Related papers: A converse to a construction of Eisenbud-Shamash

Let R be an n-dimensional Cohen-Macaulay local ring and Q a parameter ideal of R. Suppose that an acyclic complex (F_{\bullet}, \varphi_{\bullet}) of length n of finitely generated free R-modules is given. We put M = Im \varphi_{1}, which…

The primary goal of this paper is to investigate the structure of irreducible monomorphisms to and irreducible epimorphisms from finitely generated free modules over a noetherian local ring. Then we show that over such a ring,…

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying…

Let $R$ be a domain of Krull dimension one, we study when the class $\mathcal{F}$ of modules over $R$ that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If $R$ is local, we show that…

Let $R=\Bbbk[x_1,\..., x_n]$ and $M=R^s/I$ a multigraded squarefree module. We discuss the construction of cochain complexes associated to $M$ and we show how to interpret homological invariants of $M$ in terms of topological computations.…

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.

Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show…

A module over a ring $R$ is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings.…

For a finitely generated, non-free module $M$ over a CM local ring $(R,\fm,k)$, it is proved that for $n\gg 0$ the length of $\tor 1RM{R/\fm^{n+1}}$ is given by a polynomial of degree $\dim R-1$. The vanishing of $\tor iRM{N/\fm^{n+1}N}$ is…

We present a general construction of eventually periodic projective resolutions for modules over quotients of rings of finite left global dimension by a regular central element. Our approach utilizes a construction of Shamash, combined with…

Let $G$ be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of $G$. This resolution is used to define…

For an ideal $I_{m,n}$ generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this…

We present a procedure that constructs, in a combinatorial manner, a chain complex of free modules over a polynomial ring in finitely many variables, modulo an ideal generated by quadratic monomials. Applying this procedure to two specific…

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure…

Let $(R,\fm)$ be a commutative Noetherian local ring. Suppose that $M$ and $N$ are finitely generated modules over $R$ such that $M$ has finite projective dimension and such that $\Tor^R_i(M,N)=0$ for all $i>0$. The main result of this note…

Let A be a noetherian commutative ring, and let I be an ideal in A. We study questions of flatness and I-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and I-adically free A-module.

We show how to obtain minimal projective resolutions of finitely generated modules over an idempotent subring $\Gamma_e := (1-e)R(1-e)$ of a semiperfect noetherian basic ring $R$ by a construction inside $\mathsf{mod} R$. This is then…

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q…

Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that $\m S = \n$ and the induced map on residue fields $R/\m \to S/\n$ is an isomorphism. Given a finitely generated…