---
title: Schemes
description: Schemes generalize algebraic varieties by gluing affine schemes together, allowing the study of geometric spaces over arbitrary commutative rings. Learners will understand structure sheaves, morphisms of schemes, and fiber products, providing the foundation for modern arithmetic geometry.
category: mathematics
subcategory: algebraic-geometry
difficulty: beginner, intermediate, advanced
url: /subject/schemes
---

# Schemes

Schemes generalize algebraic varieties by gluing affine schemes together, allowing the study of geometric spaces over arbitrary commutative rings. Learners will understand structure sheaves, morphisms of schemes, and fiber products, providing the foundation for modern arithmetic geometry.

## Available Resources

2 Courses • 3 Websites

## Courses

### 1. Algebraic Geometry

Learn schemes with Prof. Borcherds' Algebraic Geometry course! Explore this foundational topic via lectures on YouTube.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PL8yHsr3EFj53j51FG6wCbQKjBgpjKa5PX

**Tags:** algebraic-geometry, varieties, commutative-algebra, graduate-mathematics

### 2. Math 216: Foundations of Algebraic Geometry (The Rising Sea)

**Author:** Ravi Vakil

Ravi Vakil's Stanford course site hosting free drafts of The Rising Sea, a graduate text on Grothendieck-style algebraic geometry: sheaves, schemes, morphisms, dimension, smoothness, quasicoherent sheaves and cohomology, with extensive exercises. Readers learn to work fluently with schemes and the modern foundations of the field.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://math216.wordpress.com/

**Tags:** algebraic-geometry, schemes, sheaves, cohomology, commutative-algebra

## Websites

### 1. Stacks Project

**Author:** The Stacks Project Authors

A collaborative, open-source reference on algebraic geometry run from Columbia University, with thousands of pages building from commutative algebra to schemes and algebraic stacks. Every result has a stable tag, so readers can look up precise statements and complete proofs.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://stacks.math.columbia.edu/

**Tags:** algebraic-geometry, schemes, commutative-algebra, algebraic-stacks

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 3. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

---

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