Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

McDaniel, Chris; Chen, Shujian; Iarrobino, Anthony; Marques, Pedro Macías

doi:10.1080/03081087.2019.1598930

Cited by 7 publications

(2 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Jordan type P ℓ of a non-homogeneous element ℓ ∈ m A may be the same as that would be expected for a strong Lefschetz element, even though A may have no linear strong Lefschetz elements, so A is not SL. This we first noticed on the following example of relative covariants proposed by the third author (see [MCIM,Example 3.7]).…”

Section: Evidently We Havementioning

confidence: 84%

Artinian algebras and Jordan type

Iarrobino¹,

Marques²,

McDaniel³

2018

Preprint

Self Cite

View full text Add to dashboard Cite

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal m of A, acting on M . Here, the Jordan type of ℓ ∈ m A is the partition giving the Jordan blocks of the multiplication map m ℓ : M → M . In particular, we consider the Jordan type of a generic linear element ℓ in A 1 , or in the case of a local ring A, that of a generic element ℓ ∈ m A , the maximum ideal.We often take M = A, the graded algebra, or M = A a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M . However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M , except, importantly, in modular invariant theory, or in the study of commuting Jordan types.We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of ℓ ∈ m A and a certain local Hilbert function.In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.

show abstract

Section: Evidently We Havementioning

confidence: 84%

Artinian algebras and Jordan type

Iarrobino¹,

Marques²,

McDaniel³

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We consider this relative coinvariant ring in[MCIM, Example 3.1] but there we do not discuss the dual generator for I C in Q S .…”

mentioning

confidence: 99%

Artinian Gorenstein algebras that are free extensions over ${\sf k}[t]/(t^n)$, and Macaulay duality

Iarrobino¹,

Marques²,

McDaniel³

2018

Preprint

Self Cite

View full text Add to dashboard Cite

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras C over a base A with fibre B. The free extensions are deformations of the usual tensor product; when C is also Gorenstein, so are A and B, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator F for C as a homogeneous "polynomial" in T and the dual variables for B, and given the dual generator for B, we give sufficient conditions on F that ensure that C is a free extension of A = k[t]/(t n ) with fiber B. We give examples exploring

show abstract