Let $R$ be a polynomial ring, $I$ a homogeneous ideal, and let $\operatorname{pd}(R/I)$ denote its projective dimension. Is there a characterization of homogeneous elements $a\in R\setminus I$ for which we have strict inequality: $$(\dagger) \ \ \operatorname{pd}(R/I)<\operatorname{pd}(R/(I+(a)))?$$
For example I think that if $a$ is not a zero divisor for $R/I$ then $(\dagger)$ holds by Corollary 4.3.14 in Weibel's book. I'm wondering if there are other examples of $a$'s where $(\dagger)$ holds? Or is it an if and only if, i.e. $(\dagger)$ holds if and only if $a$ is a nonzero divisor for $R/I$? Any examples or references or comments would be greatly appreciated.