Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

It is usual in algebraic geometry to represent morphisms by vertical arrows pointing downwards, like that :

$$\begin{matrix} X \\\\ \downarrow \\\\ S \end{matrix}$$

I suppose this stemmed from Grothendieck's amazingly original idea that a morphism of schemes should always be considered as some sort of fibre bundle, even in cases apparently very distant from the bundles considered in topology.
Many geometers have since adopted these vertical arrows, which they find suggestive and psychologically helpful.
My question is simply whether anybody had drawn maps vertically before Grothendieck a) in topology b) in algebraic geometry.

While on the subject I can't resist telling an anecdote I heard, according to which in some seminar led by Grothendieck, a joker ( Serre?) always drew the vertical morphism above on the blackboard just before Grothendieck arrived. So an auxiliary question might be: c) is this true?

share|cite|improve this question
Illusie says he thinks Grothendieck introduced the vertical arrow: –  Moosbrugger Oct 31 '11 at 13:13
This ia a great opportunity to write to Serre and suggest he join MO :) –  Mariano Suárez-Alvarez Oct 31 '11 at 13:14
Vertical arrows already appear in Cartan-Eilenberg's book "Homological Algebra" (1956). I think they come naturally when one tries to prove some kind of lifting theorem –  Francesco Polizzi Oct 31 '11 at 13:20
Dear users, I find your comments very interesting. Will you allow me to put a little friendly pressure on you in order that you transform them into genuine answers? –  Georges Elencwajg Oct 31 '11 at 13:38
@Ryan Budney. Grothendieck drew vertical arrows on blackboards because he felt it emphasised the fiberish nature of all scheme morphisms. But in the written accounts the arrows were typeset horizontally, precisely because of typographically difficulties.So I happen to disagree with you. What saddens me, however, is the deliberately insulting comparison of Grothendieck's credo with talk "about what kind of footwear X wears". MathOverflow is a place where we can benefit from the expertise of competent mathematicians, like you, in gentle, friendly exchanges and I hope it remains so. –  Georges Elencwajg Nov 1 '11 at 9:16

3 Answers 3

up vote 7 down vote accepted

In Hasse's school of number theory, it was quite common to represent an extension of fields by writing the bigger field above the smaller one and drawing a line segment between them, without an arrowhead. This is one possible source of Grothendieck's notation.

share|cite|improve this answer
Dear Chandan, I had completely forgotten this, even though I have often used this graphic convention in my courses, and even mentioned it stemmed from Hasse! Thanks for this very relevant answer: we'll see if anybody finds an older ancestor to vertical arrows... –  Georges Elencwajg Oct 31 '11 at 14:18

Vertical arrows appear everywhere in the book by Cartan and Eilenberg "Homological Algebra" (1956).

For instance, at page 5 one finds the statement of the $5$-lemma, with the usual commutative diagram, and a page 6 there is the definition of projective module, again with a diagram containing a vertical arrow.

I do not know where vertical arrows originated from. My guess is that they arise naturally in Topology when one tries to prove some kind of lifting theorem.

share|cite|improve this answer
Dear Francesco, in a previous draft of my question I had added a little cautionary sentence that commutative diagrams should be excluded because the vertical arrows were there for other reasons and that horizontal and vertical arrows were distributed according to typographical rather than geometric reasons. However I deleted that sentence because I thought this was perhaps mere prejudice on my part. Anyway, I'm happy to read about your take on this, and would encourage other users to comment. –  Georges Elencwajg Oct 31 '11 at 14:14
The picture of a vertical arrow is already included in the term "lifting". If you keep drawing arrows as $X \to S$, then you pull something back. –  Martin Brandenburg Oct 31 '11 at 15:48
Well, what I meant to say is the following. Assume that I want to prove a statement like this: I have two maps $f \colon A \to B$ and $g \colon C \to B$, and then I want to find a map $h \colon A \to C$ such that $g \circ h =f$. Then, after some attempt, I discover that the best thing to do in order to visualize the statement is writing the arrow of $f$ horizontally and the arrow of $g$ vertically, so that the arrow of $h$ "lifts" the one of $f$. –  Francesco Polizzi Oct 31 '11 at 17:14
In other words, MAYBE people trying to prove these kind of results came up with the "vertical arrow" visualization, and then called them "lifting theorems". But it is just a guess –  Francesco Polizzi Oct 31 '11 at 17:18

Steenrod, Eilenberg. Foundations of Algebraic Topology, p.26 in Russian ed.

share|cite|improve this answer
Do you mind telling us what's in there? :) –  Martin Brandenburg Oct 31 '11 at 15:46
Sorry, now I found the English original… There is a diagram with vertical arrows on the page 9. –  Boris Novikov Oct 31 '11 at 16:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.