\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

The stem does not explicitly state the claimed counting result. It introduces the objects and asks which conclusion is valid, without directly revealing that the counts must equal the determinant-defined quantities.

The intended correct option appears to state the theorem almost verbatim: exact formulas for the counts of available and free integers. The alternatives add some variation in quantifier strength and count interpretation, but the item is still close to a direct theorem restatement.

A student must distinguish universal from existential claims and separate counts of free, available, and singly covered integers. However, this is still mostly theorem recognition rather than substantial generative reasoning or derivation.

Several distractors are plausible mathematical confusions: swapping free and singly covered counts, altering the quantifier, or changing the relation between the two counts. But choice C is a weaker statement that would also be true if A is true, which creates ambiguity and lowers distractor quality.

Moderate-quality MCQ: it avoids overt answer leakage and has some logical discrimination, but it leans heavily on theorem recall and is weakened by an apparently true weaker distractor.


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 $r_1,\dots,r_k$?

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,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=f_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=a_k.$$

The stem introduces the objects and asks for a universal conclusion, but it does not state or strongly hint at the exact counting formula in choice A. The determinant definitions alone do not leak the answer.

The correct option is essentially the full theorem packaged as a choice, so the item is close to theorem recognition rather than an independently formulated problem. However, the presence of weaker, stronger, and tampered variants makes it more than a pure verbatim restatement.

There is some reasoning pressure because the student must distinguish the exact universal claim from a weaker true statement, an overstrong inequality/equality claim, and a swapped-count trap. Still, the item mainly tests recognition of the strongest correct statement rather than substantial generation or derivation.

The distractors are mathematically meaningful and target realistic failure modes: adding an unnecessary restriction (B), selecting a weaker true consequence (C), overstrengthening the conclusion (D), and confusing the two counts (E). They are distinct and plausible.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but only moderate generative depth because the correct choice is essentially the theorem itself.


\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

The stem does not reveal the correct option explicitly or by wording patterns. It presents hypotheses and asks for the valid conclusion, without signaling which strengthening/weakening is intended.

This is essentially a theorem-recall item: under the stated hypotheses, the correct choice is the theorem’s main base-change conclusion almost verbatim.

There is some reasoning required to reject subtle overstatements and understatements (converse for all affine base changes, extension to all of D_qc, dropping hypotheses, merely exactness). But the item mainly tests recognition of the precise theorem statement rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: unjustified converse, weakening to a merely exact functor, upgrading from bounded coherent to all quasi-coherent derived categories, and removing needed hypotheses.

A mathematically careful MCQ with strong distractors, but it is largely a direct theorem-restatement question rather than a high-generative-reasoning item.


\begin{theorem}\label{global_existence} For $d=2, 3$,
there exists a diffeomorphism $\xi_0: \mathbb{R}^d \to \mathbb{R}^d$  {and a small constant $\varepsilon>0$} such that if the initial data $(\rho_0, u_0, F_0)$, with
$(\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\in H^3_x(\mathbb{R}^d)$, satisfies
{\begin{eqnarray*}\label{decay_estiwmates_1}
    \begin{cases}
      \rho_0(x) \mathrm{det}F_0(x)=1,\,\,\,\,F_0(x) = \nabla_y \xi_0(y),\,\,\,\,  y =\xi_0^{-1}(x), \ \forall x\in\mathbb{R}^d, \\[2mm]
      \|\rho_0(x) -1\|_{H^3_x} + \|u_0(x) -x\|_{H^3_x} + \|F_0(x) -I\|_{H^3_x} + \|\xi_0^{-1}(x)-x\|_{H^4_x}\le \varepsilon,
    \end{cases}
\end{eqnarray*}}then the Cauchy problem \eqref{elasticity_equation_1} admits a unique global strong solution
$(\rho, u, F)$ such that $\big(\rho(x,t) -\frac{1}{(1+t)^d}, u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big) \in C\big([0, \infty); H^3_x(\mathbb{R}^d)\big)$. Moreover, $(\rho, u, F)$ satisfies the following estimates:
\begin{eqnarray*}\label{decay_estimates_1}
  \begin{cases}
    \|\nabla_x^{i}(\rho(x,t) -\frac{1}{(1+t)^d})\|_{L^2_x}\le C\varepsilon(1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
     \|\nabla_x^{i}\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
  \end{cases}
\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon$ and $t$.}
\end{theorem}

