Algebraic geometry 38: The Zariski tangent space (replacement)

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the Zariski tangent space, and describes some other ways of viewing tangent spaces.

(This is a replacement for the original video, which had poor audio quality.)

[jeparlefrancais]

Huge thanks for the great work you are doing. Wanna give you a big hug.

[acmusic]

Thank you so much sir, I really enjoy and appreciate your content! Keep up the great work 👏

[hausdorffm]

10:18 Dual of m/m^2 = Hom(R, k[e]/e^2) over k=R/m….. I do not understand.

Intrinsic definition of tangent space without embedding to affine space

Definition. Tangent space at a point p of variety is the dual of the cotangent bundle, where cotangent space is quotient module m/m^2, where m is a maximal ideal of local ring at p. Note that local ring is the set of the form f/g, g is not zero at p.

For simplicity, we take p = 0 as a point of variety of zeros of (f1, f2, …) in A^n, and let m=(x1, …, xn) be a maximal ideal of the local ring of the point p=0. The intrinsic definition of cotangent space is
m/m^2 = (x1, .., xn)/(f1, f2, …)+(all monomials of degree >1)
= A vector space spanned by symbols {x1, .., , xn} quotient out by linear parts of (f1, f2, …)
I do not understand this formula.


Example. A circle x^2 + y^2 -2x = (x-1)^2 + y^2 -1 = 0 tangent to y axis at (0, 0).
Intrinsic cotangent space
= m/m^2 = (x, y)/( x^2 + y^2 -2x, x^2, xy, y^2, x^3, x^2y, ….)
= vector space spanned by symbols x, y with relation -2x = 0
= vector space spanned by symbol y.
= y-axis.
The result is reasonable.




11:00 Topic of Kaehler differential.
Let A be a commutative ring with identity and let B be an A-algebra.
Next, we show that the operator d: B -> W satisfies the Leibniz rule.
That is,
PROPOSTION.
For any elements r, s of B, d(rs) = sdr + rds.
PROOF.
Note that dr*ds = 0 in W = I/I^2, which means that
from which we get
On the other hand,
Substituting (1) to (2), we get
sdr + rds = 1 @(rs) – (rs) @1 in W,
which is the desired Leibniz rule.
Q.E.D.
Example.
Let V be a real smooth manifold, then A = smooth functions, and B = A, and W = cotangent vector fields, and d; B -> W; f ->df.
Tangent vector fields of V is Hom(W, R) over R.