nLab The Rising Sea



The “rising sea” is a metaphor due to Alexander Grothendieck (see the quote below), meaning to illuminate how the development of general abstract theory eventually brings with it effortless solutions to concrete particular problems, much like a hard nut may be cracked not immediately by sheer punctual force, but eventually by gently immersing it into a whole body of water.

Following this metaphor, “The Rising Sea” was chosen as the name of a website of Daniel Murfet with notes on algebraic geometry as developed in the school of Grothendieck.

Additionally, “The Rising Sea” is the name of a textbook on algebraic geometry by Ravi Vakil.


As a metaphor for his approach of problem solving via theory building, Alexander Grothendieck once referred to the how the rising sea slowly but eventually surrounds solid land masses. This metaphor became more or less proverbial as “The Rising Sea” analogy.

From Grothendieck 85, pp. 552-3-1:

I can illustrate the second approach with the same image of a nut to be opened.

The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better,and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

A different image came to me a few weeks ago.

The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it.. yet it finally surrounds the resistant substance.

For more see (McLarty 03).


Daniel Murfet has written a body of useful notes on fundamentals of algebraic geometry, commutative algebra, and category theory, which can be found at his website The Rising Sea.

Grothendieck’s metaphor is also discussed at

Other writings by Grothendieck:


category: reference

Last revised on November 19, 2022 at 18:47:25. See the history of this page for a list of all contributions to it.