Reference Request: tangent space defined via cotangent space

Question

Wikipedia gives several ways of defining the tangent space, the first two of which I have most popularly seen in books. However, the third one---where $T_pM = (I/I^2)^*$ with $I=\mathcal C^\infty(M)$ functions which vanish on $p$---I have yet to see as the choice definition for the tangent space (rather than an equivalence).

Are there any good references which begin with this definition and proceed with results? I had a look through my references and could not find any. I understand from the link that this is useful for algebraic geometry so I will leave it open as to exactly what field it is from. I am moreso interested in how they develop it from this definition. Thanks!

I have found in another answer on stackexchange these notes which have been insightful but didn't provide a reference like the one I am looking for (Warner does it via derivations).

EDIT: I have answered with a few algebraic geometry books, though I would certainly appreciate more references, especially interested to see if there are any differential geometry books which decide to develop from this definition.

Answer 1

As mentioned in my question and comments, this definition is more common in algebraic geometry so some references that I found in my library include:

• Joe Harris's book, Algebraic Geometry: A first course, pg. 174-5;
• Ravi Vakil's notes, Foundations of Algebraic Geometry (Dec. 29, 2015 draft), pg. 101;
• Qing Liu's book, Algebraic Geometry and Arithmetic Curves, pg. 126.