---
tag: K1AZ4YF2RE
kind: concept
name: Motive
msc: [14C15, 14F42]
sources:
  - url: https://en.wikipedia.org/wiki/Motive_(algebraic_geometry)
same_as:
  wikidata: Q840594
  wikipedia: Motive (algebraic geometry)
---

## Summary

A motive is an object of a category built from algebraic varieties and [[algebraic cycles|9VFWPZEVGN]] on their products.
Grothendieck proposed this category as a universal cohomology theory for varieties.
Every good cohomology theory, such as [[ℓ-adic cohomology|E1DVHN06CR]], [[algebraic de Rham cohomology|13F5JZPDD0]] or singular cohomology, should factor through the category of motives.
Pure motives over a field are well defined, with morphisms given by correspondences modulo an equivalence relation such as [[rational equivalence|K76N026JFK]].
Many of their expected properties depend on the [[standard conjectures|D8GA2R3QT5]].

## Explanations

### L1 human

Different cohomology theories of the same variety give groups that look alike, with the same dimensions and similar structures.
Grothendieck asked for a single object behind all of them, the motive of the variety, of which each cohomology is a "realization".
Motives can be split into pieces even when the variety cannot.
For example, the motive of the projective line is the sum of the motive of a point and the Lefschetz motive $\mathbb{L}$.
The piece $\mathbb{L}$ accounts for the cohomology in degree $2$.