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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 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:
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The Chow ring $\CH(X)=\bigoplus_{i\geq 0} \CH^i(X)$ of a variety $X$ is a graded commutative ring, where $\CH^i(X)$ denotes the group of classes of cycles of codimension $i$. In earlier publications, the notation $\mathrm{A}(X)$ is sometimes used instead of $\CH(X)$.
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For a morphism $f:X \to Y$ the inverse-image homomorphism $f^*:\CH(Y) \to \CH(X)$ is a homomorphism of rings, and for $f$ proper, the direct-image homomorphism $f_*: \CH(X)\to \CH(Y)$ is a homomorphism of $\CH(Y)$-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>
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$$f_*(f^*(y)\cdot x) = y\cdot f_*(x), \quad x \in CH(X), \quad y \in CH(Y)$$
  
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
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The Chow ring is the domain of values for the classical theory of Chern classes of vector bundles (cf. [[#References|[1]]]). More precisely, if $E$ is a locally free sheaf of rank $r$ over a variety $X$, if $\pi:P(E) \to X$ is its projectivization and if $\zeta \in \CH^1(P(E))$ is the class of the divisor corresponding to the invertible sheaf $\mathcal{O}_{P(E)}(1)$, then $\pi^*$ is injective and the Chow ring $\CH(P(E))$ may be identified with the quotient ring of the polynomial ring $\CH(X)[\zeta]$ 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>
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$$\zeta^r -c_1(E)\zeta^{r-1}+\cdots + (-1)^r c_r(E).$$
  
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" />.
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The coefficient $c_k(E)\in \CH^k(X)$ is called the $k$-th Chern class of the locally free sheaf $E$.
  
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.
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In the case of a variety over the field of complex numbers, there is a homomorphism $\CH(X) \to \mathrm{H}(X,\mathbb Z)$ 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]]]).
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If $X$ is a singular quasi-projective variety, then its Chow ring $\CH(X)$ is defined as the direct limit of rings $\CH(X)=\varinjlim \CH(Y)$ over all morphisms $f:X \to Y$, where $Y$ is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. [[#References|[3]]]).
  
 
====References====
 
====References====
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====Comments====
 
====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]]]
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For $X$ a Noetherian scheme (or ring), let $K_n(X)$ denote the $n$-th $K$-group of (the category of) finitely-generated projective modules over $X$; cf. [[Algebraic K-theory|Algebraic K-theory]]. Let $\mathcal{K}$ denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf $U \mapsto K_n(U)$ where $U$ runs through the open (affine) subschemes of $X$. 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>
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$$\CH^p(X) \simeq H^p(X, \mathcal{K}_p)
  
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" />.
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providing a link between the Chow groups of $X$ and the cohomology of $X$ with values in the $\mathcal{K}$-sheaves of $X$. Using results on the algebraic K-theory of fields, [[#References|[a2]]], this can be used to obtain results on Chow groups, in particular on $\CH^2$, [[#References|[a3]]].
  
 
Cf. [[Sheaf theory|Sheaf theory]] for the notions of sheafification, pre-sheaf, sheaf, and cohomology with values in a sheaf.
 
Cf. [[Sheaf theory|Sheaf theory]] for the notions of sheafification, pre-sheaf, sheaf, and cohomology with values in a sheaf.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Bloch, "Lectures on algebraic cycles" , Dept. Math. Duke Univ. (1980) {{MR|0558224}} {{ZBL|0436.14003}} </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 {{MR|}} {{ZBL|0525.18008}} {{ZBL|0525.18007}} </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) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}} </TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Bloch, "Lectures on algebraic cycles" , Dept. Math. Duke Univ. (1980) {{MR|0558224}} {{ZBL|0436.14003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.S. Merkur'ev, A.A. Suslin, "K-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 {{MR|}} {{ZBL|0525.18008}} {{ZBL|0525.18007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.-L. Colliot-Thélène, "Hilbert's theorem 90 for $K_2$ 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) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}} </TD></TR></table>

Revision as of 20:15, 23 April 2012


$ \newcommand{\CH}{\mathrm{CH}} $

The ring of rational equivalence classes of algebraic cycles (cf. 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).

