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''of an additive category''
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An Abelian group that is assigned to an additive category by a universal additive mapping property. More exactly, let $C$ be a small [[Additive category|additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A mapping $\phi: \mathrm{Ob}(C) \to G$ is said to be additive if for any exact sequence $0 \to L \to M \to N \to 0$ in $C$, the relation $\phi(M)=\phi(L)+\phi(N)$ is valid. There exists a group $K(C)$, called the Grothendieck group of $C$, and an additive mapping $k:\mathrm{Ob}(C)\to K(C)$, known as the universal mapping, such that for any additive mapping $\mathrm{Ob}(C) \to G$, there exists a unique homomorphism $\xi: K(C) \to G$ that satisfies the condition $\phi=\xi \circ k$.
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The ''Grothendieck group of an additive category''
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is an Abelian group that is assigned to an additive category by a universal additive mapping property. More exactly, let $C$ be a small [[Additive category|additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A mapping $\phi: \mathrm{Ob}(C) \to G$ is said to be additive if for any exact sequence $0 \to L \to M \to N \to 0$ in $C$, the relation $\phi(M)=\phi(L)+\phi(N)$ is valid. There exists a group $K(C)$, called the Grothendieck group of $C$, and an additive mapping $k:\mathrm{Ob}(C)\to K(C)$, known as the universal mapping, such that for any additive mapping $\mathrm{Ob}(C) \to G$, there exists a unique homomorphism $\xi: K(C) \to G$ that satisfies the condition $\phi=\xi \circ k$.
  
This construction was first studied by A. Grothendieck for the categories of coherent and locally free sheaves on schemes in proving the Riemann–Roch theorem. See [[K-functor|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517013.png" />-functor]] in algebraic geometry. The group $K(C)$ is uniquely defined (up to isomorphism) and can be given by generators — to each object $L \in C$, there corresponds a generator $[L]$ — and by the relations $[L]-[N]-[M]=0$ for each exact sequence $0 \to L \to M \to N \to 0$.
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This construction was first studied by A. Grothendieck for the categories of coherent and locally free sheaves on schemes in proving the Riemann–Roch theorem. See [[K-functor|$K$-functor]] in algebraic geometry. The group $K(C)$ is uniquely defined (up to isomorphism) and can be given by generators — to each object $L \in C$, there corresponds a generator $[L]$ — and by the relations $[L]-[N]-[M]=0$ for each exact sequence $0 \to L \to M \to N \to 0$.
  
 
If $X$ is a topological space, then the Grothendieck group of the additive category of vector bundles over $X$ is an invariant of the space, studied in (topological) [[K-theory|K-theory]]. If $C$ is the category of non-degenerate symmetric bilinear forms on linear spaces over a field $k$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. [[Witt ring|Witt ring]]).
 
If $X$ is a topological space, then the Grothendieck group of the additive category of vector bundles over $X$ is an invariant of the space, studied in (topological) [[K-theory|K-theory]]. If $C$ is the category of non-degenerate symmetric bilinear forms on linear spaces over a field $k$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. [[Witt ring|Witt ring]]).
  
====References====
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====Comments====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Swan, "The Grothendieck ring of a finite group" ''Topology'' , '''2''' (1963) pp. 85–110 {{MR|0153722}} {{ZBL|0119.02905}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J.P. Serre, "Le théorème de Riemann–Roch" ''Bull. Soc. Math. France'' , '''86''' (1958) pp. 97–136 {{MR|116022}} {{ZBL|0091.33004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.F. Atiyah, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517026.png" />-theory: lectures" , Benjamin (1967) {{MR|224083}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Bass, "Lectures on topics in algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517027.png" />-theory" , Tata Inst. (1966)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR></table>
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One also associates a Grothendieck group $K(M)$ to any commutative monoid as the solution of the universal problem posed by additive mappings of $M$ into Abelian groups. It is the Abelian group with generators $m \in M$ and relations $m-m_1-m_2$, for all $m,m_1,m_2 \in M$ such that $m=m_1+m_2$ in $M$. Taking, for instance, the monoid of isomorphism classes of vector bundles over a topological space $X$ (with the monoid addition induced by the direct sum) one again obtains the topological K-group $K(X)$.
  