\label{section_1}
We consider the compressible {elastodynamics} in Eulerian coordinate \cite{Gurtin_1981,Dafermos_2010}:
\begin{eqnarray} \label{elasticity_equation_1}
    \begin{cases}
        \partial_t\rho+{\rm div} (\rho u) =0,\\
        \partial_t(\rho u)+{\rm div} (\rho u \otimes u)  + \nabla P(\rho)={\rm div}(\rho F F^\top),\\
        \partial_t F + u\cdot\nabla F=\nabla u F,\\
        (\rho,u,F)|_{t=0} = (\rho_0, u_0, F_0)
    \end{cases}
\end{eqnarray}
on $\mathbb{R}^d \times (0, +\infty)$, {with} $d = 2, 3${, where} $\rho=\rho(x,t)$, $u=u(x,t)$, $F=F(x,t)$ and $P(\rho)=R\rho^{\gamma}$ denotes the fluid density, the velocity field, the deformation gradient and the pressure, respectively. The $i$th component of ${\rm div}(\rho F F^\top)$ on the right-hand side of the momentum equation is $\partial_j(\rho F_{ik} F_{jk})$ and $(\nabla u F)_{ij}=\partial_k u_i F_{kj}$. The physical coefficients $R, \gamma$ satisfy $R>0$ and $\gamma>1$. Without loss of generality, we set $R=1$ throughout this paper. For smooth solutions $(\rho, u, F)$ of \eqref{elasticity_equation_1}, the physical constitution relation
\begin{align}\label{constitution_laws}
\begin{cases}
\rho\  \mathrm{det}F=1,\\
F_{j}\cdot\nabla F_{k}=F_{k}\cdot\nabla F_{j}, \quad 1\leq   j, k\leq d
\end{cases}
\end{align}
holds true for all $t\ge 0$, see \cite{Hu_2011, Qian_2010}. As a consequence of  \eqref{constitution_laws}, it is verified in \cite{Hu_2020, Hu_2020_1} that
\begin{align*}\label{constitution_laws_1}
{\rm div}(\rho F^\top)=0.
\end{align*}

We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \cite{Sideris_1996,Sideris_2000} and Agemi \cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \cite{Wang_2017}. {All these references studied the stability issue of \eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\hat{\rho}, \hat{u}, \hat{F})=(1,0, I).$$ Indeed, the linearized system of \eqref{elasticity_equation_1}
at $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form
\begin{equation}\label{wave}
\partial_t^2 \omega-\Delta \omega=0,
\end{equation}
 where $\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \cite{Sideris_2005, Lei_2016} or the space-time resonance method \cite{Wang_2017}.}

{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \eqref{elasticity_equation_1} around some non-constant equilibrium in
the communities of both physics and mathematics. As an example of non-constant equilibrium, in }\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form
$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\in\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by
\begin{equation}\label{1-1}
    \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t},  \bar{F}= (1+t)I,
  \end{equation}
 with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \eqref{1-1} is justified in \cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer
 by establishing}

We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \cite{Sideris_1996,Sideris_2000} and Agemi \cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \cite{Wang_2017}. {All these references studied the stability issue of \eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\hat{\rho}, \hat{u}, \hat{F})=(1,0, I).$$ Indeed, the linearized system of \eqref{elasticity_equation_1}
at $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form
\begin{equation}\label{wave}
\partial_t^2 \omega-\Delta \omega=0,
\end{equation}
 where $\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \cite{Sideris_2005, Lei_2016} or the space-time resonance method \cite{Wang_2017}.}

{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \eqref{elasticity_equation_1} around some non-constant equilibrium in
the communities of both physics and mathematics. As an example of non-constant equilibrium, in }\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form
$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\in\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by
\begin{equation}\label{1-1}
 \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t}, \bar{F}= (1+t)I,
 \end{equation}
 with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \eqref{1-1} is justified in \cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer
 by establishing}

{With the absence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation, the system \eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \cite{Sideris_2017}. Hadzic and Jang in \cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\gamma\in(1,\frac53]$, and the full range of $\gamma\in (1,\infty)$ is studied in \cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\rightarrow\infty$. In this sense, Theorem \ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.

