  We develop the Hadamard renormalization of the stress-energy tensor for a
massive scalar field theory defined on a general spacetime of arbitrary
dimension. Our formalism could be helpful in treating some aspects of the
quantum physics of extra spatial dimensions. More precisely, for spacetime
dimension up to six, we explicitly describe the Hadamard renormalization
procedure and for spacetime dimension from seven to eleven, we provide the
framework permitting the interested reader to perform this procedure explicitly
in a given spacetime. We complete our study (i) by considering the ambiguities
of the Hadamard renormalization of the stress-energy tensor and the
corresponding ambiguities for the trace anomaly, (ii) by providing the
expressions of the gravitational counterterms involved in the renormalization
process (iii) by discussing the connections between Hadamard renormalization
and renormalization in the effective action. All our results are expanded on
standard bases for Riemann polynomials constructed from group theoretical
considerations and thus given on irreducible forms.
