A note on cancellation of reflexive modules

On the number of indecomposable totally reflexive modules

In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive module.

关于Matlis自反模的注记

Notes on Matlis reflexive modules

本文研究局部Noether环上Matlis自反模，证明了自反模的拟完备性，改进Zöschinger数值判据，并刻画Enochs数为1的条件。

A note on cancellation of projective modules

Let $A$ be a ring of dimension $d$. Assume that for every finite extension ring $R$ of $A$, E_{d+1}(R) acts transitively on Um_{d+1}(R). Then we prove that E(A\oplus P) acts transitively on Um(A\oplus P), for any projective A-module P of rank d. As a consequence of this, we generalise some results of Gubeladze.

Note on the candidate counter-example in the cancellation problem for affine spaces posed by Arno Van den Essen

We have proved the following Problem:{\it Let $R$ be a $\mathbb{C}$-affine domain, let $T$ be an element in $R \setminus \mathbb{C}$ and let $i : \mathbb{C}[T] \hookrightarrow R$ be the inclusion. Assume that $R/TR \cong_{\mathbb{C}} \mathbb{C}^{[n-1]}$ and that $R_T \cong_{\mathbb{C}[T]} \mathbb{C}[T]_T^{[n-1]}$. Then $R \cong_{\mathbb{C}} \mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {\it Let $A:=\mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be…

关于UFD的一个注记

A note on UFD

研究高度为一的理想为主理想的条件，通过局部上同调技术简化Auslander-Buchsbaum定理的证明，并解决Samuel问题，证明重数至多为3的局部环是超曲面，肯定Braun关于有限内射维数自反理想的猜想，简化Hartshorne的一个结果。

Cancellation of homothetic modules

Let R be a commutative ring, M an R-module, and N a finitely presented R-module such that the intersection of Max(R) and Supp(N) is finite-dimensional and Noetherian. Suppose also that N is homothetic; in other words, suppose that the natural ring homomorphism from R to the R-endomorphism ring of N is surjective. Working under these conditions, we describe various ways to guarantee the existence of a split surjective map in a specified coset of Hom(M,N). Using these results, we yield an extension of Bass's Cancellation Theorem that gives sufficient conditions for cancelling N or, more generall…