Motive (algebraic geometry)

In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In the formulation of Grothendieck for smooth projective varieties, a motive is a triple ${\displaystyle (X,p,m)}$, where ${\displaystyle X}$ is a smooth projective variety, ${\displaystyle p:X\vdash X}$ is an idempotent correspondence, and m an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from ${\displaystyle (X,p,m)}$ to ${\displaystyle (Y,q,n)}$ is given by a correspondence of degree ${\displaystyle n-m}$. A more object-focused approach is taken by Pierre Deligne in Le Groupe Fondamental de la Droite Projective Moins Trois Points. In that article, a motive is a "system of realisations" – that is, a tuple

${\displaystyle \left(M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p},\operatorname {comp} _{\mathrm {DR} ,B},\operatorname {comp} _{\mathbb {A} ^{f},B},\operatorname {comp} _{\operatorname {cris} p,\mathrm {DR} },W,F_{\infty },F,\phi ,\phi _{p}\right)}$

consisting of modules

${\displaystyle M_{B},M_{\mathrm {DR} },M_{\mathbb {A} ^{f}},M_{\operatorname {cris} ,p}}$

over the rings

${\displaystyle \mathbb {Q} ,\mathbb {Q} ,\mathbb {A} ^{f},\mathbb {Q} _{p},}$

respectively, various comparison isomorphisms

${\displaystyle \operatorname {comp} _{\mathrm {DR} ,B},\operatorname {comp} _{\mathbb {A} ^{f},B},\operatorname {comp} _{\operatorname {cris} p,\mathrm {DR} }}$

between the obvious base changes of these modules, filtrations ${\displaystyle W,F}$, a ${\displaystyle \operatorname {Gal} ({\overline {\mathbb {Q} }},\mathbb {Q} )}$-action ${\displaystyle \phi }$ on ${\displaystyle M_{\mathbb {A} ^{f}},}$ and a "Frobenius" automorphism ${\displaystyle \phi _{p}}$ of ${\displaystyle M_{\operatorname {cris} ,p}}$. This data is modeled on the cohomologies of a smooth projective ${\displaystyle \mathbb {Q} }$-variety and the structures and compatibilities they admit, and gives an idea about what kind of information is contained in a motive.