Algebra & Number Theory

When are permutation invariants Cohen–Macaulay over all fields?

Ben Blum-Smith and Sophie Marques

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We prove that the polynomial invariants of a permutation group are Cohen–Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The “if” direction of the argument uses Stanley–Reisner theory and a recent result of Christian Lange in orbifold theory. The “only if” direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen–Macaulayness.

Article information

Algebra Number Theory, Volume 12, Number 7 (2018), 1787-1821.

Received: 26 February 2018
Revised: 16 May 2018
Accepted: 17 June 2018
First available in Project Euclid: 9 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
Secondary: 05E40: Combinatorial aspects of commutative algebra

invariant theory modular invariant theory henselization Stanley–Reisner Cohen–Macaulay commutative ring finite group


Blum-Smith, Ben; Marques, Sophie. When are permutation invariants Cohen–Macaulay over all fields?. Algebra Number Theory 12 (2018), no. 7, 1787--1821. doi:10.2140/ant.2018.12.1787.

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