over $Y_2$.
\end{lemma}

\begin{proposition}
    \label{prop:left_adjoint_pullback}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_1$. Then $\Phi_{\mathbf{L}(t^\prime)^\ast K^\prime}$ is left adjoint to $\Phi_{\mathbf{L} (t^\prime)^\ast K}$ on $D_{\operatorname{qc}}$ where $K^\prime:= \mathbf{R}\operatorname{\mathcal{H}\! \mathit{om}} (  K,  (f^\prime_2)^! \mathcal{O}_{Y_1}) $. In particular, $K^\prime\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$.
\end{proposition}

\begin{proposition}
    \label{prop:right_adjoint_pullback}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension (e.g.\ $t=1_S$). Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. Then $\Phi_{\mathbf{L}(t^\prime)^\ast K^\prime}$ is right adjoint to $\Phi_{\mathbf{L} (t^\prime)^\ast K}$ on $D_{\operatorname{qc}}$ where $K^\prime:= \mathbf{R}\operatorname{\mathcal{H}\! \mathit{om}} (  K,  (f^\prime_1)^! \mathcal{O}_{Y_2})$. In particular, $K^\prime\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$.
\end{proposition}

In practice, many schemes we may consider are separated, and hence, semiseparated. As a pleasant consequence, this allows us to partially generalize Orlov's result to the singular setting.

\begin{corollary}
    [cf.\ {\cite[Lemma 2.12]{Orlov:2002}}]
    \label{cor:orlov_varieties}
    Consider proper schemes $Y_1$ and $Y_2$ over a field $k$. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_k Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{K_L}$ for any field extension $L/k$.
\end{corollary}

Our proofs require careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change. However, additional subtleties also arise in this context. In particular, we need to work in the larger category $ D_{\operatorname{qc}} $ in order to have a firm grasp on ascending fully faithfulness; see e.g.\ \Cref{prop:fully_faithful_on_small_categories} and \Cref{thm:ascending}. Notably, the techniques we use are independent of those appearing in \cite[Lemma 2.12]{Orlov:2002}.

\subsection{Affine variations}
\label{sec:intro_affine_variations}

Naturally, we sought analogs of \Cref{introthm:fully_faithfulness_equivalence} in which $t$ need not be locally of finite tor-dimension. Ideally, such a framework would include morphisms from special fibers as in \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. Accordingly, we work in the setting of affine morphisms to the base, while imposing a mild restriction on one of the $Y_i \to S$.

The following result is a slight generalization of \cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. In particular, we perform base change along affine morphisms rather than closed immersions of closed points. Note that a key difference between what follows and \Cref{sec:preserve_adjoints,sec:fully_faithful_equivalence} is that we impose one of the $Y_i \to S$ be locally projective (as opposed to proper like our earlier results).

\begin{theorem}
    [see \Cref{cor:ascending_faithfully_affine}]
    \label{introthm:ascending_faithfully_affine}
    Consider affine morphism $t\colon T \to S$ to a semiseparated Noetherian scheme. Suppose $Y_1\to S$ is flat, locally projective and $Y_2\to S$ is flat, proper. Denote by $t^\prime \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{theorem}

It is worthwhile to note that \Cref{introthm:ascending_faithfully_affine} yields an independent proof of the ascending full faithfulness or equivalences in \cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}. To do so, we show in suitable cases that the unit of an adjoint pair of integral transforms is preserved under base change when the left adjoint is fully faithful (see \Cref{prop:adjoints_generalized_affine} and \Cref{prop:invariance_under_unit_preserve_adjointness}). However, it is not clear to us whether this remains true when the left adjoint is not fully faithful a priori. This condition seems necessary (at least to us) to establish the `descent' part of \cite[Proposition 2.15]{Ruiperez/Hernandez/Martin/SanchodeSalas:2009} in the case of special fibers.

