View source for Chow ring

The ring of rational equivalence classes of algebraic cycles (cf. [[Algebraic cycle|Algebraic cycle]]) on a non-singular quasi-projective algebraic variety. Multiplication in this ring is defined in terms of intersections of cycles (cf. [[Intersection theory|Intersection theory]]).

The Chow ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221601.png" /> of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221602.png" /> is a graded commutative ring, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221603.png" /> denotes the group of classes of cycles of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221604.png" />. For a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221605.png" /> the inverse-image homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221606.png" /> is a homomorphism of rings, and the direct-image homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221607.png" /> is (for proper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221608.png" />) a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c0221609.png" />-modules. This means that there is a projection formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216010.png" /></td> </tr></table>

The Chow ring is the domain of values for the theory of Chern classes of vector bundles (cf. [[#References|[1]]]). More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216011.png" /> is a locally trivial sheaf of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216012.png" /> over a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216013.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216014.png" /> is its projectivization, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216015.png" /> is the canonical projection, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216016.png" /> is the class of divisors corresponding to the invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216018.png" /> is an imbedding and the Chow ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216019.png" /> may be identified with the quotient ring of the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216020.png" /> by the ideal generated by the polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216021.png" /></td> </tr></table>

The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216022.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216024.png" />-th Chern class of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216025.png" />.

In the case of a variety over the field of complex numbers, there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216026.png" /> into the singular cohomology ring that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216027.png" /> is a singular quasi-projective variety, then its Chow ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216028.png" /> is defined as the direct limit of rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216029.png" /> over all morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216031.png" /> is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  "Anneaux de Chow et applications" , ''Sem. Chevalley''  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Fulton,   "Rational equivalence on singular varieties"  ''Publ. Math. IHES'' , '''45'''  (1975)  pp. 147–167</TD></TR></table>



====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216032.png" /> a Noetherian scheme (or ring), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216033.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216034.png" />-groups of (the category of) finitely-generated projective modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216035.png" />; cf. [[Algebraic K-theory|Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216036.png" />-theory]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216037.png" /> denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216038.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216039.png" /> runs through the open (affine) subschemes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216040.png" />. One then has the Bloch formula [[#References|[a1]]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216041.png" /></td> </tr></table>

providing a link between the Chow groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216042.png" /> and the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216043.png" /> with values in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216044.png" />-sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216045.png" />. Using results on the algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216046.png" />-theory of fields, [[#References|[a2]]], this can be used to obtain results on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216047.png" />, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216048.png" />, [[#References|[a3]]]. Another often used notation for the Chow group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216049.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216050.png" />.

Cf. [[Sheaf theory|Sheaf theory]] for the notions of sheafification, pre-sheaf, sheaf, and cohomology with values in a sheaf.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bloch,   "Lectures on algebraic cycles" , Dept. Math. Duke Univ.  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S. Merkur'ev,   A.A. Suslin,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216051.png" />-cohomology of Severi–Brauer varieties and norm residue homomorphism"  ''Math. USSR Izv.'' , '''21'''  (1983)  pp. 307–340  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''46''' :  5  (1982)  pp. 1011–1046</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-L. Colliot-Thélène,   "Hilbert's theorem 90 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022160/c02216052.png" /> with application to the Chow groups of rational surfaces"  ''Inv. Math.'' , '''71'''  (1983)  pp. 1–20</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Fulton,   "Intersection theory" , Springer  (1984)</TD></TR></table>