Weak and global dimension for a field and the integers

Under The Axiom of Choice, Every vector space has a basis supplies a basis for any $k$-module $V$, identifying it with the free module on that basis. By Free modules are projective, with the exact choice boundary, $V$ is projective: this second use of AC chooses preimages of basis values in each lifting problem. It is flat by Projective left and right modules are flat over an arbitrary ring. Thus $0\to V\stackrel{\mathrm{id}}{\to }V\to 0$ is both a length-zero projective and a length-zero flat resolution. For a supplied finite basis, the same free-projectivity theorem uses only finite choice, so these conclusions hold without AC.

Under AC, The integers have weak and global dimension one proves all four integer dimensions equal one, using actual length-one resolutions and the nonflat module $\mathbb{Z}\mathrm{/}2\mathbb{Z}$. No independent undeclared Tor or Ext calculation is needed here.

The definitions Projective dimension of an object and The flat dimension of a module assign dimension zero to the modules in step 1.1. Their suprema, as in Left and right global dimension of a ring and Left and right weak global dimension, are therefore zero. Because $k$ is commutative, the right-module assertion is the same statement with the scalar action written on the other side. Together with step 1.2 this gives all the claimed values.