\subsection{Applications}
\label{sec:intro_applications}

To conclude our work, we discuss the ramifications of \Cref{introthm:fully_faithfulness_equivalence,introthm:ascending_faithfully_affine}. Our results extend the literature on schemes of finite type over algebraically closed fields to the more general setting of arbitrary fields. They reveal an interplay between the intuition of \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2007,Ruiperez/Hernandez/Martin/SanchodeSalas:2009} concerning special fibers and our work on faithfully flat descent along affine morphisms. In particular, certain singularities are shown to be derived invariants for fibrations over Noetherian schemes, broadening the scope to include arithmetically flavored situations of potential interest for future research.

For a pair of geometrically integral projective curves over a field $k$ of characteristic zero, Fourier--Mukai partnership implies they are $k$-forms (see \Cref{prop:curves_isomorphic_after_base_change}). Here, a $k$-form means there exists a finite extension $L/k$ such that their base changes along $L/k$ become isomorphic. The result holds for singular curves and enriches algebraically closed case of \cite{Spence:2023}. In \Cref{ex:curves_isomorphic_after_base_change}, we exhibit a pair of non-isomorphic elliptic curves over $\mathbb{Q}$ that are nevertheless $\mathbb{Q}$-forms. For fibrations of relative dimension one over a Noetherian $\mathbb{Q}$-scheme, Fourier--Mukai partnership forces the genera of special fibers to coincide (see \Cref{cor:special_fibers_genii_invariance}).

Smoothness for proper schemes over an arbitrary field is likewise invariant under Fourier--Mukai partnership (see \Cref{prop:smoothness_derived_invariance}). This subtlety arises especially over imperfect fields. Since regularity is characterized by $D^b_{\operatorname{coh}} = \operatorname{Perf}$, one might expect this condition to persist after a change of field extensions, but there exist regular projective curves that become singular after such a base change (see e.g.\ \cite[Remark 16]{Kollar:2011}).

In the setting of projective fibrations, smoothness of the total space is also a derived invariant (see \Cref{cor:smooth_morphisn_derived_invariance}). Finally, generalizing \cite{Ruiperez/Hernandez/Martin/SanchodeSalas:2009}, we show that the Cohen--Macaulay and Gorenstein properties of a morphism remain invariant under Fourier--Mukai partnership for Noetherian base schemes (see \Cref{prop:CM_or_Gorensteinnes_morphisn_derived_invariance}).

\subsection*{Notation}
\label{sec:intro_notation}

Let $X$ be a Noetherian scheme. We work with the following triangulated categories:
\begin{enumerate}
    \item $D(X):=D(\operatorname{Mod}(X))$ is the derived category of $\mathcal{O}_X$-modules.
    \item $D_{\operatorname{qc}}(X)$ is the (strictly full) subcategory of $D(X)$ consisting of complexes with quasi-coherent cohomology.
    \item $D_{\operatorname{coh}}^b(X)$ (resp.\ $D_{\operatorname{qc}}^b(X)$) is the (strictly full) subcategory of $D(X)$ consisting of complexes having bounded and coherent (resp.\ quasi-coherent) cohomology.
    \item $\operatorname{Perf}(X)$ is the (strictly full) subcategory of $D_{\operatorname{qc}}(X)$ consisting of the perfect complexes on $X$.
\end{enumerate}
At times if $X$ is affine, we abuse notation and write $D(R):=D_{\operatorname{qc}}(X)$ where $R:=H^0(X,\mathcal{O}_X)$ are the global sections; similar conventions will occur for the other categories.

\begin{ack}
    Lank was supported under the ERC Advanced Grant 101095900-TriCatApp. Part of this work was completed while Lank visited the Basque Center for Applied Mathematics (BCAM) and thanks them for their hospitality. The authors thank Dylan Spence, Ana Cristina L\'{o}pez-Mart\'{i}n, and El\'{i}as Guisado Villalgordo for discussions in an earlier version of the work.
\end{ack}

The post-theorem introduction does not give a step-by-step proof of \Cref{introthm:fully_faithfulness_equivalence}, but it indicates the main technical ingredients: the proofs involve “careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change,” and “additional subtleties” mean one “need[s] to work in the larger category $D_{\operatorname{qc}}$ in order to have a firm grasp on ascending fully faithfulness; see e.g.\ \Cref{prop:fully_faithful_on_small_categories} and \Cref{thm:ascending}.” It is also noted that “the techniques we use are independent of those appearing in \cite[Lemma 2.12]{Orlov:2002}.”

