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\fancyhead[L]{\small Nonexistence of lim Cohen--Macaulay modules}
\fancyhead[R]{\small Duality and truncation}
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\hypersetup{
 pdftitle={Nonexistence of lim Cohen-Macaulay modules following Chen and Anghel},
 pdfsubject={Local cohomology approximation in arbitrary dimension and its applications},
 pdfkeywords={lim Cohen-Macaulay sequences, local cohomology, canonical module, approximation}
}
\title{\bfseries Nonexistence of lim Cohen--Macaulay modules\\
following \cite{Chen} and \cite{Anghel}}
\author{}
\date{22 October 2026}
\begin{document}
\maketitle

\begin{abstract}
Using the inequality stated in \cite[Main Claim 1.1]{Chen}, we prove
nonexistence of lim
Cohen--Macaulay sequences for the completed section rings of smooth
projective surfaces over \(\C\) with an ample globally generated divisor
\(H\) satisfying \(K_Y\equiv4H\) and
\(15H^2/8>\chi(\OO_Y)\).
Such a pair is exhibited in \cite[Corollary 4.3]{Anghel}, with
\(H^2=1080\) and \(\chi(\OO_Y)=1926\).
For a complete local domain \((R,\mm)\) with
\(\ell_R\HH^j_{\mm}(R)<\infty\) for \(0\le j<\dim R\),
we prove that a lim Cohen--Macaulay sequence in the sense
of Bhatt, Hochster, and Ma \cite{BHM} exists if and only if a lim
\(\Gamma_{\mm}\)-Cohen--Macaulay sequence exists
(\cref{def:errors}). These may be different sequences.
Chen's geometric inequality \cite[Main Claim 1.1]{Chen} obstructs lim
\(\Gamma_{\mm}\)-Cohen--Macaulay sequences over the completed section
rings above. The comparison therefore rules out lim Cohen--Macaulay
sequences over those rings.
\end{abstract}

\section*{Introduction}

For a Noetherian local ring \((R,\mm,k)\), Bhatt, Hochster, and Ma
\cite[Definition 1.1]{BHM} define lim Cohen--Macaulay sequences of
nonzero finite discrete modules \(M_n\in\Mod_R\). For a fixed
system of parameters and each positive degree, the Koszul homology
length divided by \(\mu_R(M_n)\) must tend to zero, where
\(\mu_R(M_n)\) is the minimal number of generators of \(M_n\).
A lim
\(\Gamma_{\mm}\)-Cohen--Macaulay sequence satisfies the corresponding
condition for lower local cohomology; both definitions are given in
\cref{def:errors}.

For complete local domains \((R,\mm)\) satisfying
\(\ell_R\HH^j_{\mm}(R)<\infty\) for \(0\le j<\dim R\),
\cref{cor:existence} shows that the existence of these two different kinds
of sequences is equivalent, up to changing the sequence.
A given lim Cohen--Macaulay sequence need not itself be lim
\(\Gamma_{\mm}\)-Cohen--Macaulay; see \cref{rem:different-sequences}.
Chen's geometric inequality \cite[Main Claim 1.1]{Chen} obstructs lim
\(\Gamma_{\mm}\)-Cohen--Macaulay sequences for the completed section
rings considered here. The comparison then excludes lim Cohen--Macaulay
sequences.

\subsection*{Conventions}

All rings are discrete commutative Noetherian rings. The inputs and
witnesses in the statements are finite discrete modules. All tensor
products, mapping objects, local cohomology, and fiber sequences are taken
in the stable category \(\Mod_R\). The symbol \(\Hom_R\) denotes the
mapping spectrum with its natural \(R\)-module structure. Write
\(H^i(X)=\pi_{-i}(X)\); a discrete module shifted by \([s]\) lies in
homotopy degree \(s\). The truncations \(\tau_{\ge i}\) and
\(\tau_{\le i}\) use homotopy degrees. Free resolutions are indexed
homologically, with differential from degree \(i\) to degree \(i-1\).
Kernels and cokernels of maps between discrete modules are taken in
the heart \(\Mod_R^\heartsuit\).

\section{Definitions}

In the definitions below, \((R,\mm,k)\) is a complete local ring
of dimension \(d\ge1\).

\begin{definition}[Koszul and local cohomology errors]\label{def:errors}
Fix a system of parameters \(\mathbf q=(q_1,\ldots,q_d)\). For a finite discrete
\(M\in\Mod_R\), let \(\mu_R(M)\) be its minimal number of generators
and put
\[
 K(\mathbf q;M)=M\otimes_R\bigotimes_{i=1}^d
       \operatorname{cofib}(q_i:R\longrightarrow R),
 \qquad \HH^j_{\mm}(M)=\pi_{-j}\Gamma_{\mm}(M).
\]
Here \(\Gamma_{\mm}:\Mod_R\to\Mod_R\) is the local cohomology
functor. Define
\[
 \kappa_{\mathbf q}(M)=\max_{1\le i\le d}\ell_R\pi_iK(\mathbf q;M),
 \qquad
 \lambda_R(M)=\max_{0\le j<d}\ell_R\HH^j_{\mm}(M),
\]
allowing infinite length in the second expression. A nonzero \(M\)
is \emph{\(\varepsilon\)-CM} if
\(\kappa_{\mathbf q}(M)\le\varepsilon\mu_R(M)\), and is
\emph{\(\varepsilon\)-local-cohomology-CM} if
\(\lambda_R(M)\le\varepsilon\mu_R(M)\).
A sequence of nonzero finite discrete modules \(M_n\in\Mod_R\) is a
\emph{lim Cohen--Macaulay sequence} \cite[Definition 1.1]{BHM} if
\[
 \frac{\kappa_{\mathbf q}(M_n)}{\mu_R(M_n)}\longrightarrow0.
\]
It is a \emph{lim \(\Gamma_{\mm}\)-Cohen--Macaulay sequence} if
\(\lambda_R(M_n)<\infty\) for every \(n\), and
\[
 \frac{\lambda_R(M_n)}{\mu_R(M_n)}\longrightarrow0.
\]
\end{definition}

