Let $(A,\mathfrak{m})$ be a Gorenstein local ring with $G(A)$ Cohen-Macaulay, and let $M$ be a Cohen-Macaulay $A$-module of finite projective dimension. In \cite{Quasipure}, the authors proved that $e_1(M)\geq \binom{c+1}{2}$, where $c=\operatorname{reg}G(A)$ and $e_i(M)$ is the $i$th Hilbert coefficient of $M$. We first show that this bound remains valid when $A$ is Cohen-Macaulay. We then study upper bounds for $e_2(M)$ when $e_1(M)=\binom{c+1}{2}+i$ for $i=1,2$, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the $h$-polynomial of $G(M)$. Finally, we extend these results to strict complete intersection rings without assuming that $M$ has finite projective dimension.
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of p...
Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let $\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $\Phi \colon\underline{CM}(A) \rightarrow \underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ a...
Let $R$ be a commutative ring and $f(a_1, \ldots, a_n) = \sum_{i=1}^k r_{i1}^{a_1} \cdots r_{in}^{a_n} m_i$ be a linear-exponential map over an $R$-module $M$. Dong and Shafrir (2026) showed that, when $\ell M = 0$ for some $\ell \in \mathbb{N}_{>0}$, the zero set of $f$ is the intersection of effectively computable $p...
Let $X$ be a smooth complex projective variety of dimension $n$, and let $L$ be an ample Cartier divisor. We prove that $K_X+mL$ is globally generated for every integer $m\geq\lceil C_0n\rceil$, where $C_0=1.77629\ldots$ is an explicit constant. In particular, $K_X+2nL$ is globally generated. We also prove that if $Z$...
Let $R$ be a Noetherian standard $\mathbb{N}$-graded ring, and let $I_1,\ldots,I_r$ be homogeneous ideals of $R$. Let $L,M$, and $N$ be finitely generated $\mathbb{Z}$-graded $R$-modules with $N\subseteq M$. For $\underline{n} :=(n_1,\dots,n_r)\in\mathbb N^r$, set ${\bf I}^{\underline{n}}:=I_1^{n_1}\cdots I_r^{n_r}$. W...
Let $A$ be an abelian group and $E=\Endo_{\Z}(A)$. We give a module-theoretic description of the singular $E$-submodules asked for in Fuchs'Problem~1.2. For a unital ring $R$ and a left $R$-module $M$, put $W=I_R({}_RR)\oplus I_R(M)$ and $J=\Jaco(\Endo_R(W))$, and let $u$ be the image of $1_R$. The classical essential-...
Jiang Yang, Xin Zhang· 0 citations
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