Full-Trace Modules
Ela Celikbas, Olgur Celikbas, Jürgen Herzog, Shinya Kumashiro
TL;DR
This work introduces full-trace modules, defined by $\operatorname{tr}_R(M)=\mathfrak{m}$, and links them to nearly Gorenstein rings via trace ideals. It classifies when syzygies of the residue field $k$, namely $\Omega_R^{i}(k)$, are full-trace across regular, principal, and non-principal rings, using Koszul and Gulliksen–Tate constructions to illuminate the structure of traces. The paper further develops full-trace Ulrich modules over Cohen–Macaulay rings, establishing that minimal multiplicity is equivalent to the existence of such modules and providing a precise decomposition in dimension one for numerical semigroup rings. It also furnishes higher-dimensional examples to delimit the scope of these decompositions, situating the results within trace-ideal theory and endomorphism algebras. Overall, the results advance the understanding of trace ideals and their role in classifying and characterizing local rings via homological and Ulrich-theoretic properties.
Abstract
Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the existence of such modules and prove that, over rings that are neither regular nor principal ideal rings, every positive syzygy module of the residue field is full-trace. Moreover, over Cohen-Macaulay rings, we study full-trace Ulrich modules--that is, maximally generated maximal Cohen-Macaulay modules that are full-trace. We establish the following characterization: a non-regular Cohen-Macaulay local ring has minimal multiplicity if and only if it admits a full-trace Ulrich module. Finally, for numerical semigroup rings with minimal multiplicity, we show that each full-trace Ulrich module decomposes as the direct sum of the maximal ideal and a module that is either zero or Ulrich.