The post-theorem introduction does not give a step-by-step proof of \Cref{introthm:fully_faithfulness_equivalence}, but it indicates the main technical ingredients: the proofs involve “careful bookkeeping of how the (co)units of adjunctions for integral transforms behave under base change,” and “additional subtleties” mean one “need[s] to work in the larger category $D_{\operatorname{qc}}$ in order to have a firm grasp on ascending fully faithfulness.” In particular, one uses the following proposition.

\begin{proposition}
    \label{prop:fully_faithful_on_small_categories}
    Consider \Cref{setup:fm_cube_pullback} where $t=1_S$ and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. Denote by $K^\prime$ for the kernel of the right adjoint for $\Phi_K$ obtained in \Cref{prop:right_adjoint_pullback}. Set $\eta$ to be the unit of the adjunction $\Phi_K$ and $\Phi_{K^\prime}$ on $D_{\operatorname{qc}}$. Then the following are equivalent:
    \begin{enumerate}
        \item \label{prop:fully_faithful_on_small_categories1} $\Phi_K$ is fully faithful on $D_{\operatorname{qc}}$
        \item \label{prop:fully_faithful_on_small_categories2} $\eta_E$ is an isomorphism for all $E\in D^-_{\operatorname{coh}}(Y_1)$
        \item \label{prop:fully_faithful_on_small_categories3} $\eta_E$ is an isomorphism for all $E\in D^b_{\operatorname{coh}}(Y_1)$
        \item \label{prop:fully_faithful_on_small_categories4} $\eta_E$ is an isomorphism for all $E\in \operatorname{Perf}(Y_1)$
    \end{enumerate}
    Additionally, if $K$ is relatively perfect over $Y_1$, then these are equivalent to $\Phi_K$ is fully faithful on $D^b_{\operatorname{coh}}$.
\end{proposition}

One also uses the following ascent result.

\begin{theorem}
    \label{thm:ascending}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ is semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over $Y_2$. If $\Phi_K$ is fully faithful (resp.\ an equivalence) on $D_{\operatorname{qc}}$, then so is $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{theorem}

It is also noted that “the techniques we use are independent of those appearing in \cite[Lemma 2.12]{Orlov:2002}.”

\begin{corollary}
    \label{cor:ascending_faithfully}
    Consider \Cref{setup:fm_cube_pullback} where $t$ is an affine morphism locally of finite tor-dimension and $S$ be semiseparated. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}(t^\prime)^\ast K}$.
\end{corollary}

\label{introthm:fully_faithfulness_equivalence}
Consider affine morphism $t\colon T \to S$ locally of finite tor-dimension of Noetherian schemes whose target is semiseparated. Suppose $Y_1$ and $Y_2$ are proper and flat $S$-schemes. Denote by $\pi \colon Y_1\times_S Y_2 \times_S T \to Y_1\times_S Y_2$ the natural morphism. Let $K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$ be relatively perfect over each $Y_i$. If $\Phi_K$ restricts to a fully faithful functor (resp.\ an equivalence) on $D^b_{\operatorname{coh}}$, then so does $\Phi_{\mathbf{L}\pi^\ast K}$.,

