EGA Daily #1: Rings of Fractions (§1.0-1.7)

December 01, 2025

Day 1: Setting Up The Machinery

Okay, we’re diving into EGA proper now! After yesterday’s introduction, we hit Chapter 0, §1: Anneaux de Fractions (Rings of Fractions). This is Grothendieck laying down all the conventions and machinery before we even get to schemes.

The Ground Rules (1.0)

First thing: Grothendieck is extremely careful about conventions. All rings have unit elements, all ring homomorphisms preserve the unit, subrings contain the ambie When they say “anneau” without qualification, they mean commutative ring.

There’s also this interesting notion of “di-homomorphisms” - thinking of (ring, module) pairs as objects in their own category. The categorical perspective is already showing through.

Nilradicals: Prime Factors of Non-Reduced Schemes (1.1)

Here’s where it clicked for me: the radical is like taking prime factors of non-reduced schemes.

For an ideal a, the radical r(a) consists of all x where \(x^n \in \mathfrak{a}\) for some \(n\). So for the ideal \((x^2)\), we get \(r((x^2)) = (x)\).

Geometrically:

\(\text{Spec}(k[x]/(x^2))\) is a “double point” - it has nilpotent fuzz, like \(dx^2\)
\(\text{Spec}(k[x]/(x))\) is just a reduced point
Taking the radical removes the multiplicity

The key fact: \(r(\mathfrak{a})\) is exactly the intersection of all prime ideals containing \(\mathfrak{a}\). So the radical really is the “squarefree part.”

The nilradical \(\sqrt{(0)}\) removes ALL nilpotents, giving you the reduced ring \(A/\sqrt{(0)}\). This becomes \(X_{\text{red}}\) for schemes - the honest geometry underlying any scheme when you ignore infinitesimal thickening.

Localization: Zooming In (1.2-1.6)

A multiplicative subset \(S \subset A\) is closed under multiplication and contains 1. The two key examples:

\(S_f = \{1, f, f^2, \ldots\}\) gives you \(A_f\) - geometrically this is “zoom in where \(f \neq 0\)”
\(S = A - \mathfrak{p}\) for prime \(\mathfrak{p}\) gives you \(A_\mathfrak{p}\) - the local ring at \(\mathfrak{p}\)

The direct/inverse limit machinery in §1.4-1.6 is setting up something crucial: the connection between artinian local rings and complete local rings.

The tower:

A/m     (just the point)
A/m²    (point + tangent data)
A/m³    (point + tangent + quadratic)
A/m⁴    (point + tangent + quadratic + cubic)
...

The inverse limit gives you formal power series - the completion \(\hat{A}\).

Jets and Infinitesimals

This connects to thinking about jets geometrically. I was trying to understand: can I think of the second-order jet as a quadratic surface?

YES! For a point in \(\mathbb{A}^3\):

1-jet = tangent plane
2-jet = tangent plane + osculating paraboloid (best quadratic approximation)
3-jet = + cubic corrections

Each quotient by \((x,y,z)^{n+1}\) gives you polynomial approximation up to degree \(n\).

Artinian local = finite thickening (truncated Taylor series up to degree \(n\)) Complete local = full Taylor series (all orders)

When you’re feeling an algebraic curve:

1-jet = which way is it pointing? (velocity)
2-jet = how is it bending? (curvature)
3-jet = how is the bending changing? (jerk)

Each jet level adds the next polynomial approximation surface.

Support of Modules (1.7)

I struggled with this until I understood: what modules are we even talking about?

For an A-module M, the support is:

\[\text{Supp}(M) = \{\mathfrak{p} \in \text{Spec}(A) : M_\mathfrak{p} \neq 0\}\]

Key examples:

M = A itself: Supp(A) = all of Spec(A)
M = k[x,y]/(x): This is the structure sheaf of the line x = 0
- \(\text{Supp}(M) = V(x)\) = the y-axis
M = k[x,y]/(x,y): The skyscraper sheaf at origin
- \(\text{Supp}(M) = \{(x,y)\}\) = just the origin
M = k[x,y]/(x²,y): A fat point at origin
- \(\text{Supp}(M)\) = origin with nilpotent thickening in x-direction

For finitely generated M, \(\text{Supp}(M) = V(\text{Ann}(M))\) - exactly the closed subscheme defined by the annihilator!

Spec vs Supp

This was confusing at first: how is support different from Spec?

Spec(A) = the whole space (all prime ideals)

Supp(M) = a subset where module M “lives”

Example: In \(A = k[x,y]\):

\(\text{Spec}(A)\) = the whole affine plane \(\mathbb{A}^2\)
For \(M = k[x,y]/(x)\), \(\text{Supp}(M)\) = just the y-axis

The philosophy:

Spec gives you the stage
Supp tells you where the actors are

Spec(A) is fixed - it’s your geometric space. Supp(M) varies with M - it’s where M lives as a sheaf on that space.

That’s §1! We now have: localization, radicals, and supports. The foundation for turning algebra into geometry.

Tomorrow: §2 on irreducible spaces and Noetherian spaces.

This is part of my EGA Daily project where I’m working through Grothendieck’s Éléments de Géométrie Algébrique page by page.