Free extensions and Lefschetz properties, with an application to rings of relative coinvariants. (English) Zbl 1461.13024
Broadly speaking, for a graded Artinian algebra \(A\), “Lefschetz properties” (weak and strong) refer to the study of the extent that multiplication by powers \(L^d\) of a general linear form \(L\), from any component \([A]_i\) of \(A\) to \([A]_{i+d}\), behave in an expected way: being either injective or surjective, depending on the dimensions of the components. (“Weak Lefschetz Property” is the case \(d=1\).) One way of measuring this extent is by studying the “Jordan type” of \(A\). This arises by looking at the multiplication map \(\times \ell : A \rightarrow A\), which is a nilpotent linear transformation, and studying its Jordan canonical form, encoded in a partition of \(N = \dim_k(A)\). This partition is called the Jordan type. From the introduction: “The goals of this paper are, first, to survey what we know about the Lefschetz and Jordan type properties for free extensions, and to present new results, especially about strong Lefschetz Jordan type ... Then we apply these results to relative coinvariant rings,” namely the quotient of the invariant ring of a finite subgroup by the ideal of invariants of a larger finite group containing it.
Reviewer: Juan C. Migliore (Notre Dame)
MSC:
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14C05 | Parametrization (Chow and Hilbert schemes) |
| 14L35 | Classical groups (algebro-geometric aspects) |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Keywords:
Artinian algebra; coinvariant; free extension; Hilbert function; invariant; Jordan type; Lefschetz property; reflection group; tensor productReferences:
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