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When considering an additive category $C$ in which not every short exact sequence splits, there are two possible natural associated Grothendieck groups. Both are Abelian groups generated by the isomorphism classes of objects of $C$. For the first there is a relation $[M]-[M_1]-[M_2]$ whenever $M$ is isomorphic to $M_1 \oplus M_2$, and for the second there is a relation $[M]-[M_1]-[M_2]$ whenever there is a short exact sequence $0 \to M_1 \to M \to M_2 \to 0$. Both notions occur in the literature.
  
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The Grothendieck group $K_0(A)$ defined by the additive category of finitely-generated projective modules over a ring $A$ (in which every short exact sequence splits of course) is sometimes called the Grothendieck group of the ring $A$. Cf. also [[Algebraic K-theory|Algebraic K-theory]]. Another important example of a Grothendieck group is the [[Picard group|Picard group]] $\mathrm{Pic}(A)$ of a ring (or of a scheme). It is the Grothendieck group associated to the commutative monoid of isomorphism classes of rank 1 projective modules over $A$ with the addition induced by the tensor product over $A$.
  
====Comments====
 
One also associates a Grothendieck group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517028.png" /> to any commutative monoid as the solution of the universal problem posed by additive mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517029.png" /> into Abelian groups. It is the Abelian group with generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517030.png" /> and relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517031.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517034.png" />. Taking, for instance, the monoid of isomorphism classes of vector bundles over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517035.png" /> (with the monoid addition induced by the direct sum) one again obtains the topological <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517036.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517037.png" />.
 
 
When considering an additive category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517038.png" /> in which not every short exact sequence splits, there are two possible natural associated Grothendieck groups. Both are Abelian groups generated by the isomorphism classes of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517039.png" />. For the first there is a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517040.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517041.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517042.png" />, and for the second there is a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517043.png" /> whenever there is a short exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517044.png" />. (Such a short exact sequence splits if there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517046.png" />.) Both notions occur in the literature.
 
 
The Grothendieck group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517047.png" /> defined by the additive category of finitely-generated projective modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517048.png" /> (in which every short exact sequence splits of course) is sometimes called the Grothendieck group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517049.png" />. Cf. also [[Algebraic K-theory|Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517050.png" />-theory]]. Another important example of a Grothendieck group is the [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517051.png" /> of a ring (or of a scheme). It is the Grothendieck group associated to the commutative monoid of isomorphism classes of rank 1 projective modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517052.png" /> with the addition induced by the tensor prodect over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517053.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bass, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517054.png" />-theory" , Benjamin (1968) {{MR|249491}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Karoubi, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517055.png" />-theory" , Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Berrick, "An approach to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045170/g04517056.png" />-theory" , Pitman (1982) {{MR|649409}} {{ZBL|}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|At}}||valign="top"| M.F. Atiyah, "K-theory: lectures", Benjamin (1967) {{MR|224083}} {{ZBL|0159.53302}}
 +
|-
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|valign="top"|{{Ref|Ba}}||valign="top"| H. Bass, "Lectures on topics in algebraic K-theory", Tata Inst. (1966) {{MR|0279159}} {{ZBL|0226.13006}}
 +
|-
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|valign="top"|{{Ref|Ba2}}||valign="top"| H. Bass, "Algebraic K-theory", Benjamin (1968) {{MR|249491}} {{ZBL|0174.30302}}
 +
|-
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|valign="top"|{{Ref|Be}}||valign="top"| J. Berrick, "K-theory"-theory", Pitman (1982) {{MR|649409}} {{ZBL|0382.55002}}
 +
|-
 +
|valign="top"|{{Ref|BoSe}}||valign="top"| A. Borel, J.P. Serre, "Le théorème de Riemann–Roch" ''Bull. Soc. Math. France'', '''86''' (1958) pp. 97–136 {{MR|116022}} {{ZBL|0091.33004}}
 +
|-
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|valign="top"|{{Ref|Ka}}||valign="top"| M. Karoubi, "K-theory", Springer (1978) {{MR|0488029}} {{ZBL|0382.55002}}
 +
|-
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|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}}
 +
|-
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|valign="top"|{{Ref|Sw}}||valign="top"| R. Swan, "The Grothendieck ring of a finite group" ''Topology'', '''2''' (1963) pp. 85–110 {{MR|0153722}} {{ZBL|0119.02905}}
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|-
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|}