\begin{definition}[Duality]
Fix a normalized dualizing object
\(\omega_R\in\Mod_R\), so \(\Hom_R(k,\omega_R)\simeq k\), and set
\[
 D_R(X)=\Hom_R(X,\omega_R).
\]
\end{definition}
For finite discrete \(M\), one has \(\pi_iD_R(M)=0\) for
\(i\notin[0,d]\), and biduality gives
\(D_R(D_R(M))\simeq M\).

\begin{definition}[Reflexive modules]
For a finite discrete \(V\in\Mod_R\), put
\(V^\vee=\tau_{\ge0}\Hom_R(V,R)\).
We call \(V\) \emph{reflexive} if the evaluation map
\(V\to V^{\vee\vee}\) is an isomorphism.
\end{definition}

\section{\texorpdfstring{From lim Cohen--Macaulay to\\
lim \(\Gamma_{\mm}\)-Cohen--Macaulay sequences}{From lim Cohen-Macaulay to lim Gamma-m-Cohen-Macaulay sequences}}
\label{sec:equivalence}

\begin{definition}[The replacement]\label{def:replacement}
Let \((R,\mm,k)\) be complete local of dimension \(d\ge1\).
Fix a normalized dualizing object \(\omega_R\in\Mod_R\), and let
\(M\in\Mod_R\) be finite and discrete.
Choose a minimal free resolution \(P_\bullet\simeq D_R(M)\), with
each \(P_i\) finite free and \(P_i=0\) for \(i<0\). Let
\(\sigma_{<d}P_\bullet\) retain degrees \(0,\ldots,d-1\), and put
\[
 Q_M=\sigma_{\ge d}P_\bullet
   :=P_\bullet/\sigma_{<d}P_\bullet\in\Mod_R.
\]
Thus the retained degrees are \(d,d+1,\ldots\), including degree
\(d\); the differential out of degree \(d\) is replaced by zero.
This is the brutal truncation of the chosen resolution. It has exactly
one possibly nonzero homotopy module:
\[
 \pi_dQ_M=\coker(P_{d+1}\longrightarrow P_d),
 \qquad \pi_iQ_M=0\quad(i\ne d).
\]
Consequently \(\pi_iD_R(Q_M)=0\) for \(i\notin[-d,0]\).
Define the finite discrete replacement and its map by
\begin{equation}\label{eq:explicit-N}
 \boxed{\quad
 N_M=\tau_{\ge0}D_R(\sigma_{\ge d}P_\bullet)
       \xrightarrow{\ \rho_M\ }M.
 \quad}
\end{equation}
The map is the composite
\[
 N_M\longrightarrow D_R(Q_M)\longrightarrow D_R(D_R(M))\simeq M,
\]
obtained by dualizing the quotient \(D_R(M)\to Q_M\).
The connective cover retains \(\pi_0D_R(Q_M)\); the negative homotopy
groups of \(D_R(Q_M)\) need not vanish. The estimates below control the error caused by
this final truncation.
\end{definition}

\begin{remark}[Injective resolutions in the torsion category]\label{rem:injective}
Under the hypotheses of \cref{def:replacement}, work in the presentable
stable category
\[
 \mathcal T=\Mod_{R,\mm}
   :=\{X\in\Mod_R:\Gamma_{\mm}X\simeq X\}
\]
of modules supported at \(\mm\). The standard \(t\)-structure on
\(\Mod_R\) restricts to \(\mathcal T\); its heart consists of
discrete \(\mm\)-power torsion modules and has enough injectives.
Let \(E\in\mathcal T^\heartsuit\) be the injective hull of \(k\).
Choose a minimal injective resolution in \(\mathcal T\),
\[
 \Gamma_{\mm}M\simeq
   (J^0\longrightarrow J^1\longrightarrow J^2\longrightarrow\cdots),
 \qquad J^j\simeq E^{\oplus t_j},\quad t_j<\infty,
\]
with \(J^j\) placed in homotopy degree \(-j\). Here minimality
means that \(\ker(J^j\to J^{j+1})\to J^j\) is an injective hull
in \(\mathcal T^\heartsuit\) for every \(j\ge0\).
Retain the terms beginning with \(J^d\), deleting its incoming
differential. The resulting object
\[
 T_M=(0\longrightarrow J^d\longrightarrow J^{d+1}
          \longrightarrow\cdots)\in\mathcal T
\]
maps to \(\Gamma_{\mm}M\). Since \(\HH^j_{\mm}(M)=0\) for
\(j>d\), its only possibly nonzero homotopy module is
\[
 C_M:=\pi_{-d}T_M=\ker(J^d\longrightarrow J^{d+1})
       \in\mathcal T^\heartsuit,
 \qquad T_M\simeq C_M[-d].
\]
The module \(C_M\) need not be injective.