\section{Global well-posedness in Eulerian coordinate}\label{section_4}
This section is devoted to the completion of the proof of Theorem \ref{global_existence}. Throughout this section, C denotes a generic positive constant that depends on
some known constants, but independent of $\varepsilon$, $\varepsilon_1$ and $t$.
\subsection{Regularity estimates}\label{subsection_7}
With Theorem \ref{global_existence_Lagrangian}, we are going to construct a global-in-time solution to the system
\eqref{elasticity_equation_1} in Eulerian coordinate. For any $(u_0, \xi_0)$ given in Theorem \ref{global_existence}, we define $V_0$ as follows
 \begin{equation}\label{V_0}
 V_0(y)=u_0(\xi_0(y)).
\end{equation}
{Combining \eqref{V_0} with the assumptions of Theorem \ref{global_existence}, we deduce
\begin{equation*}\label{1}
 \begin{split}
 \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le C\varepsilon + C\varepsilon^3 \le C\varepsilon.
 \end{split}
\end{equation*}
Then, letting $\varepsilon = \frac{\varepsilon_1}{C}$ yields
\begin{equation}\label{2}
 \begin{split}
 \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le \varepsilon_1.
 \end{split}
\end{equation}
Substituting $V_0(y)$ and $\xi_0(y)$, that satisfy \eqref{V_0} and \eqref{2}, into Theorem \ref{global_existence_Lagrangian}, we obtain the existence of a unique global solution $(\xi,V)$ to the system \eqref{compressible_elasticity_equation_3}.

