Constructing Big Indecomposable modules

Constructing indecomposable integrally closed modules over a two-dimensional regular local ring

In this article, we construct integrally closed modules of rank two over a two-dimensional regular local ring. The modules are explicitly constructed from a given complete monomial ideal with respect to a regular system of parameters. Then we investigate their indecomposability. As a consequence, we have a large class of indecomposable integrally closed modules whose Fitting ideal is not simple. This gives an answer to Kodiyalam's question.

Ulrich bundles on non-special surfaces with $p_g=0$ and $q=1$

Let $S$ be a surface with $p_g(S)=0$, $q(S)=1$ and endowed with a very ample line bundle $\mathcal O_S(h)$ such that $h^1\big(S,\mathcal O_S(h)\big)=0$. We show that such an $S$ supports families of dimension $p$ of pairwise non-isomorphic, indecomposable, Ulrich bundles for arbitrary large $p$. Moreover, we show that $S$ supports stable Ulrich bundles of rank $2$ if the genus of the general element in $\vert h\vert$ is at least $2$.

分次环上分次模的不可分解性

Indecomposability of graded modules over a graded ring

本文证明了在 Henselian 局部环上的非负分次环及其有限型分次代数中，模的不可分解性在分次、局部化和完备化意义下等价，并给出了分次模分解的 Krull-Schmidt 唯一性定理及其在特征 p 环 FFRT 性质上的应用。

A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity

Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Maca…

分解有限 $\mathbb{Z}$-代数

Decomposing Finite $\mathbb{Z}$-Algebras

本文针对有限生成的加法群非结合环（有限 $\mathbb{Z}$-代数），提出基于标量环分解与提升的算法，在弱假设下实现了 $R/{\rm Ann}(R)$ 的直接不可分解分解，且具有概率多项式时间复杂度。