TITLE: Weak Global Dimension and Global Dimension
QUESTION [10 upvotes]: Let $R$ be a commutative unit ring (not necessarily Noetherian). Is there an example such that weak global dimension of $R$ is finite but the global dimension of $R$ is infinite? Can we find such an example if $R$ is a local ring?

REPLY [2 votes]: The answer of rschweib refers to a result of Osofsky that gives a ring with infinite global dimension but finite weak global dimension.
Another result of Osofsky gives a local example.
In Corollary 2 of
Osofsky, B. L., Global dimension of valuation rings, Trans. Am. Math. Soc. 127, 136-149 (1967). ZBL0145.27602
she shows that there is a valuation domain $R$ with infinite global dimension.
But valuation domains are precisely the commutative local rings of weak global dimension at most one. See, for example, Corollary 4.2.6 of
Glaz, Sarah, Commutative coherent rings, Lecture Notes in Mathematics. 1371. Berlin etc.: Springer-Verlag. xi, 347 p. (1989). ZBL0745.13004.