---
tag: 9VFWPZEVGN
kind: concept
name: Algebraic cycle
msc: [14C25]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_cycle
same_as:
  wikidata: Q4723996
  wikipedia: Algebraic cycle
---

## Summary

An algebraic cycle on a variety $X$ is a finite formal sum, with integer coefficients, of irreducible closed subvarieties of $X$.
Cycles of codimension one are the [[Weil divisors|Z9DN23DAE2]].
Cycles modulo [[rational equivalence|K76N026JFK]] form the [[Chow groups|EBEF6PEK4N]], the basic objects of [[intersection theory|4HV4WTAX9P]].
The question of which cohomology classes come from cycles is the subject of the [[Hodge conjecture|XAEEDWV3PM]] and of the [[standard conjectures|D8GA2R3QT5]].

## Explanations

### L1 human

A cycle is a list of subvarieties, each with a whole number attached, such as "two copies of this curve minus one copy of that curve".
Cycles let one add and subtract subvarieties as if they were numbers.
For example, a line in the plane usually meets a cubic curve in the cycle $[P] + [Q] + [R]$ of three points.
If the line is tangent to the cubic at $P$, the intersection is $2[P] + [R]$.
The coefficients record multiplicities in this way.