Related papers: Wiles defect for modules and criteria for freeness

It is proved that if $\varphi\colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated $B$-module $N$ whose flat dimension over $A$ is at most $\mathrm{edim}\, A - \mathrm{edim}\, B$, is free…

Let $A\to B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond's Theorem 2.1 from his…

Let $O$ be a discrete valuation ring and $A := O[X_{m \times n}]/I_{m}(X)$ the determinantal ring of maximal minors. We consider algebra maps $\lambda \colon A \to O$, which is tantamount to choosing rank-deficient matrices $a \in O^{m…

Let $(Q, \mathfrak{n})$ be a regular local ring and let $f_1, \ldots, f_c \in \mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, \mathfrak{m}) = (Q/(\mathbf{f}), \mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms $f_1^*,…

A ring with a test module of finite upper complete intersection dimension is complete intersection.

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…

Let $M$ be a finitely generated module over a ring $\Lambda$. With certain mild assumptions on $\Lambda$, it is proven that $M$ is a reflexive $\Lambda$-module, once $M \cong M^{**}$ as a $\Lambda$-module.

We prove a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient…

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed…

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective…

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…

Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local…

Let $M$ be a module over a Noetherian local ring $A$. We study $M$-independent sequences of elements of $\mathfrak{m}_A$ in the sense of Lech and Hanes. The main tool is a new characterization of the $M$-independence of a sequence in terms…

A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in…

The relationship between ${\cal D}$-modules and free divisors has been studied in a general setting by L. Narv\'aez and F.J. Calder\'on. Using the ideas of these works we prove in this article a duality formula between two ${\cal…

Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a…

Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module…

In this paper we give elementary conditions completely characterising when the theory of modules of a Pr\"ufer domain is decidable. Using these results, we show that the theory of modules of the ring of integer valued polynomials is…

In this note, we establish conditions under which the union of an increasing sequence of completely decomposable modules over domains are again completely decomposable. In our investigation, the condition of purity of modules is crucial. In…