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Let $(A,\mathfrak m)$ be a Noetherian local ring of dimension $d$, $I\subset A$ an $\mathfrak m$-primary ideal, and $M$ a finitely generated $A$-module. Writing $J_n:=\overline{I^n}$ for the integral-closure filtration, we study the asymptotic homological complexity of the quotients $M/J_nM$ via the syzygy growth functions \[ f_i(n)\;:=\;\mu_A\!\big(\Syz_i(M/J_nM)\big),\qquad i\ge1. \] Our first main result establishes eventual polynomial control: for each $i\ge 1$ there exists a polynomial $P_i(t)\in\mathbb{Q}[t]$ with \[ f_i(n)\ \le\ P_i(n)\quad\text{for all $n\gg 0$,}\qquad \deg P_i\le d-1, \] and, under the natural depth hypotheses $\depth M\ge i$ and $\depth\gr_{J_\bullet}(A)\ge 2$, the refined bound $\deg P_i\le d-1-i$ holds. For $i=1$ the leading term of $P_1$ is controlled by Hilbert--Samuel data: its leading coefficient is comparable to $e(I;M)$ with constants depending only on the Rees valuation data of $I$. In fact, the leading coefficient is identified explicitly as $\mathrm{LC}(P_{1})=\frac{e^{\,d-1,1}(I,\mathfrak m;M)}{(d-1)!}$. Our second main result proves \emph{depth stability}: $\depth(M/J_nM)$ is eventually constant in $n$, and the syzygies $\operatorname{Syz}_i(M/J_nM)$ admit uniform annihilators independent of $n$. The method combines valuation-theoretic control of $J_n$ via Rees valuations (yielding linear comparability with ordinary powers) with a graded-transfer mechanism to $\gr_{J_\bullet}(-)$ and exact $\Tor$ sequences, complemented by Artin--Rees type estimates for syzygies. Several classes of ideals illustrate sharpness of the bounds. The results establish a common mechanism linking multiplicity and Rees data to asymptotic syzygy growth and depth behavior along integral-closure filtrations.
Kundnani R. T., Ojha Sh. K., Alam K., Marimuthu V. 2026. Asymptotic Syzygy Growth and Depth Stability Along Integral-Closure Filtrations. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113835
