  Normality of the Dirac operator is shown to be necessary for chiral
properties.
  From the global chiral Ward identity, which in the continuum limit gives the
index theorem, a sum rule results which constrains the spectrum. The
Ginsparg-Wilson relation is to be restricted to its simple form and is a member
of a set of spectral constraints. A family of alternative chiral
transformations is introduced. The one of L\"uscher is a special case which
transports only the anomaly term to the measure. An alternative transformation
would also be needed to correct Fujikawa's path-integral approach. From a
general function of the hermitean Wilson-Dirac operator the one of Neuberger
follows.