For the second term, we similarly obtain
 \begin{equation}\label{Euler_part_17}
 \begin{split}
 &\int_{\mathbb{R}^d} \Big|{\rm adj}[\nabla_y \xi] - (1+t)^{d-1}I\Big|^2\,dy\\ &\le C \Big(\|\nabla_y \xi\|_{L^\infty_y}^{2(d-2)} + (1+t)^{2(d-2)}\Big)\|\nabla_y \xi -(1+t)I\|_{L^2_y}^2\\ & \le C\varepsilon^2 (1+t)^{2d-3}.
 \end{split}
 \end{equation}
Substituting \eqref{Euler_part_16} and \eqref{Euler_part_17} into \eqref{Euler_part_14} and \eqref{Euler_part_15} gives
 \begin{equation}\label{Euler_part_18}
 \begin{split}
 \|\rho(x,t) -\frac{1}{(1+t)^d}\|_{L^2_x}^2 \le C\varepsilon^2 (1+t)^{-d-1}.
 \end{split}
 \end{equation}
 Next, we consider the estimate of $\|\nabla \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$ and $\|\nabla^2 \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$. Combining \eqref{decay_estimates_Lagrangian}, \eqref{Euler_part_4_1}, \eqref{Euler_part_nn_2}-\eqref{Euler_part_nn_3}, {$\varepsilon = \frac{\varepsilon_1}{C}$} with $\nabla_y B_{ij}= -B_{il}\partial_l\nabla_y\xi_{k}B_{kj}$ and $\nabla_y \mathcal{J}= \mathcal{J} B_{ij}\partial_j\nabla_y \xi_{i}$ in \eqref{flow_map_5_1}, we have
 \begin{equation}\label{Euler_part_10}
 \begin{split}
 &\|\nabla_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2 \\ &\le C(1+t)^d \int_{\mathbb{R}^d} |B\nabla_y(\mathcal{J}^{-1})|^2\,dy\\
 &\le C(1+t)^d\int_{\mathbb{R}^d} |B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2\,dy\\
 &\le C(1+t)^d|B|^4 |\mathcal{J}|^{-2} \int_{\mathbb{R}^d} |\nabla^2_y \xi|^2\,dy\\
 &\le C \varepsilon^2 (1+t)^{-d-3}
 \end{split}
 \end{equation}
 and
 \begin{equation}\label{Euler_part_11}
 \begin{split}
 &\|\nabla^2_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2\\
 &\le C(1+t)^d\int_{\mathbb{R}^d} \big|B\nabla_y[B\nabla_y(\mathcal{J}^{-1})]\big|^2\,dy \\
 &\le C(1+t)^d\Big[\int_{\mathbb{R}^d} |B\nabla_y B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy + \int_{\mathbb{R}^d} |BB \mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy\\
 & +\int_{\mathbb{R}^d} |BB\mathcal{J}^{-2}\nabla_y^2\mathcal{J}|^2 \,dy\Big]\\
 &\le C \varepsilon^2 (1+t)^{-d-6} + C \varepsilon^2 (1+t)^{-d-6} +C \varepsilon^2 (1+t)^{-d-5} \\
 &\le C \varepsilon^2 (1+t)^{-d-5}.
 \end{split}
 \end{equation}
 Finally, we estimate $\|\nabla^3_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$. Combining \eqref{decay_estimates_Lagrangian}, \eqref{Euler_part_4_1}, \eqref{Euler_part_nn_4}, $\varepsilon = \frac{\varepsilon_1}{C}$ with $\nabla_y B_{ij}= -B_{il}\partial_l\nabla_y\xi_{k}B_{kj}$ and $\nabla_y \mathcal{J}= \mathcal{J} B_{ij}\partial_j\nabla_y \xi_{i}$ in \eqref{flow_map_5_1}, we have
 \begin{align}\label{Euler_part_11_1}
 &\notag\|\nabla^3_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2\\ \notag &\le C(1+t)^d\int_{\mathbb{R}^d} \Big|B\nabla_y\big\{B\nabla_y\big[B\nabla_y(\mathcal{J}^{-1})\big]\big\}\Big|^2\,dy\\ &
 \le C(1+t)^d\Big[\int_{\mathbb{R}^d} |B\nabla_y B\nabla_y B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy + \int_{\mathbb{R}^d} |BB \nabla_y^2 B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy\notag\\ & +\int_{\mathbb{R}^d} |BB\nabla_y B\mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy +\int_{\mathbb{R}^d} |BB\nabla_y B\mathcal{J}^{-2}\nabla_y^2\mathcal{J}|^2 \,dy\\ & +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-4}\nabla_y\mathcal{J}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y^2\mathcal{J}|^2 \,dy\notag\\ & \notag +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-2}\nabla_y^3\mathcal{J}|^2\,dy\Big]\\ &\notag
 \le\, C \varepsilon^2 (1+t)^{-d-9} + C \varepsilon^2 (1+t)^{-d-8} + C \varepsilon^2 (1+t)^{-d-9}+ C \varepsilon^2 (1+t)^{-d-8} \notag \\ &+ C \varepsilon^2 (1+t)^{-d-9} + C \varepsilon^2 (1+t)^{-d-8} + C \varepsilon^2 (1+t)^{-d-7}\notag\\ &\le \,C \varepsilon^2 (1+t)^{-d-7}.\notag
\end{align}
 From \eqref{Euler_part_ww_1}-\eqref{Euler_part_ww_4}, \eqref{Euler_part_ee_1}-\eqref{Euler_part_ee_4}, and \eqref{Euler_part_18}-\eqref{Euler_part_11_1}, we conclude that
 \begin{eqnarray*}\label{decay_1}
 \begin{cases}
 \|\nabla_x^i \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}\le C\varepsilon (1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
 \|\nabla_x^i\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
 \end{cases}
 \end{eqnarray*}
 for all $t \ge 0$ and $i=0,1,2,3$. The proof of Theorem \ref{global_existence} is complete.

We prefer to work in the framework of $H^3$-spaces in order to keep the strategy as clear as possible. With our strategy in hand, it is possible to reduce the regularity of solutions to some Besov spaces, however the presentation would be more technical and complicated, and we leave it for the interested readers.

{With the absence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation, the system \eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \cite{Sideris_2017}. Hadzic and Jang in \cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\gamma\in(1,\frac53]$, and the full range of $\gamma\in (1,\infty)$ is studied in \cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\rightarrow\infty$. In this sense, Theorem \ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.

