Poincaré duality in modular coinvariant rings (Q2827361)

The present article is devoted to the modular representations of the cyclic group \(C_p\) of prime order \(p\). Recall that if \(V\) is a finite dimensional representation of a finite group \(G\) over a field \(k\), then the representation is called modular if the characteristic of \(k\) divides the order of \(G\). The induced action on \(V^*\) extends naturally to \(k[V] := S(V^*)\) by graded algebra automorphisms. Let \(k[V]^G := \{f \in k[V ]\mid g(f) = f, \forall g\in G\}\) denote the subalgebra of invariant polynomials in \(k[V ]\). A classical problem is to characterize the representations whose invariant rings are polynomial. In this paper the authors study the ring of coinvariants which is the quotient ring \(k[V]_G := k[V ]/I\), where \(I\) is the Hilbert ideal of \(V\) in \(k[V ]\) generated by invariants of positive degree. Since \(G\) is finite, \(k[V ]_G\) is a finite dimensional vector space. Coinvariants provide information about the invariants and often play an important role in the construction of the invariant ring. In the main result, Theorem 1, is proven a necessary and sufficient condition for the ring of coinvariants \(k[V ]_{C_p}\) to be Poincaré duality algebra. Moreover, Poincaré duality coinvariant rings that are listed in the theorem are actually complete intersections. For other representations is shown that the dimension of the top degree of the coinvariants grows at least linearly with respect to the number of summands of dimension at least four in the representation.