Let \((R,{\mathfrak m})\) be a noetherian, local Cohen-Macaulay ring. A finitely generated module \(\omega \) is called dualizing if the following three conditions hold:
(i) The natural map \(\omega \to \text{End}_R(\omega )\) is bijective.
(ii) For every \(i\geq 1\) one has \(\text{Ext}^i_R(\omega ,\omega )=0\).
(iii) The injective dimension of \(\omega \) is finite.
A dualizing module exists precisely iff \(R\) is a homomorphic image of a local Gorenstein ring. If it exists it is unique up to isomorphism. \(\omega \) is called semidualizing (or spherical) if (i) and (ii) hold. Vasconcelos asked (1) if the set of isomorphism classes of such semidualizing modules is finite and (2) if it is true that this set has even cardinality when it contains more than one element.
In this paper it is shown that (1) has a positive answer if \(R\) is equicharacteristic and (2) has a positive answer if \(R\) is equicharacteristic and has a dualizing module.