  We investigated Relations Among Green Functions defined in an alternative
strategy for coping with the divergences, also called the Implicit
Regularization Method (IREG): the mathematical content (divergent and finite)
will remain intact until the calculations end. The divergent part will be
organized through standardized objects free of physical quantities. In
contrast, the finite part is projected in a class of well-behaved functions
that carry all the amplitudes' physical content. That relations arise in
fermionic amplitudes in even space-time dimensions, where anomalous tensors
connect to finite amplitudes as in the bubbles and triangles in two and four
dimensions. Those tensors depend on surface terms, whose non-zero values arise
from finite amplitudes as requirements of consistency with the linearity of
integration and uniqueness. Maintaining these terms implies breaking
momentum-space homogeneity and, in a later step, the Ward identities.
Meanwhile, eliminating them allows more than one mathematical expression for
the same amplitude. That is a consequence of choices related to the involved
Dirac traces. Independently of divergences, it is impossible to satisfy all
symmetry implications by simultaneously requiring vanishing surface terms and
linearity. Then we approach the 1-loop level fermionic correction for the
propagation of the graviton in a space-time D=1+1 through the action of a Weyl
fermion in curved space-time. In this context, gravitational anomalies arise,
and the amplitudes investigated have the highest degree of divergence
quadratic. That imposes a substantial algebraic effort; however, the
conclusions are in agreement with the non-gravitational amplitudes. At the end
of the calculations, we show how it is possible to fix the value of the
divergent part through the relations imposed for amplitudes.