Write \(\Lambda_{\mm}:\Mod_R\to\Mod_R\) for completion, given by
\(\Lambda_{\mm}X=\Hom_R(\Gamma_{\mm}R,X)\)
(\stacks{0A6V}). Then the replacement and its map are
\[
 \boxed{\quad N_M\simeq\tau_{\ge0}\Lambda_{\mm}(C_M[-d])\quad},
 \qquad
 N_M\longrightarrow\Lambda_{\mm}T_M
       \longrightarrow\Lambda_{\mm}\Gamma_{\mm}M\simeq M.
\]
Indeed, termwise Matlis duality gives a minimal free resolution
\(P_i=\Hom_R(J^i,E)\) of \(D_R(M)\). For this choice of
\(P_\bullet\), the quotient \(D_R(M)\to Q_M\) is dual to the
inclusion \(T_M\to\Gamma_{\mm}M\). Local duality and completion
therefore identify
\[
 T_M\simeq\Hom_R(Q_M,E)\simeq\Gamma_{\mm}D_R(Q_M),
 \qquad \Lambda_{\mm}T_M\simeq D_R(Q_M)
\]
(\stacks{0A81}). Thus this is the construction in
\cref{def:replacement}, including its map to \(M\). The negative
homotopy groups of \(\Lambda_{\mm}T_M\) can be nonzero; the final
connective cover removes them and is finite and discrete.
\end{remark}

\begin{proposition}[Discrete quantitative comparison]
\label{prop:equivalence}
Let \((R,\mm,k)\) be a complete local domain of dimension \(d\ge1\)
such that
\begin{equation}\label{eq:ring-cohomology}
 h:=\sum_{j=0}^{d-1}\ell_R\HH^j_{\mm}(R)<\infty.
\end{equation}
Fix a system of parameters \(\mathbf q\). There is a constant
\(C=C(R,\mathbf q)\ge1\) such that the replacement
\(\rho_M:N_M\to M\) in \eqref{eq:explicit-N} satisfies, for every
finite discrete \(M\in\Mod_R\),
\begin{equation}\label{eq:approximation}
 \begin{gathered}
 \rank_RN_M\ge\rank_RM,\qquad
 \HH^j_{\mm}(N_M)=0\quad(0\le j<\min\{2,d\}),\\
 \ell_R\coker(\rho_M)
   +\sum_{j=0}^{d-1}\ell_R\HH^j_{\mm}(N_M)
       \le C\kappa_{\mathbf q}(M).
 \end{gathered}
\end{equation}
For every finite discrete \(M\) with \(\lambda_R(M)<\infty\),
\begin{equation}\label{eq:local-to-koszul}
 \kappa_{\mathbf q}(M)\le C\lambda_R(M).
\end{equation}
In particular,
\begin{equation}\label{eq:normalized}
 \begin{gathered}
 \mu_R(M)>0,\qquad 0\le\varepsilon\le\tfrac12,\qquad
 \kappa_{\mathbf q}(M)\le\varepsilon\mu_R(M)\\
 \Longrightarrow\quad
 N_M\ne0,\qquad
 \lambda_R(N_M)\le C\varepsilon\mu_R(N_M).
 \end{gathered}
\end{equation}
\end{proposition}

For the proof of \cref{prop:equivalence}, work under its hypotheses
and choose a finite local inclusion
\((A,\nn,k)\subset(R,\mm,k)\), with \(A\) complete regular of
dimension \(d\), and regular parameters \(\mathbf x\) of \(A\)
(\stacks{0323}, Lemma 10.160.11). These are also parameters of \(R\),
and finite lengths agree over \(A\) and \(R\). Put
\[
 \beta_i(M)=\dim_k\pi_i(k\otimes_A M),\qquad
 b=\sum_{i=0}^d\beta_i(R).
\]
The parameter comparison in \cite[Theorem 4.6]{BHM} gives constants
\(u,v\ge1\), independent of \(M\), such that
\begin{equation}\label{eq:parameters}
 \max_{i>0}\beta_i(M)\le u\kappa_{\mathbf q}(M),\qquad
 \kappa_{\mathbf q}(M)\le v\max_{i>0}\beta_i(M).
\end{equation}
Indeed, \(k\otimes_A M\simeq K(\mathbf x;M)\), and that theorem
compares the sums of positive Koszul homology lengths.

