How to Understand Functions on Algebraic Prevariety Product Spaces

What We're Solving:

You're exploring how functions behave on the product of algebraic prevarieties, specifically looking for an intuitive understanding of what the tensor product of structure sheaves actually means geometrically, rather than just working with the formal algebraic definition.

The Approach:

This is a beautiful question that gets to the heart of why algebraic geometry is so powerful! We want to build intuition by:

• Starting with familiar examples (like polynomial rings)
• Understanding what "functions on a product" should mean geometrically
• Seeing how the tensor product captures this intuition algebraically
• Connecting the abstract sheaf language to concrete function behavior

Step-by-Step Solution:

Step 1: Start with the concrete case Let's begin with affine varieties. If you have two affine varieties X = Spec(A) and Y = Spec(B), then their product should be X × Y = Spec(A ⊗ B).

Why this makes sense: A function on X × Y should depend on coordinates from both X and Y. If f ∈ A is a function on X and g ∈ B is a function on Y, then f ⊗ g should be the function on X × Y that takes a point (x,y) to f(x)·g(y).

Step 2: Think about what functions on products should do Geometrically, a regular function on X × Y should:

• Be determined by polynomial expressions in coordinates from both spaces
• Allow you to "separate variables" when possible
• Reduce to functions on X when you fix coordinates on Y (and vice versa)

Step 3: Connect to the sheaf picture For general prevarieties X and Y, the structure sheaf O_{X×Y} should satisfy:

• Global sections should be "polynomial-like" combinations of functions from X and Y
• Local behavior should respect the product structure
• The tensor product O_X ⊗ O_Y captures exactly this intuition

Step 4: Work through a specific example Consider X = A¹ (affine line) and Y = A¹. Then:

• Functions on X are polynomials in one variable: k[x]
• Functions on Y are polynomials in another variable: k[y]
• Functions on X × Y = A² should be polynomials in both: k[x,y] = k[x] ⊗ k[y]

Step 5: Understand the sheaf-theoretic meaning The tensor product of sheaves means that locally, functions on the product look like finite sums of products: Σ f_i ⊗ g_i where f_i are functions on X and g_i are functions on Y.

The Answer:

The intuitive interpretation is that functions on X × Y are built from functions that can be written as finite sums of "separable" pieces - each piece being a product of a function depending only on the X-coordinates times a function depending only on the Y-coordinates. The tensor product O_X ⊗ O_Y precisely encodes this geometric intuition algebraically, ensuring that:

• 1. You can multiply functions from different factors
• 2. The result behaves correctly under localization
• 3. Morphisms and geometric properties are preserved

Memory Tip:

Think of it like this: "Functions on a product space are recipes made from ingredients from each factor." Just as you might combine flour (from one pantry) with sugar (from another pantry) to bake something new, functions on X × Y combine functions from X with functions from Y via the tensor product structure!

The beauty is that this intuitive idea - "mix and match functions from different spaces" - translates perfectly into the precise language of sheaf tensor products.