One of the first remarkable results in representation theory artin algebras, due to Auslander and Ringel-Tachikawa, is characterisation when an algebra representation-finite. In this paper, we investigate aspects representation-finiteness general context exact categories sense Quillen. framework, introduce “big objects” prove Auslander-type “splitting-big-objects” theorem. Our approach generali...

Let R be a commutative noetherian ring. $$\mathsf {H}(R)$$ the quotient of Grothendieck group finitely generated R-modules by subgroup pseudo-zero modules. Suppose that $$\mathbb {R}$$ -vector space {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb has finite dimension. {C}(R)$$ (resp. {C}_r(R)$$ ) convex cone in spanned maximal Cohen–Macaulay rank r). We explore interior, closure an...

We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even li...

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded k-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically CohenMacaulay one-dimensional schemes of ...

For a polynomial ring R = k[x1, ..., xn], we present a method to compute the characteristic cycle of the localization Rf for any nonzero polynomial f ∈ R that avoids a direct computation of Rf as a D-module. Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules H I (R) for any ideal I ⊆ R using the Čech complex. The algorithm, in ...

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...