Finite duality identifies \(D_R(X)\) with \(\Hom_A(X,A)[d]\)
after restricting to \(\Mod_A\). For the discrete injective hull
\(E=E_R(k)\in\Mod_R\), local duality gives
\begin{equation}\label{eq:local-duality}
 \Gamma_{\mm}D_R(X)\simeq\Hom_R(X,E),\qquad
 \sum_{r=0}^{d-1}\ell_R\pi_r\omega_R=h
\end{equation}
for \(X\in\Mod_R\) with finitely many nonzero homotopy modules, each finite
(\stacks{0A81}). The regular subring \(A\) is used only in the
estimates; the construction \eqref{eq:explicit-N} does not use it.

\begin{lemma}[The ranks below degree \(d\)]\label{lem:free}
For the chosen minimal resolution of \(D_R(M)\), put
\(t_i=\rank_RP_i\). Then
\begin{equation}\label{eq:generator-bound}
 \sum_{i=0}^{d-1}t_i
       \le a\kappa_{\mathbf q}(M),\qquad
 a=d u(1+b)^{d-1}.
\end{equation}
\end{lemma}
\begin{proof}
Construct the resolution from degree zero upwards. If
\(Y_i\in\Mod_R\) is the remaining object at the \(i\)-th step,
minimality gives fiber sequences
\[
 P_i[i]\longrightarrow Y_i\longrightarrow Y_{i+1},
 \qquad Y_0=D_R(M),\quad Y_i\in\Mod_R^{\ge i},\quad
 t_i=\mu_R(\pi_iY_i).
\]
Since the residue fields agree,
\(t_i\le\dim_k\pi_i(k\otimes_A Y_i)\).
The finite \(A\)-module \(M\) is perfect over \(A\), so
\[
 k\otimes_A D_R(M)
     \simeq\Hom_k(k\otimes_A M,k)[d].
\]
Tensor the preceding fiber sequences with \(k\) over \(A\). Their
homotopy sequences give, for \(0\le i<d\),
\[
 t_i\le\beta_{d-i}(M)
          +\sum_{j<i}\beta_{i-j-1}(R)t_j
     \le\beta_{d-i}(M)+b\sum_{j<i}t_j.
\]
Induction yields
\(\sum_{i<d}t_i\le(1+b)^{d-1}\sum_{q=1}^d\beta_q(M)\).
Now apply \eqref{eq:parameters}.
\end{proof}

\begin{lemma}[Dualizing the truncation]\label{lem:dual}
The map \(\rho_M:N_M\to M\) satisfies
\[
 \ell_R\coker(\rho_M)
   +\sum_{j=0}^{d-1}\ell_R\HH^j_{\mm}(N_M)
       \le h\sum_{i=0}^{d-1}t_i.
\]
Moreover, \(\rank_RN_M\ge\rank_RM\), and
\(\HH^j_{\mm}(N_M)=0\) for \(0\le j<\min\{2,d\}\).
\end{lemma}
\begin{proof}
Let \(G_M=\sigma_{<d}P_\bullet\in\Mod_R\). It is perfect of
Tor-amplitude \([0,d-1]\), and the quotient gives a fiber sequence
\begin{equation}\label{eq:dual-approximation}
 G_M\longrightarrow D_R(M)\longrightarrow Q_M.
\end{equation}
Dualizing and using biduality gives
\begin{equation}\label{eq:replacement-map}
 D_R(Q_M)\longrightarrow M\longrightarrow D_R(G_M).
\end{equation}
The finite filtration of \(G_M\) has factors \(P_i[i]\).
Thus \(D_R(G_M)\) has a finite filtration with factors
\(\omega_R^{t_i}[-i]\), where \(0\le i<d\). By
\eqref{eq:local-duality},
\begin{equation}\label{eq:dual-error}
 \sum_{s\le0}\ell_R\pi_sD_R(G_M)
       \le h\sum_{i<d}t_i.
\end{equation}

Set \(W_M=\tau_{\le-1}D_R(Q_M)\in\Mod_R\). Because \(M\) is
discrete, the homotopy sequence of \eqref{eq:replacement-map} gives
\[
 0\longrightarrow\coker(\rho_M)
  \longrightarrow\pi_0D_R(G_M)
  \longrightarrow\pi_{-1}W_M\longrightarrow0,
 \qquad
 \pi_sW_M\simeq\pi_{s+1}D_R(G_M)\quad(s\le-2).
\]
Consequently all these modules have finite length, and
\begin{equation}\label{eq:finite-error}
 \ell_R\coker(\rho_M)+\sum_{s<0}\ell_R\pi_sW_M
       \le h\sum_{i<d}t_i.
\end{equation}
In particular \(\Gamma_{\mm}W_M\simeq W_M\), and \(\rho_M\)
is surjective at the generic point. This proves the rank inequality.