The presence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation makes the wave operator non-degenerate. Motivated by the study of compressible Euler equations in \cite{Shkoller_2019}, we turn to employing a Lagrangian transformation and consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$. Compared with the Euler equations, the presence of elastic stress introduces an additional diffusion mechanism, see~\eqref{compressible_elasticity_equation_5}. However, this feature requires a different analytical framework, and the framework developed in \cite{Shkoller_2019} cannot be applied in a straightforward manner. Fortunately, the expanding factor on $t$ in the affine solution \eqref{1-1} contributes a damping term for the velocity in linearized system of \eqref{elasticity_equation_1} in Lagrangian coordinate, see \eqref{compressible_elasticity_equation_5}. However, due to the expanding factor in $t$ of the affine solution,  the coefficients of the linearized system of  \eqref{elasticity_equation_1} involves functions depending on $t$. This kind of
hyperbolic systems with variable coefficients excludes the applicability of both the vector field method and the space-time resonance method, which are designed for quasilinear wave equations with constant variables. In order to overcome this difficulty, we turn to the $t-$dependent damping effect of the velocity. Indeed, with this damping at hand, different weights for unknowns will be constructed in the form of $\mathcal{L}(t)$, and  the differential inequality \eqref{est_L_1} is verified. More precisely, since we are concerned with the stability of a non-constant equilibrium, it is necessary to introduce different temporal weights in a series of uniform estimates. For instance, the terms in $\mathcal{L}(t)$ in \eqref{L} carry distinct weights {depending on $\gamma$ and the dimension}. Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all {unknowns} become uniform. This uniformity plays a key role in establishing the global well-posedness of the compressible elastodynamics system in both two and three dimensions for all $\gamma>1$ within a unified framework. As a consequence, the global existence in Lagrangian coordinate would be established in Theorem \ref{global_existence_Lagrangian} with the help of }the local well-posedness  and the global-in-time estimate of the quantity $\mathcal{L}(t)$.

The global well-posedness in Lagrangian coordinate will be transferred back into the Eulerian coordinate as stated in Theorem \ref{global_existence} via the flow map. We remark that in Lagrangian coordinate, the impact of differentiation on time decay is largely concealed, whereas in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$. Moreover, the $L^2$ norms of $\rho(x,t) -\frac{1}{(1+t)^d}$, $u(x,t)-\frac{x}{1+t}$ and  $F(x,t) -(1+t)I$ decay at faster rates compared with those obtained in \cite{Hu_preprint}.

The rest of the paper is organized as follows. In Section \ref{section_3},
we will reformulate the Cauchy problem \eqref{elasticity_equation_1} in the Lagrangian coordinate and establish the global well-posedness of system \eqref{compressible_elasticity_equation_5} through energy estimates. Building upon Theorem \ref{global_existence_Lagrangian}, Section \ref{section_4} is devoted to the construction of the global solution of system \eqref{elasticity_equation_1}
and the proof of Theorem \ref{global_existence}.

To prove Theorem~\ref{global_existence}, the authors:
\begin{itemize}
\item \emph{Employ a Lagrangian transformation} ("we turn to employing a Lagrangian transformation") and "consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$."
\item Use that the affine expansion yields a \emph{$t$-dependent damping} in the Lagrangian linearized system ("the expanding factor on $t$ in the affine solution \eqref{1-1} contributes a damping term for the velocity in linearized system ... see \eqref{compressible_elasticity_equation_5}").
\item Since the coefficients depend on $t$ ("the coefficients ... involves functions depending on $t$"), they do not use vector-field/space-time resonance methods; instead they "turn to the $t$-dependent damping effect of the velocity" and construct "different weights for unknowns ... in the form of $\mathcal{L}(t)$" and verify "the differential inequality \eqref{est_L_1}."
\item Introduce "different temporal weights in a series of uniform estimates"; "the terms in $\mathcal{L}(t)$ in \eqref{L} carry distinct weights". "Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform," and this "plays a key role in establishing the global well-posedness ... in both two and three dimensions". This yields global existence in Lagrangian variables ("the global existence in Lagrangian coordinate would be established in Theorem~\ref{global_existence_Lagrangian} with the help of the local well-posedness and the global-in-time estimate of the quantity $\mathcal{L}(t)$").
\item \emph{Transfer back to Eulerian coordinates} ("transferred back into the Eulerian coordinate as stated in Theorem~\ref{global_existence} via the flow map"), and obtain the stated decay, noting that "in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$."
\end{itemize}