Question:
Let $t\colon T\to S$ be an affine morphism locally of finite tor-dimension between Noetherian schemes, and assume that $S$ is semiseparated. Let $Y_1$ and $Y_2$ be proper and flat $S$-schemes. Write
$$\pi\colon Y_1\times_S Y_2\times_S T\to Y_1\times_S Y_2$$
for the natural morphism, and let
$$K\in D^b_{\operatorname{coh}}(Y_1\times_S Y_2)$$
be relatively perfect over each $Y_i$ (with respect to the natural projections $Y_1\times_S Y_2\to Y_i$). For a kernel $L$ on a fiber product, let the associated integral transform be
$$\Phi_L:=\mathbf{R}(p_2)_*\big(\mathbf{L}p_1^*(-)\otimes^{\mathbf L} L\big),$$
where $p_1,p_2$ are the projection morphisms. Suppose that
$$\Phi_K\colon D_{\operatorname{qc}}(Y_1)\to D_{\operatorname{qc}}(Y_2)$$
restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)
$$D^b_{\operatorname{coh}}(Y_1)\to D^b_{\operatorname{coh}}(Y_2).$$
Which conclusion about the pullback kernel $\mathbf{L}\pi^*K$ is valid?

The base-changed integral transform with kernel $\mathbf{L}\pi^*K$,
$$\Phi_{\mathbf{L}\pi^*K}\colon D_{\operatorname{qc}}(Y_1\times_S T)\to D_{\operatorname{qc}}(Y_2\times_S T),$$
also restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence):
$$D^b_{\operatorname{coh}}(Y_1\times_S T)\to D^b_{\operatorname{coh}}(Y_2\times_S T).$$

Choices:
B.
The base-changed integral transform with kernel $\mathbf{L}\pi^*K$,
$$\Phi_{\mathbf{L}\pi^*K}\colon D_{\operatorname{qc}}(Y_1\times_S T)\to D_{\operatorname{qc}}(Y_2\times_S T),$$
also restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)
$$D^b_{\operatorname{coh}}(Y_1\times_S T)\to D^b_{\operatorname{coh}}(Y_2\times_S T),$$
and the converse implication also holds for every affine morphism $t$ locally of finite tor-dimension.
C.
The base-changed integral transform with kernel $\mathbf{L}\pi^*K$,
$$\Phi_{\mathbf{L}\pi^*K}\colon D_{\operatorname{qc}}(Y_1\times_S T)\to D_{\operatorname{qc}}(Y_2\times_S T),$$
restricts on bounded coherent complexes to an exact functor
$$D^b_{\operatorname{coh}}(Y_1\times_S T)\to D^b_{\operatorname{coh}}(Y_2\times_S T).$$
D.
The base-changed integral transform with kernel $\mathbf{L}\pi^*K$,
$$\Phi_{\mathbf{L}\pi^*K}\colon D_{\operatorname{qc}}(Y_1\times_S T)\to D_{\operatorname{qc}}(Y_2\times_S T),$$
is fully faithful (respectively, an equivalence) on all of $D_{\operatorname{qc}}(Y_1\times_S T)\to D_{\operatorname{qc}}(Y_2\times_S T)$ whenever $\Phi_K$ restricts on bounded coherent complexes to a fully faithful functor (respectively, to an equivalence)
$$D^b_{\operatorname{coh}}(Y_1)\to D^b_{\operatorname{coh}}(Y_2).$$
E. The same conclusion holds after base change for $\Phi_{\mathbf{L}\pi^*K}$ provided only that $t\colon T\to S$ is locally of finite tor-dimension; the assumptions that $t$ be affine and that $S$ be semiseparated are unnecessary.

faithfully-flat converse replaced by converse for all affine lftd base changes

dropped fully faithful/equivalence conclusion, keeping only preservation of $D^b_{\operatorname{coh}}$

small-category hypothesis upgraded directly to a $D_{\operatorname{qc}}$ conclusion after base change

removed affineness and semiseparatedness assumptions needed in ascent/bookkeeping of adjunctions