Apply \(\Gamma_{\mm}\) to the fiber sequence
\(N_M\to D_R(Q_M)\to W_M\). Local duality identifies the result
with
\[
 \Gamma_{\mm}N_M\longrightarrow\Hom_R(Q_M,E)
       \longrightarrow W_M.
\]
The only possibly nonzero homotopy module of the middle object is in
degree \(-d\). Therefore
\[
 \HH^j_{\mm}(N_M)\simeq\pi_{1-j}W_M\qquad(0\le j<d).
\]
The asserted vanishing and length bound follow from
\(W_M\in\Mod_R^{\le-1}\) and \eqref{eq:finite-error}.
\end{proof}

\begin{lemma}[Local cohomology controls Koszul homology]\label{lem:local}
For every finite discrete \(M\) with \(\lambda_R(M)<\infty\),
\[
 \kappa_{\mathbf q}(M)\le2^d v\lambda_R(M).
\]
\end{lemma}
\begin{proof}
The natural map \(k\otimes_A\Gamma_{\mm}M\to k\otimes_A M\)
is an isomorphism. Filter \(\Gamma_{\mm}M\) by its homotopy
truncations. The factor \(\HH^d_{\mm}(M)[-d]\) has no positive
homotopy groups after tensoring with \(k\), since \(\pd_A k=d\).
For a finite-length discrete \(A\)-module \(T\), a composition
series gives
\(\dim_k\pi_p(k\otimes_A T)\le\binom dp\ell_A(T)\).
It follows that, for \(i>0\),
\[
 \beta_i(M)\le\sum_{j=0}^{d-1}\binom d{i+j}\ell_R\HH^j_{\mm}(M)
       \le2^d\lambda_R(M),
\]
with binomial coefficients outside \([0,d]\) equal to zero.
Use \eqref{eq:parameters}.
\end{proof}

\begin{proof}[Proof of \cref{prop:equivalence}]
The three lemmas give \eqref{eq:approximation} with constant \(ha\)
and \eqref{eq:local-to-koszul} with constant \(2^dv\).
For the normalization, put \(e_{\mathbf q}=e((\mathbf q),R)\).
The following rank bound holds for every finite discrete \(M\):
\begin{equation}\label{eq:rank}
 \mu_R(M)\le e_{\mathbf q}\rank_RM+\kappa_{\mathbf q}(M).
\end{equation}
Indeed, write \(h_i=\ell_R\pi_iK(\mathbf q;M)\). The multiplicity
formula and nonnegativity of the second partial Euler characteristic
give
\[
 h_0-h_1+\underbrace{(h_2-h_3+\cdots)}_{\chi_2\ge0}
       =e_{\mathbf q}\rank_RM.
\]
The right side is zero when \(M\) has rank zero. The nonnegativity
is Serre's theorem; see \cite{Puth}. Since \(\mu_R(M)\le h_0\),
this proves \eqref{eq:rank}.
If \(M\ne0\) and \(\kappa_{\mathbf q}(M)\le\varepsilon\mu_R(M)\) with
\(\varepsilon\le1/2\), then
\[
 \mu_R(N_M)\ge\rank_RN_M\ge\rank_RM
       \ge\frac{\mu_R(M)}{2e_{\mathbf q}}>0.
\]
Hence \(\lambda_R(N_M)\le2e_{\mathbf q}ha\varepsilon\mu_R(N_M)\).
Take \(C=\max\{1,ha,2^dv,2e_{\mathbf q}ha\}\).
\end{proof}

\begin{corollary}\label{cor:existence}
For a complete local domain satisfying \eqref{eq:ring-cohomology},
the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item For every \(\varepsilon>0\), there is a nonzero finite discrete
 \(\varepsilon\)-CM module for one fixed system of parameters.
 \item For every \(\varepsilon>0\), there is a nonzero finite discrete
 \(\varepsilon\)-local-cohomology-CM module.
\end{enumerate}
The first condition is independent of the system of parameters.
Equivalently, \(R\) admits a lim Cohen--Macaulay sequence if and only
if it admits a lim \(\Gamma_{\mm}\)-Cohen--Macaulay sequence.
\end{corollary}
\begin{proof}
Apply the proposition with input error
\(\min\{1/2,\varepsilon/C\}\). The converse uses
\eqref{eq:local-to-koszul} on the same module. Parameter independence
follows from \cite[Theorem 4.6]{BHM}. Choose errors tending to zero
for the assertion about sequences.
\end{proof}

\begin{remark}[Cohen--Macaulay approximations]\label{rem:AB}
If \(M\) is maximal Cohen--Macaulay, then \(\pi_iD_R(M)=0\) for
\(i\ne d\), so minimality gives \(P_i=0\) for \(i<d\).
Thus \(Q_M\simeq D_R(M)\), and \(\rho_M:N_M\to M\) is an
isomorphism.

If \(R\) is Cohen--Macaulay, then \(h=0\), and the construction
gives an exact sequence of finite discrete modules
\[
 0\longrightarrow L_M\longrightarrow N_M
    \xrightarrow{\rho_M}M\longrightarrow0,
 \qquad N_M\text{ maximal Cohen--Macaulay}.
\]
The module \(L_M\) has finite injective dimension. Indeed,
\(W_M=0\) by \eqref{eq:finite-error}, and
\(D_R(G_M)\simeq L_M[1]\). Its finite filtration has factors
\(K_R^{t_i}[d-i]\), where \(K_R=H^{-d}(\omega_R)\) is the
canonical module, which has finite injective dimension.
This is an Auslander--Buchweitz maximal Cohen--Macaulay approximation
\cite{AB}.
\end{remark}

