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Well known theorem due to Stallings (finished by Swan) characterises free groups as those with $cd_{\mathbb Z} \leq 1$. We can treat it as a model case and try to extrapolate somehow to other categories and homological (feel free to interprete this adjective ad lib) functors — can we characterise free objects as "1-dimensional" ones? Here are some examples with different outcomes.

  • groups, homological dimension — yes for finitely presented (by duality), false for arbitrary ($\mathbb Q$), open for finitely generated
  • Lie algebras, Chevalley cohomology — yes for 2-generated (Feldman, 1983), no for char $\geq$ 3 (Mikhalev, 90s), open otherwise
  • restricted Lie algebras, Chevalley cohomology — sort of, the only example is 1-dimensional vector space (Zusmanovich, 2010s)
  • restricted Lie algebras, restricted cohomology ($HH^*$ of univ. p-enveloping) — 0-dimensional are finite dimensional tori (exercise in Weibel's book), conjectured by Zusmanovich that any 1-dimensional has solvable series with 1 free factor and arbitrary number of tori
  • associative algebras, Hochschild cohomolology — measures different property (formal smoothness), same story but somewhat different details for commutative case

Finally, a few questions.

  1. Are there some other interesting categories with "Stallings" property? (I'm particularly interested in the case of (pre)crossed modules with something like triple cohomology — which is not very good for intended purpose, but everything goes.)
  2. Can we fit this zoo in a unified framework?
  3. Back to beginning — every proof of Stallings' thm I'm avare of is geometric or combinatorial in nature and usually relies on Grushko rank theorem which is again an essentially combinatorial statement. Is there a purely homological one?
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    $\begingroup$ Re the last question: Gromov gave a non-combinatorial proof (details checked by Kapovich arxiv.org/pdf/0707.4231.pdf), but it uses harmonic functions ... $\endgroup$
    – David Snyder
    Commented Apr 2, 2017 at 4:08
  • $\begingroup$ Novikov showed a cancellative monoid of cohomological dimension one embeds in a free group. In general cohomological dimension one says nothing. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 2, 2017 at 11:16

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A finite dimensional algebra has global dimension one if and only if it is Morita equivalent to a path algebra of an acyclic quiver by Gabriel's theorem. A path algebra is just the algebra of a free category on the quiver.

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  • $\begingroup$ Nice! But free category on a quiver is very far from being as free as a free group, because Cat and Quiv are very similar, unlike Grp and Set. In my opinion, this result is closer to nonexistence of 1-dimensional p-Lie algebras than to "Stallings-y" ones — in a sense, gd is too restrictive. $\endgroup$
    – Denis T
    Commented Apr 2, 2017 at 17:21
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This is more of a comment than an answer, but the question is closely related to this question. Particularly relevant to your question is John Klein's answer, and even more specifically, the second part, where he discusses Daniel Cohen's Lecture notes (with pointers to more algebraic approaches).

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