To prove the main theorem, the authors:
\begin{itemize}
\item \emph{Employ a Lagrangian transformation} ("we turn to employing a Lagrangian transformation") and "consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$."
\item Use that the affine expansion yields a \emph{$t$-dependent damping} in the Lagrangian linearized system ("the expanding factor on $t$ in the affine solution
\begin{equation}\label{1-1}
    \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t},  \bar{F}= (1+t)I,
  \end{equation}
contributes a damping term for the velocity in linearized system ... see
\begin{eqnarray} \label{compressible_elasticity_equation_5}
	\begin{cases}
		\partial_t \widetilde{\eta}=v,\\
		\partial_t v_{i} + \frac{2}{1+t} v_{i} + \frac{1}{(1+t)^{d(\gamma -1)+2}} a_{il}\partial_l q -\Delta \widetilde{\eta}_{i} =0,\\
		(v,\widetilde{\eta})|_{t=0} = \big(v_0, \widetilde{\eta}_0\big)
	\end{cases}
  \end{eqnarray}
).
\item Since the coefficients depend on $t$ ("the coefficients ... involves functions depending on $t$"), they do not use vector-field/space-time resonance methods; instead they "turn to the $t$-dependent damping effect of the velocity" and construct "different weights for unknowns ... in the form of $\mathcal{L}(t)$" and verify the differential inequality
\begin{equation}\label{est_L_1}
  \begin{split}
  \udt \mathcal{L}(t) + \frac{1}{1+t} \mathcal{L}(t)\le \frac{C}{(1+t)^{2}}\big(\mathcal{L}(t) + \mathcal{L}^{\frac{7}{2}}(t)\big).
  \end{split}
\end{equation}
\item Introduce "different temporal weights in a series of uniform estimates"; "the terms in $\mathcal{L}(t)$ in
\begin{equation}\label{L}
	\begin{split}
    \mathcal{L}(t) := &\|v(t)\|_{H^3}^2 + \frac{2}{(1+t)^2}\|\widetilde{\eta}(t)\|_{H^3}^2 + \|\nabla \widetilde{\eta}(t)\|_{H^3}^2+ \frac{2}{1+t}\sum_{i = 0}^{3} \langle\nabla^{i} v(t),\nabla^{i} \widetilde{\eta}(t)\rangle \\ &+ \frac{1}{\gamma(1+t)^{d(\gamma -1)+2}} \int_{\mathbb{R}^d}J^{\gamma+1}(t) |\nabla^3 q(t)|^2 \,dy.
	\end{split}
\end{equation}
carry distinct weights". "Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform," and this "plays a key role in establishing the global well-posedness ... in both two and three dimensions". This yields global existence in Lagrangian variables; more precisely, they establish the following theorem.

\begin{theorem}\label{global_existence_Lagrangian}
       For $d=2,3$,  there exists a sufficiently small  constant $\varepsilon_1>0$ such that the Cauchy problem \eqref{compressible_elasticity_equation_3} admits a global strong solution
       $(V,\xi)$ satisfying $(V(y,t)-y, \xi(y,t) - (1+t)y)$ in $C\big([0, \infty); H^3_y(\mathbb{R}^d)\times H^4_y(\mathbb{R}^d)\big)$,  provided that
          \begin{equation*} \label{initial_assumption_Lagrangian}
              \begin{split}
              \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le \varepsilon_1.
              \end{split}
          \end{equation*}
Moreover, $(V,\xi)$ satisfies the following estimates:
\begin{eqnarray}\label{decay_estimates_Lagrangian}
  \begin{cases}
    \|V(y,t) -y\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{1}{2}},\\[1mm]
    \|\xi(y,t) - (1+t)y\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{3}{2}},\\[1mm]
     \|\nabla_y \xi(y,t) - (1+t)I\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{1}{2}}
  \end{cases}
\end{eqnarray}
for all $t>0$, where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon_1$ and $t$.
\end{theorem}

\item \emph{Transfer back to Eulerian coordinates} ("transferred back into the Eulerian coordinate as stated in the main theorem via the flow map"), and obtain the stated decay, noting that "in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$."
\end{itemize}

