TITLE: A name for "not quite saturated" graded modules
QUESTION [8 upvotes]: Let $M$ be a finitely generated graded module over a graded ring $R$. Let $\mathcal{F}$ be the corresponding coherent sheaf on $\operatorname{Proj} R$.  There is a natural map of graded $R$-modules
$$\phi \colon M \to \Gamma^*(\mathcal{F}) := \bigoplus_{n} \Gamma(\operatorname{Proj} R, \mathcal{F}(n)).$$
If I recall Ravi Vakil's notes correctly, $M$ is called saturated if $\phi$ is an isomorphism.

Is there a term (perhaps semi-saturated, or some such) for modules $M$ such that $\phi$ is injective?

This concept is appealing for several reasons.  For one thing, it is easier to test "semi-saturatedness" than saturatedness; e.g., unless I am mistaken, $\phi$ is automatically injective if $M$ admits any positive-degree homogeneous nonzerodivisor.  For another, at least if $R$ is a polynomial ring, $M = R/I$ is "semi-saturated" iff $I$ is a saturated ideal of $R$.  (Note that the definition of "saturated ideal" is different from the definition given above for "saturated module", and I do not think the two are equivalent for ideals.)

REPLY [3 votes]: As Hailong wrote, the injectivity means the vanishing of $H^0_m(M)$. But $\operatorname{depth}(m,M)$ is the least non-vanishing local cohomology, thus the map $M \to H^0_*(\tilde{M})$ is injective iff $M$ has depth $\ge$ 1.
If $M$ has finite projective dimension, e.g. if R is regular, then this happens iff $M$ has projective dimension at most depth $R-1$ by the Auslander-Buchsbaum formula.