Latest revision as of 06:11, 26 March 2023

2020 Mathematics Subject Classification: Primary: 18-XX Secondary: 55N15 [MSN][ZBL]

The Grothendieck group of an additive category is an Abelian group that is assigned to an additive category by a universal additive mapping property. More exactly, let $C$ be a small additive category with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A mapping $\phi: \mathrm{Ob}(C) \to G$ is said to be additive if for any exact sequence $0 \to L \to M \to N \to 0$ in $C$, the relation $\phi(M)=\phi(L)+\phi(N)$ is valid. There exists a group $K(C)$, called the Grothendieck group of $C$, and an additive mapping $k:\mathrm{Ob}(C)\to K(C)$, known as the universal mapping, such that for any additive mapping $\mathrm{Ob}(C) \to G$, there exists a unique homomorphism $\xi: K(C) \to G$ that satisfies the condition $\phi=\xi \circ k$.

This construction was first studied by A. Grothendieck for the categories of coherent and locally free sheaves on schemes in proving the Riemann–Roch theorem. See $K$-functor in algebraic geometry. The group $K(C)$ is uniquely defined (up to isomorphism) and can be given by generators — to each object $L \in C$, there corresponds a generator $[L]$ — and by the relations $[L]-[N]-[M]=0$ for each exact sequence $0 \to L \to M \to N \to 0$.

If $X$ is a topological space, then the Grothendieck group of the additive category of vector bundles over $X$ is an invariant of the space, studied in (topological) K-theory. If $C$ is the category of non-degenerate symmetric bilinear forms on linear spaces over a field $k$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. Witt ring).

Comments

One also associates a Grothendieck group $K(M)$ to any commutative monoid as the solution of the universal problem posed by additive mappings of $M$ into Abelian groups. It is the Abelian group with generators $m \in M$ and relations $m-m_1-m_2$, for all $m,m_1,m_2 \in M$ such that $m=m_1+m_2$ in $M$. Taking, for instance, the monoid of isomorphism classes of vector bundles over a topological space $X$ (with the monoid addition induced by the direct sum) one again obtains the topological K-group $K(X)$.

When considering an additive category $C$ in which not every short exact sequence splits, there are two possible natural associated Grothendieck groups. Both are Abelian groups generated by the isomorphism classes of objects of $C$. For the first there is a relation $[M]-[M_1]-[M_2]$ whenever $M$ is isomorphic to $M_1 \oplus M_2$, and for the second there is a relation $[M]-[M_1]-[M_2]$ whenever there is a short exact sequence $0 \to M_1 \to M \to M_2 \to 0$. Both notions occur in the literature.

The Grothendieck group $K_0(A)$ defined by the additive category of finitely-generated projective modules over a ring $A$ (in which every short exact sequence splits of course) is sometimes called the Grothendieck group of the ring $A$. Cf. also Algebraic K-theory. Another important example of a Grothendieck group is the Picard group $\mathrm{Pic}(A)$ of a ring (or of a scheme). It is the Grothendieck group associated to the commutative monoid of isomorphism classes of rank 1 projective modules over $A$ with the addition induced by the tensor product over $A$.


References

[At] M.F. Atiyah, "K-theory: lectures", Benjamin (1967) MR224083 Zbl 0159.53302
[Ba] H. Bass, "Lectures on topics in algebraic K-theory", Tata Inst. (1966) MR0279159 Zbl 0226.13006
[Ba2] H. Bass, "Algebraic K-theory", Benjamin (1968) MR249491 Zbl 0174.30302
[Be] J. Berrick, "K-theory"-theory", Pitman (1982) MR649409 Zbl 0382.55002
[BoSe] A. Borel, J.P. Serre, "Le théorème de Riemann–Roch" Bull. Soc. Math. France, 86 (1958) pp. 97–136 MR116022 Zbl 0091.33004
[Ka] M. Karoubi, "K-theory", Springer (1978) MR0488029 Zbl 0382.55002
[La] S. Lang, "Algebra", Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[Sw] R. Swan, "The Grothendieck ring of a finite group" Topology, 2 (1963) pp. 85–110 MR0153722 Zbl 0119.02905
How to Cite This Entry:
Grothendieck group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Grothendieck_group&oldid=24624
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article