The Chow ring $\CH(X)=\bigoplus_{i\geq 0} \CH^i(X)$ of a variety $X$ is a graded commutative ring, where $\CH^i(X)$ denotes the group of classes of cycles of codimension $i$. In earlier publications, the notation $\mathrm{A}(X)$ is sometimes used instead of $\CH(X)$.

For a morphism $f:X \to Y$ the inverse-image homomorphism $f^*:\CH(Y) \to \CH(X)$ is a homomorphism of rings, and for $f$ proper, the direct-image homomorphism $f_*: \CH(X)\to \CH(Y)$ is a homomorphism of $\CH(Y)$-modules. This means that there is a projection formula:

$$f_*(f^*(y)\cdot x) = y\cdot f_*(x), \quad x \in CH(X), \quad y \in CH(Y)$$

The Chow ring is the domain of values for the classical theory of Chern classes of vector bundles (cf. [1]). More precisely, if $E$ is a locally free sheaf of rank $r$ over a variety $X$, if $\pi:P(E) \to X$ is its projectivization and if $\zeta \in \CH^1(P(E))$ is the class of the divisor corresponding to the invertible sheaf $\mathcal{O}_{P(E)}(1)$, then $\pi^*$ is injective and the Chow ring $\CH(P(E))$ may be identified with the quotient ring of the polynomial ring $\CH(X)[\zeta]$ by the ideal generated by the polynomial

$$\zeta^r -c_1(E)\zeta^{r-1}+\cdots + (-1)^r c_r(E).$$

The coefficient $c_k(E)\in \CH^k(X)$ is called the $k$-th Chern class of the locally free sheaf $E$.

In the case of a variety over the field of complex numbers, there is a homomorphism $\CH(X) \to \mathrm{H}(X,\mathbb Z)$ into the singular cohomology ring that preserves the degree and commutes with the inverse-image and direct-image homomorphisms.

If $X$ is a singular quasi-projective variety, then its Chow ring $\CH(X)$ is defined as the direct limit of rings $\CH(X)=\varinjlim \CH(Y)$ over all morphisms $f:X \to Y$, where $Y$ is non-singular. One obtains a contravariant functor into the category of graded rings, satisfying the projection formula (cf. [3]).

References

[1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[2] "Anneaux de Chow et applications" , Sem. Chevalley (1958) Zbl 0098.13101
[3] W. Fulton, "Rational equivalence on singular varieties" Publ. Math. IHES , 45 (1975) pp. 147–167 MR0404257 Zbl 0332.14002


Comments

For $X$ a Noetherian scheme (or ring), let $K_n(X)$ denote the $n$-th $K$-group of (the category of) finitely-generated projective modules over $X$; cf. Algebraic K-theory. Let $\mathcal{K}$ denote the sheaf obtained by sheafifying (in the Zariski topology) the pre-sheaf $U \mapsto K_n(U)$ where $U$ runs through the open (affine) subschemes of $X$. One then has the Bloch formula [a1]

$$\CH^p(X) \simeq H^p(X, \mathcal{K}_p)

providing a link between the Chow groups of $X$ and the cohomology of $X$ with values in the $\mathcal{K}$-sheaves of $X$. Using results on the algebraic K-theory of fields, [a2], this can be used to obtain results on Chow groups, in particular on $\CH^2$, [a3].

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

References

[a1] S. Bloch, "Lectures on algebraic cycles" , Dept. Math. Duke Univ. (1980) MR0558224 Zbl 0436.14003
[a2] A.S. Merkur'ev, A.A. Suslin, "K-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 Zbl 0525.18008 Zbl 0525.18007
[a3] J.-L. Colliot-Thélène, "Hilbert's theorem 90 for $K_2$ with application to the Chow groups of rational surfaces" Inv. Math. , 71 (1983) pp. 1–20
[a4] W. Fulton, "Intersection theory" , Springer (1984) MR0735435 MR0732620 Zbl 0541.14005
How to Cite This Entry:
Chow ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chow_ring&oldid=23781
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article