arXiv subjects

Monalisa Dutta

Publications and source records attributed to Monalisa Dutta.

3 recordsLinked to original sources

arXiv2024

On containment of trace ideals in ideals of finite homological dimension

Motivated by recent result of Pérez and R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by Smith that parameter test ideal cannot be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. One of our results says that the trace ideal of any big Cohen--Macaulay module over a Gorenstein complete local domain cannot be contained in any ideal of finite projective dimension, thereby generalizing Smith's result in this case. As consequences of our results, we give upper bounds on $\mathfrak m$-adic order of trace ideals of certain modules over local Cohen--Macaulay rings. We also prove analogous results for ideal of entries of maps in a free resolution of modules.

math.AC↗

arXiv2023

Ulrich split rings

A local Cohen--Macaulay ring is called Ulrich-split if any short exact sequence of Ulrich modules split. In this paper we initiate the study of Ulrich split rings. We prove several necessary or sufficient criteria for this property, linking it to syzygies of the residue field and cohomology annihilator. We characterize Ulrich split rings of small dimensions. Over complex numbers, $2$-dimensional Ulrich split rings, which are normal and have minimal multiplicity, are precisely cyclic quotient singularities with at most two indecomposable Ulrich modules up to isomorphism. We give several ways to construct Ulrich split rings, and give a range of applications, from test ideal of the family of maximal Cohen--Macaulay modules, to detecting projective/injective modules via vanishing of $\operatorname{Ext}$.

math.AC↗

arXiv2023

Exact subcategories, subfunctors of $\operatorname{Ext}$, and some applications

Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of $\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in details, and find a range of applications from detecting regularity to understanding Ulrich modules.

math.CT↗