\begin{remark}[The witnesses may change]
\label{rem:different-sequences}
The proposition compares existence conditions; it does not bound the two
errors on the same module. For example, let \(S=\C[[u,v,w]]\) and
\[
 P_n=\coker\bigl(S\xrightarrow{(u^n,v^n,w^n)}S^3\bigr),
 \qquad M_n=S^n\oplus P_n.
\]
The presentation and local duality give
\[
 \mu_S(M_n)=n+3,\qquad \kappa_{(u,v,w)}(M_n)=1,\qquad
 \ell_S\HH^2_{(u,v,w)}(M_n)=n^3.
\]
Thus the normalized Koszul error tends to zero while the normalized
local cohomology error tends to infinity. For these modules,
\eqref{eq:explicit-N} gives \(N_{M_n}\simeq S^{n+3}\).
\end{remark}

\begin{remark}[Extending lower bounds to arbitrary modules]
\label{rem:remove-reflexivity}
Let \((R,\mm)\) be any three-dimensional Noetherian normal local
domain.
For every finite discrete \(M\in\Mod_R\), the natural map
to its reflexive hull induces a surjection
\begin{equation}\label{eq:hull-surjection}
 \HH^2_{\mm}(M)\longrightarrow\HH^2_{\mm}(M^{\vee\vee})
          \longrightarrow0,
 \qquad \rank_RM^{\vee\vee}=\rank_RM.
\end{equation}
To prove this, let \(T\subset M\) be its torsion submodule and put
\(M'=M/T\), with both regarded as discrete \(R\)-modules.
Then \(\dim\Supp T\le2\), so \(\HH^3_{\mm}(T)=0\) and
\(\HH^2_{\mm}(M)\to\HH^2_{\mm}(M')\) is surjective.
The injection \(M'\to M^{\vee\vee}\) is an isomorphism at primes
of height at most one (\stacks{0AVC}).
Its cokernel has support of dimension at most one, so its second
local cohomology vanishes. The second surjection follows, proving
\eqref{eq:hull-surjection}.

Consequently, a bound
\[
 \ell_R\HH^2_{\mm}(V)\ge\delta\rank_RV
\]
for finite discrete reflexive \(V\) with finite-length
\(\HH^2_{\mm}(V)\) implies the same bound for every finite discrete
\(M\) with finite-length \(\HH^2_{\mm}(M)\).
For rank zero the bound is automatic. One may also allow infinite
length, in which case the inequality is automatic whenever the
left side is infinite.
This assertion concerns the lower bound; it does not assert that
\(\HH^2_{\mm}(M)\) has finite length for arbitrary \(M\).
\end{remark}

\begin{corollary}[A uniform obstruction]\label{cor:gap}
Let \(R\) satisfy the hypotheses on the ring in
\cref{prop:equivalence}, and fix a system of parameters \(\mathbf q\).
Suppose \(\delta>0\) and
\[
 \sum_{j=0}^{d-1}\ell_R\HH^j_{\mm}(N)\ge\delta\rank_RN
\]
for every finite discrete \(N\in\Mod_R\) satisfying
\(\ell_R\HH^j_{\mm}(N)<\infty\) for \(0\le j<d\).
Then there is \(\eta>0\) such that every nonzero
finite discrete \(M\in\Mod_R\) satisfies
\begin{equation}\label{eq:gap}
 \kappa_{\mathbf q}(M)\ge\eta\mu_R(M).
\end{equation}
In particular, \(R\) admits no lim Cohen--Macaulay sequence.
\end{corollary}
\begin{proof}
Use \cref{prop:equivalence}, and let \(C\) be its
constant. The module \(N\) in \eqref{eq:approximation} gives
\[
 \delta\rank_RM\le\delta\rank_RN
       \le\sum_{j=0}^{d-1}\ell_R\HH^j_{\mm}(N)
       \le C\kappa_{\mathbf q}(M).
\]
With \(e_{\mathbf q}=e((\mathbf q),R)\), \eqref{eq:rank} implies
\[
 \mu_R(M)\le e_{\mathbf q}\rank_RM+\kappa_{\mathbf q}(M)
       \le(e_{\mathbf q}C/\delta+1)\kappa_{\mathbf q}(M).
\]
Take \(\eta=(e_{\mathbf q}C/\delta+1)^{-1}\). This also covers rank zero.
\end{proof}

\section{\texorpdfstring{Nonexistence of lim
\(\Gamma_{\mm}\)-Cohen--Macaulay sequences from Chen's inequality}{Nonexistence of lim Gamma-m-Cohen-Macaulay sequences from Chen's inequality}}
\label{sec:application}

Let \(Y\) be a smooth connected projective surface over \(\C\), and
let \(H\) be an ample globally generated Cartier divisor such that
\[
 K_Y\equiv4H,\qquad
 \delta=\frac{15}{8}H^2-\chi(\OO_Y)>0.
\]
In discrete commutative \(\C\)-algebras, put
\[
 B=\bigoplus_{n\ge0}H^0(Y,\OO_Y(nH)),
 \qquad
 B_+=\bigoplus_{n>0}B_n,
 \qquad
 R=\widehat{B_{B_+}},
 \qquad \mm=B_+R.
\]
We use the geometric inequality stated in \cite[Main Claim 1.1]{Chen}:
\begin{equation}\label{eq:geometric}
 \ell_R\HH^2_{\mm}(V)\ge\delta\rank_RV
\end{equation}
for every nonzero finite discrete reflexive \(V\in\Mod_R\).
The length on the left is finite by \cite[Lemma 2.5]{Chen}.

