EGA Daily: Introduction — Forget What You Know
A page-a-day journey through Grothendieck’s Éléments de Géométrie Algébrique, with Claude.
The Dedication
À Oscar Zariski et André Weil.
The two mathematicians who shaped algebraic geometry before Grothendieck. He’s dedicating to them the work that will supersede their foundations.
Forget What You Know
The first paragraph makes a striking claim: this treatise presupposes no knowledge of algebraic geometry, and in fact prior knowledge might actually hurt you—specifically the “birational point of view” that dominated classical approaches.
What he’s saying is: put away the equations and tricks and calculations you learned in school. Let’s build this up from nothing. And hopefully what you’ll end up with is a much richer understanding, as these fine elements fill in all the nooks and crannies that we were missing when we were just focused on a few things we knew how to solve well.
The Prerequisites
You don’t need algebraic geometry, but you do need:
Commutative algebra — basically factorizing polynomials. This makes sense because the beautiful things in nature have integral models, and then you combine them with operations like multiplication for intersections, composition, convolutions.
Homological algebra — really this is basic topology. What can you smoothly deform into what? What’s the topological essence of a situation?
Sheaf theory — patching together local information. Like if different countries each have a map of their own territory plus a bit of the neighbors—can you patch all those local maps into a single global map? Also: sheaves of functions, where you start with polynomials and as you shrink the domain you allow poles away from it, and in the limit you get germs at a point.
Functorial language — categories. How things relate to each other matters more than what they “are.”
Nilpotents Are In
Grothendieck says that accepting nilpotent elements in local rings is essential, and this “necessarily relegates the notion of rational map to second place, in favor of regular maps or morphisms.”
What do nilpotents see? Tangent vectors and infinitesimal information.
The classic example: Spec(k[ε]/(ε²)). That’s a “point with a tangent direction.” The ε is nilpotent (ε² = 0), and a map from this gadget into a scheme X is the same thing as a tangent vector on X.
Or: two lines meeting transversally intersect in a point. But two lines tangent to each other? The intersection scheme is Spec(k[x]/(x²)) - a point with nilpotent fuzz. The nilpotent remembers the tangency, the multiplicity. If you throw away nilpotents, you lose that.
A “nilpotent thickening” is the same underlying space but with extra nilpotent fuzz that remembers infinitesimal directions. The x-axis defined by y = 0 is a line. The double x-axis defined by y² = 0 is topologically the same line, but it carries the information “which direction was the other line coming from?”
In the old way, singularities were problems - failures of the equations to be nice. You either avoid them (birational) or fix them (resolution). In the new way, a singular point is a scheme. It has a local ring, maybe nilpotents, a tangent cone, a formal neighborhood, deformation theory. It’s an object in its own right.
The Four-Step Method
Grothendieck sketches his technique for attacking problems:
- Localize - reduce to an affine neighborhood, replace by the local ring at a point
- Artinian case - study the problem when the ring is Artinian (maximal ideal is nilpotent)
- Complete - pass to formal power series via completion
- Algebraize - go back from formal to algebraic
This is like looking at Taylor expansions - Artinian rings are truncated Taylor series (finitely many terms before everything dies, like k[x]/(xⁿ)), completion gives you the full formal power series, and then you ask if it’s actually algebraic.
In differential geometry these are called jet bundles. The 1-jet is value plus first derivative. The 2-jet adds second derivative. The k-jet is the Taylor polynomial up to order k. So Spec(k[ε]/(ε^(n+1))) is the n-jet space.
Step 4 is where the real work happens. Roughly: if you have a proper morphism (like projective varieties), formal = algebraic. But non-proper situations can have formal deformations that don’t algebraize. Steps 1-3 are relatively systematic; step 4 is where you need theorems.
Birational vs Scheme-Theoretic
What’s the difference from old-style birational geometry?
Birational shrinking: throw away bad points, look at what’s left. Go to smaller open sets. “Ignore the singularity, work generically.”
Formal/infinitesimal: zoom into a single point, add nilpotent fuzz. Go to infinitesimal neighborhoods. “Stare at the singularity under a microscope.”
These go opposite directions. Scheme theory wants both ends: the global projective structure and the infinitesimal local structure. Birational geometry lived in the comfortable generic middle.
The old geometers had families - pencils of curves, linear systems. But ad hoc. What they couldn’t do systematically: What are all first-order deformations of this object? Is this moduli space smooth at this point? The machinery for interrogating families infinitesimally was missing.
The Psychological Difficulty
Near the end of the introduction, Grothendieck warns that readers will have trouble getting used to the language of schemes. The difficulty is that you have to transport familiar notions—Cartesian products, group laws, fibers, principal bundles—to a category quite different from sets.
But then he says: this effort of abstraction is perhaps quite minimal compared to what our mathematical fathers did when they got used to set theory.
What’s the quintessential scheme that’s not a variety? It’s $\operatorname{Spec}(\mathbb{Z})$. It has the spectrum of primes, and from that you get a family—the different finite approximations, arithmetic going off in a different dimension. Then completions, algebraic closures, p-adic stuff. Or moduli spaces: continuous families of varieties.
When set theory came in, mathematicians had to accept that everything—numbers, functions, spaces—could be reduced to sets and membership. A function isn’t really a set of ordered pairs, but you can treat it that way and reason uniformly.
Similarly with schemes: you have to stop asking “what is this thing made of” and start asking “how does it behave categorically.” A group scheme satisfies group axioms in the category of schemes. A fiber product satisfies the universal property. That’s what matters.
Next time: Chapter 0 begins—rings of fractions.