\label{global_existence} For $d=2, 3$,
there exists a diffeomorphism $\xi_0: \mathbb{R}^d \to \mathbb{R}^d$  {and a small constant $\varepsilon>0$} such that if the initial data $(\rho_0, u_0, F_0)$, with
$(\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\in H^3_x(\mathbb{R}^d)$, satisfies
{\begin{eqnarray*}\label{decay_estiwmates_1}
    \begin{cases}
      \rho_0(x) \mathrm{det}F_0(x)=1,\,\,\,\,F_0(x) = \nabla_y \xi_0(y),\,\,\,\,  y =\xi_0^{-1}(x), \ \forall x\in\mathbb{R}^d, \\[2mm]
      \|\rho_0(x) -1\|_{H^3_x} + \|u_0(x) -x\|_{H^3_x} + \|F_0(x) -I\|_{H^3_x} + \|\xi_0^{-1}(x)-x\|_{H^4_x}\le \varepsilon,
    \end{cases}
\end{eqnarray*}}then, in establishing the main theorem, the Cauchy problem
\begin{eqnarray} \label{elasticity_equation_1}
    \begin{cases}
        \partial_t\rho+{\rm div} (\rho u) =0,\\
        \partial_t(\rho u)+{\rm div} (\rho u \otimes u)  + \nabla P(\rho)={\rm div}(\rho F F^\top),\\
        \partial_t F + u\cdot\nabla F=\nabla u F,\\
        (\rho,u,F)|_{t=0} = (\rho_0, u_0, F_0)
    \end{cases}
\end{eqnarray}
admits a unique global strong solution
$(\rho, u, F)$ such that $\big(\rho(x,t) -\frac{1}{(1+t)^d}, u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big) \in C\big([0, \infty); H^3_x(\mathbb{R}^d)\big)$. Moreover, $(\rho, u, F)$ satisfies the following estimates:
\begin{eqnarray*}\label{decay_estimates_1}
  \begin{cases}
    \|\nabla_x^{i}(\rho(x,t) -\frac{1}{(1+t)^d})\|_{L^2_x}\le C\varepsilon(1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
     \|\nabla_x^{i}\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
  \end{cases}
\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon$ and $t$.}

Question:
Let $d\in\{2,3\}$, and consider the compressible elastodynamics Cauchy problem on $\mathbb{R}^d$ for unknowns density $\rho(x,t)$, velocity $u(x,t)$, and deformation gradient $F(x,t)$:
\[
\begin{cases}
\partial_t\rho+\operatorname{div}(\rho u)=0,\\
\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)+\nabla P(\rho)=\operatorname{div}(\rho F F^{\top}),\\
\partial_tF+u\cdot\nabla F=\nabla u\,F,\\
(\rho,u,F)|_{t=0}=(\rho_0,u_0,F_0).
\end{cases}
\]
Here $I$ denotes the $d\times d$ identity matrix, and $H_x^k(\mathbb{R}^d)$ is the usual Sobolev space in the $x$-variable. Assume there exist a diffeomorphism $\xi_0:\mathbb{R}^d\to\mathbb{R}^d$ and a sufficiently small constant $\varepsilon>0$ such that
\[
(\rho_0-1,\,u_0-x,\,F_0-I)\in H_x^3(\mathbb{R}^d),
\]
and, for all $x\in\mathbb{R}^d$,
\[
\rho_0(x)\det F_0(x)=1,\qquad F_0(x)=\nabla_y\xi_0(y),\qquad y=\xi_0^{-1}(x),
\]
with the smallness condition
\[
\|\rho_0-1\|_{H_x^3}+\|u_0-x\|_{H_x^3}+\|F_0-I\|_{H_x^3}+\|\xi_0^{-1}-x\|_{H_x^4}\le \varepsilon.
\]
Under these hypotheses, which conclusion about the solution holds?

The Cauchy problem admits a unique global strong solution $(\rho,u,F)$ such that
\[
\Big(\rho(x,t)-\frac{1}{(1+t)^d},\;u(x,t)-\frac{x}{1+t},\;F(x,t)-(1+t)I\Big)\in C\big([0,\infty);H_x^3(\mathbb{R}^d)\big).
\]
Moreover, for every $t>0$ and every $i=0,1,2,3$,
\[
\|\nabla_x^i\big(\rho(x,t)-\tfrac{1}{(1+t)^d}\big)\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12-i},
\]
and
\[
\|\nabla_x^i\big(u(x,t)-\tfrac{x}{1+t},\;F(x,t)-(1+t)I\big)\|_{L_x^2}\le C\varepsilon(1+t)^{\frac d2+\frac12-i},
\]
where $C$ is a positive constant independent of $\varepsilon$ and $t$.