\begin{remark}[Reflexivity is unnecessary in the numerical lower bound]
\label{rem:chen}
By \cref{rem:remove-reflexivity}, the reflexivity hypothesis in
Chen's numerical inequality \eqref{eq:geometric} can be omitted.
The same inequality holds for every finite discrete \(V\in\Mod_R\)
with finite-length \(\HH^2_{\mm}(V)\), and with the convention
of infinite length it holds for every finite discrete \(V\).
This removes a hypothesis from the lower bound, not from Chen's
separate assertion of finite length.
No reflexivity condition is therefore needed when applying the
inequality to the quantitative criterion of \cref{prop:equivalence}.
\end{remark}

\begin{corollary}
\label{cor:surface}
For the pair \((Y,H)\) above, \(R\) is a three-dimensional complete normal local domain
with no lim \(\Gamma_{\mm}\)-Cohen--Macaulay sequence and hence
no lim Cohen--Macaulay sequence.
\end{corollary}
\begin{proof}
The ring \(R\) is complete, normal, and three-dimensional by
\cite[Proposition 2.1]{Chen}.
Normality gives \(\HH^0_{\mm}(R)=\HH^1_{\mm}(R)=0\).
Applying \cite[Lemma 2.5]{Chen} to \(V=R\) shows that
\(\ell_R\HH^2_{\mm}(R)<\infty\).
Thus the ring hypotheses of \cref{prop:equivalence} hold.
By \cref{rem:chen}, the inequality supplied by
\cite[Main Claim 1.1]{Chen} applies to every finite discrete module
with finite-length second local cohomology.
Fix parameters \(\mathbf q\), put \(e_{\mathbf q}=e((\mathbf q),R)\),
and let \(C\) be the constant in \cref{prop:equivalence}.
For every nonzero finite discrete \(M\in\Mod_R\) with
\(\lambda_R(M)<\infty\), Chen's inequality gives
\(\lambda_R(M)\ge\delta\rank_R M\).
Together with \eqref{eq:rank} and \eqref{eq:local-to-koszul}, this yields
\[
 \mu_R(M)\le e_{\mathbf q}\rank_R M+\kappa_{\mathbf q}(M)
       \le\bigl(e_{\mathbf q}/\delta+C\bigr)\lambda_R(M).
\]
Thus no lim \(\Gamma_{\mm}\)-Cohen--Macaulay sequence exists.
By \cref{cor:existence}, no lim Cohen--Macaulay sequence exists either.
\end{proof}

\paragraph{The numerical condition and the example.}
If \(\sigma(Y)\) denotes the signature of the intersection form
on \(H^2(Y(\C),\mathbf R)\), the signature and Noether formulas give
\[
 \sigma(Y)=K_Y^2-8\chi(\OO_Y),\qquad
 \delta=\frac{\sigma(Y)-H^2}{8},
\]
using \(K_Y\equiv4H\).
Thus the numerical condition agrees with the degree--signature
inequality in \cite{Anghel}.

In projective \(\Q\)-schemes, let
\[
 W_{\Q}=V_+\bigl(u_3^6-u_0^6+u_1^6,\;
 u_4^6-u_0^6+u_2^6,\;
 u_5^6-u_1^6+u_2^6\bigr)\subset\mathbf P^5_{\Q}.
\]
Let \(\rho:Y_{\Q}\to W_{\Q}\) be the blow-up of the reduced
singular locus, let \(E\) be its reduced exceptional divisor, and
put \(H=3\rho^*\OO_{W_{\Q}}(1)-E\).
After extension to \(\C\), the resulting pair satisfies
\[
 \begin{gathered}
 Y\text{ smooth},\qquad H\text{ ample and globally generated},\\
 K_Y\sim4H,\qquad H^2=1080,\qquad \chi(\OO_Y)=1926.
 \end{gathered}
\]
These are the surface and polarization in
\cite[Remark 4.2 and Corollary 4.3]{Anghel} and
\cite[\S7]{Chen}. Hence \(\delta=99\).
Applying Chen's geometric inequality \cite[Main Claim 1.1]{Chen},
\cref{cor:surface} excludes
lim Cohen--Macaulay sequences over the completed vertex of its
full section ring.

\begin{remark}[Mixed characteristic]\label{rem:mixed}
By \cref{cor:surface}, the rational example above gives, for every prime \(p\), a four-dimensional complete
normal local domain of mixed characteristic \((0,p)\), with
residue field \(\mathbf F_p\), admitting no lim Cohen--Macaulay sequence.

