  Let $G$ be an almost linear Nash group, namely, a Nash group which admits a
Nash homomorphism with finite kernel to some $\GL_k(\mathbb R)$. A homology
theory (the Schwartz homology) is established for the category of smooth \Fre
representations of $G$ of moderate growth. Frobenius reciprocity and Shapiro's
lemma are proved in this category. As an application, we give a criterion for
automatic extensions of Schwartz homologies of Schwartz sections of a tempered
$G$-vector bundle.