Choices:
B.
The Cauchy problem admits a unique global strong solution $(\rho,u,F)$ such that
\[
\Big(\rho(x,t)-\frac{1}{(1+t)^d},\;u(x,t)-\frac{x}{1+t},\;F(x,t)-(1+t)I\Big)\in C\big([0,\infty);H_x^3(\mathbb{R}^d)\big).
\]
Moreover, for every $t>0$ and every $i=0,1,2,3$,
\[
\|\nabla_x^i\big(\rho(x,t)-\tfrac{1}{(1+t)^d}\big)\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12-i},
\]
and
\[
\|\nabla_x^i\big(u(x,t)-\tfrac{x}{1+t},\;F(x,t)-(1+t)I\big)\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12-i},
\]
where $C$ is a positive constant independent of $\varepsilon$ and $t$.
C.
The Cauchy problem admits a unique global strong solution $(\rho,u,F)$ such that
\[
\Big(\rho(x,t)-\frac{1}{(1+t)^d},\;u(x,t)-\frac{x}{1+t},\;F(x,t)-(1+t)I\Big)\in C\big([0,\infty);H_x^3(\mathbb{R}^d)\big).
\]
Moreover, there exists a positive constant $C$ independent of $\varepsilon$ and $t$ such that, for every $t>0$,
\[
\|\rho(x,t)-\tfrac{1}{(1+t)^d}\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12},
\]
and
\[
\|\big(u(x,t)-\tfrac{x}{1+t},\;F(x,t)-(1+t)I\big)\|_{L_x^2}\le C\varepsilon(1+t)^{\frac d2+\frac12}.
\]
D.
The Cauchy problem admits a unique global strong solution $(\rho,u,F)$ such that
\[
\Big(\rho(x,t)-\frac{1}{(1+t)^d},\;u(x,t)-\frac{x}{1+t},\;F(x,t)-(1+t)I\Big)\in C\big([0,\infty);H_x^3(\mathbb{R}^d)\big).
\]
Moreover, there exists a positive constant $C$ independent of $\varepsilon$, $t$, and the initial diffeomorphism $\xi_0$ such that, for every $t>0$ and every $i=0,1,2,3$,
\[
\|\nabla_x^i\big(\rho(x,t)-\tfrac{1}{(1+t)^d}\big)\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12-i},
\]
and
\[
\|\nabla_x^i\big(u(x,t)-\tfrac{x}{1+t},\;F(x,t)-(1+t)I\big)\|_{L_x^2}\le C\varepsilon(1+t)^{\frac d2+\frac12-i},
\]
for all sufficiently small initial data satisfying the stated assumptions.
E.
The Cauchy problem admits a unique global strong solution $(\rho,u,F)$ such that
\[
\Big(\rho(x,t)-\frac{1}{(1+t)^d},\;u(x,t)-\frac{x}{1+t},\;F(x,t)-(1+t)I\Big)\in C\big([0,\infty);H_x^4(\mathbb{R}^d)\big).
\]
Moreover, for every $t>0$ and every $i=0,1,2,3,4$,
\[
\|\nabla_x^i\big(\rho(x,t)-\tfrac{1}{(1+t)^d}\big)\|_{L_x^2}\le C\varepsilon(1+t)^{-\frac d2-\frac12-i},
\]
and
\[
\|\nabla_x^i\big(u(x,t)-\tfrac{x}{1+t},\;F(x,t)-(1+t)I\big)\|_{L_x^2}\le C\varepsilon(1+t)^{\frac d2+\frac12-i},
\]
where $C$ is a positive constant independent of $\varepsilon$ and $t$.

Eulerian derivative-transfer gives opposite behavior for velocity/deformation rates

higher-derivative decay family restricted to the zeroth-order case only

dependence of the generic constant on fixed auxiliary data such as the initial diffeomorphism

available regularity level is H^3/H^4 on flow map, not H^4 Eulerian solution with fourth-order decay estimates

The stem does not explicitly disclose the correct option. It presents the hypotheses and asks for the resulting conclusion, without giving away the exact decay rates or regularity statement.

This is essentially a theorem-recall item: the stem states the full hypotheses of a specific result and asks for the corresponding conclusion. The correct choice is basically the theorem's conclusion verbatim.

There is some reasoning/verification pressure because the distractors alter decay exponents, regularity, and uniformity of constants in subtle ways. However, the task is still primarily recognition of the exact theorem statement rather than genuine derivation.

The distractors are mathematically plausible and target realistic failure modes: incorrect decay behavior, a weaker true statement, overly strong regularity, and an unjustified uniformity claim. They are distinct and nontrivial.

A solid theorem-recall MCQ with strong distractors, but it is largely a restatement of a known result rather than a genuinely generative reasoning question.