Here are the algebraic details. Let \(B_{\Q}\) be the full section
ring of the rational pair above, let \(V=\Z_p\), \(K=\Q_p\), and
put \(B_K=B_{\Q}\otimes_{\Q}K\).
Choose homogeneous generators \(b_1,\ldots,b_t\) of strictly
positive degree for \(B_{\Q}\).
The normalization \(C\) of \(V[b_1,\ldots,b_t]\) in
\(\Frac(B_K)\) is finite by excellence.
It lies in \(B_K\), is graded (\stacks{077G}), and satisfies
\(C_0=V\) and \(C[1/p]=B_K\).
Each graded piece is finite free over \(V\), and
\(\dim(C/pC)=3\).
Lift homogeneous parameters of \(C/pC\) to \(s_0,s_1,s_2\in C\).
Degreewise Nakayama shows that \(C\) is finite over
\(V[s_0,s_1,s_2]\); the parameter degrees may exceed one.
Put
\[
 T=\widehat{C_{(p,C_+)}},\qquad
 A=V[[s_0,s_1,s_2]],\qquad
 P=C_+T.
\]
Then \(T\) is normal (\stacks{0C23}), finite over \(A\), of
dimension four, and \(T/P=V\).
The positive-degree ideal of the finite graded \(V\)-algebra
\(C/(s_0,s_1,s_2)\) is nilpotent.
Hence \(P\) is the unique prime over \((s_0,s_1,s_2)A\), and
\begin{equation}\label{eq:mixed-completion}
 \widehat{T_P}
 \cong T\otimes_A K[[s_0,s_1,s_2]]
 \cong\widehat{(B_K)_{(B_K)_+}}.
\end{equation}

For a hypothetical lim Cohen--Macaulay sequence of finite discrete
\(M_n\in\Mod_T\), write
\[
 r_n=\rank_TM_n,\qquad e=\rank_AT,\qquad
 \eps_n=\sum_{i=1}^{4}\dim_{\mathbf F_p}
              \pi_i(\mathbf F_p\otimes_AM_n).
\]
The Euler characteristic gives
\(\mu_T(M_n)\le er_n+\eps_n\).
Since \(\eps_n=o(\mu_T(M_n))\), one has \(r_n>0\) eventually
and \(\eps_n=o(r_n)\).
Localizing and completing a minimal finite \(A\)-free resolution
bounds the positive Betti numbers of
\(N_n=M_n\otimes_T\widehat{T_P}\) over \(K[[s_0,s_1,s_2]]\)
by those of \(M_n\) over \(A\).
Flatness makes \(N_n\) discrete, and
\(\rank_{\widehat{T_P}}N_n=r_n\).
Since \(\mu_{\widehat{T_P}}(N_n)\ge r_n\), these modules form
a lim Cohen--Macaulay sequence over \(\widehat{T_P}\).

Choose an abstract field embedding \(K\hookrightarrow\C\).
The map \(K[[s_0,s_1,s_2]]\to\C[[s_0,s_1,s_2]]\) is faithfully
flat and local (\stacks{00R4}).
Its base change in \eqref{eq:mixed-completion} is the completed
section ring over \(\C\).
It preserves minimal generator numbers and lengths of the positive
Koszul homotopy modules.
The resulting sequence contradicts \cref{cor:surface}.
\end{remark}

\Needspace{12\baselineskip}
\begin{thebibliography}{BHM26}
\bibitem[AB89]{AB}
Maurice Auslander and Ragnar-Olaf Buchweitz,
\emph{The homological theory of maximal Cohen--Macaulay approximations},
M\'emoires de la Soci\'et\'e Math\'ematique de France, nouvelle s\'erie
\textbf{38} (1989), 5--37.
\href{https://doi.org/10.24033/msmf.339}{doi:10.24033/msmf.339}.

\bibitem[Put04]{Puth}
Tony J. Puthenpurakal,
\emph{A short note on the non-negativity of partial Euler characteristics},
preprint, 2004.
\url{https://arxiv.org/abs/math/0409059}.


\bibitem[Ang26]{Anghel}
Cristian Anghel,
\emph{Polarized varieties without arithmetically Cohen--Macaulay
bundles and section rings without graded maximal Cohen--Macaulay
modules in characteristic zero},
preprint, arXiv:2609.15589v1, 14 September 2026.
\url{https://arxiv.org/abs/2609.15589v1}.

\bibitem[BHM26]{BHM}
Bhargav Bhatt, Mel Hochster, and Linquan Ma,
\emph{Lim Cohen--Macaulay sequences of modules},
Journal f\"ur die reine und angewandte Mathematik \textbf{835}
(2026), 111--169.
\href{https://doi.org/10.1515/crelle-2026-0021}{doi:10.1515/crelle-2026-0021}.
The numbering used here agrees with the
\href{https://sites.lsa.umich.edu/hochster/wp-content/uploads/sites/1337/2026/02/limCM-Crelle-final.pdf}{author manuscript dated 10 February 2026}.

\bibitem[Che26]{Chen}
Liang Chen,
\emph{A local cohomology obstruction to small Cohen--Macaulay modules},
preprint, arXiv:2609.24142v1, 21 September 2026.
\url{https://arxiv.org/abs/2609.24142v1}.

\bibitem[Sta26]{Stacks}
The Stacks Project Authors, \emph{The Stacks Project}, 2026.
\url{https://stacks.math.columbia.edu/}.
Specific tags are linked in the text. Accessed 22 October 2026.
\end{thebibliography}
\end{document}