Certain results of Bialynicki-Birula as well as other pieces of evidence suggest that an algebraic torus action on an affine space is always linear with respect to a suitably chosen coordinate system (cf. [l, 51). With eventual proof of this conjecture in mind, we examine in this note torus actions on a smooth affine variety over an algebraically closed ground field. It turns out that, in what we have termed unmixed cases (see 2.3), the variety in question is a vector bundle over the fixed point variety and the torus action is linear along the fibers (see Theorem 2.5 below). We have noticed that this fact is already known in essence to Bialynicki-Biruala [2; Th. 2.51. But our proof is elementary and seems simpler than his, resting upon the smoothness of fixed point schemes (Fogarty [3]) and a version of Nakayama’s Lemma for semigroup-graded rings and modules (see 1.2 and 1.4). It is clear that definiteness of a torus action as defined by Bialynicki-Birula (see [2; p. 482)) implies unmixedness, but the converse is true as well (see 1.6). Since unmixedness is a quite intrinsic condition, it seems worthwhile to point this out. As an immediate consequence of the above result and the Quillen-Suslin Theorem [9, 121, a smooth affine variety with an unmixed torus action is actually an affine space with a linear torus action, provided the fixed point variety is isomorphic to some affine space (possibly a single point). By making use of recent results of Fujita, Miyanishi and Sugie [4, 81, we show this last to be the case indeed for any unmixed torus action on an affine space with a fixed point variety of dimension 12 (see Theorem 3.4). Hyman Bass told one of the authors how a graded ring R =@,,ro R, becomes isomorphic to a symmetric algebra over Ro under a certain homological hypothesis (see 2.6). His remark was largely responsible for getting us started on the present investigation. His suggestions were also responsible for an improvement in our treat-