Related papers: Resolutions over strict complete intersections

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if…

Let $(R,\fm)$ be a local ring and $\fa$ be an ideal of $R$. The inequalities $$\begin{array}{ll} \ \Ht(\fa) \leq \cd(\fa,R) \leq \ara(\fa) \leq l(\fa) \leq \mu(\fa) \end{array}$$ are known. It is an interesting and long-standing problem to…

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of…

Let $(Q,\mathfrak n,k)$ be a commutative local Noetherian ring, $f_1,\dots, f_c$ a $Q$-regular sequence in $\mathfrak n$, and $R=Q/(f_1,\dots,f_c)$. Given a complex of finitely generated free $R$-modules, we give a construction of a complex…

In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the…

Let $A$ be a local complete intersection ring. Let $M,N$ be two finitely generated $A$-modules and $I$ an ideal of $A$. We prove that \[ \bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}\mathrm{Ass}_A\left(\mathrm{Ext}_A^i(M,N/I^n N)\right) \]…

Let $R$ be a finite ring and let $M, N$ be two finite left $R$-modules. We present two distinct deterministic algorithms that decide in polynomial time whether or not $M$ and $N$ are isomorphic, and if they are, exhibit an isomorphism. As…

The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families…

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underline{a} = a_1,\ldots,a_t$ such that $a_i \in \mathfrak{q}^{c_i}, i = 1, \ldots,t,$ and $n \in \mathbb{Z}$ we investigate a subcomplex…

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$. We investigate the structure of formal local cohomology modules of…

Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,f_c $ is an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S $-modules. It is…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M, N$ two finitely generated $R$-modules. By using a spectral sequence argument, it is shown that if either $\mathrm{dim}_RM\leq2$ and $\mathrm{H}^{i}_\mathfrak{a}(N)$…

Let A be a Cohen-Macaulay local ring of dimension d and I an ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. Let I be a locally complete intersection ideal of analytic deviation one and reduction number at most…

We prove that every quasi-complete intersection ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by-product we establish a rigidity statement for the minimal two-step Tate complex…

We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\bd C$ such that the sequence $\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all $i>0$. Over these rings we show…

It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F-purity. In this note, we prove that if (R,m) is an equidimensional and S_2 local ring that admits a…

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$)…

It is proved that a map $\varphi\colon R\to S$ of commutative noetherian rings that is essentially of finite type and flat is locally complete intersection if and only $S$ is proxy small as a bimodule. This means that the thick subcategory…