If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.

Nondegenerate here means that the map is injective on each factor, ie, if the image of $a\otimes b$ is zero then either $a$ or $b$ is zero.

This result is known (at least in the Algebraic Geometry community) as the "Hopf lemma" (see for instance this question). Arbarello-Cornalba-Griffits-Harris actually contextualize the statement (for $\mathbb C$) by saying "In fact, it is a remarkable theorem of H. Hopf that [theorem]. This result, which is false over $\mathbb R$, was one of the earliest applications of topology to algebra." (reference given by Yusuf Mustopa in the cited MO question).

After some unsuccessful searches, one encounters a few works of Hopf on "compositions of quadratic forms", notably

The focus is on maps of real vector spaces, and the motivation seems to be the classification of division algebras over $\mathbb R$. I haven't been able to find the result for $\mathbb C$ there. On the other hand, Daniel Shapiro in the book

  • D. B. Shapiro, Compositions of Quadratic Forms, W. de Gruyter Verlag, 2000.

(downloadable from the author's webpage) attributes the result for $\mathbb C$ to Larry Smith, and more precisely to the paper

Shapiro also includes a proof for an arbitrary algebraically closed field (Prop 14.25). Smith starts his proof (for $\mathbb C$) by saying "The following argument is inspired by the paper [9] of Hopf", ie, the 1940/41 paper.

So, did Hopf ever state the "lemma" above? If not, is it generally accepted that Smith's contribution was a minor modification, so justifying an attribution to Hopf? Or is there a reference older than Smith? The latter seems unlikely, as Shapiro's book seems very well documented. But at the same time, it is a bit strange that ACGH say "it was one of the earliest applications of topology to algebra" of a result proved in 1978.

  • 1
    $\begingroup$ In the algebraic geometry context, it is Lemma 1.5.1 in Saint-Donat's paper On Petri's Analysis of the Linear System of Quadrics through a Canonical Curve (Math. Ann. 206, 1973). It is stated as "an easy lemma" (without proof). $\endgroup$
    – abx
    Feb 4, 2014 at 15:50
  • $\begingroup$ This is really relevant, and seems to support the idea that, after Hopf's work, the result became known (even for an arbitrary algebraically closed field, not C!) But Saint-Donat does not even mention Hopf or anybody else... $\endgroup$
    – quim
    Feb 4, 2014 at 17:44
  • $\begingroup$ What's a nondegenerate linear map? With the meaning of nondegenerate that I'm familiar with (the induced map $A \to B^{\ast} \otimes C$ being an isomorphism) this seems clearly false, and I'm not aware of any other meaning in this context. $\endgroup$ Feb 4, 2014 at 22:03
  • $\begingroup$ @QiaochuYuan: added clarification in the body of the question. $\endgroup$
    – quim
    Feb 5, 2014 at 0:19


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