In mathematics, the Grothendieck group, or group of differences,[1] of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
^Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Springer. p. 50. ISBN 978-0-387-76355-2.
and 20 Related for: Grothendieck group information
mathematics, the Grothendieckgroup, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from...
Alexander Grothendieck (/ˈɡroʊtəndiːk/; German pronunciation: [ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk] ; French: [ɡʁɔtɛndik]; 28 March 1928 – 13 November 2014) was...
Grothendieck construction Grothendieck duality Grothendieck existence theorem Grothendieck fibration Grothendieck's Galois theory Grothendieckgroup Grothendieck's...
commutative monoid can be extended to an abelian group. This extension is known as the Grothendieckgroup. The extension is done by defining equivalence...
Elementary abelian group – Commutative group in which all nonzero elements have the same order Grothendieckgroup – Abelian group extending a commutative...
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets...
all free objects, direct products and direct sums, free groups, free lattices, Grothendieckgroup, completion of a metric space, completion of a ring, Dedekind–MacNeille...
this condition is also sufficient and the Grothendieckgroup of the semigroup provides a construction of the group of fractions. The problem for non-commutative...
field, is Z. The Grothendieck-Witt ring of R is isomorphic to the group ring Z[C2], where C2 is a cyclic group of order 2. The Grothendieck-Witt ring of any...
by the Dolbeault-Grothendieck lemma. The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme...
{B}})} be the Grothendieckgroup of B {\displaystyle {\mathcal {B}}} . Let A {\displaystyle A} be a ring which is free as an abelian group, and let a =...
invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieckgroup of that monoid. The forgetful functor U: Grp → Set has a left adjoint...
semigroup homomorphism from the semigroup into the group may be non-injective. Originally, the Grothendieckgroup was, more specifically, the result of this construction...
{Z} \oplus Cl(R)} , where K 0 ( R ) {\displaystyle K_{0}(R)} is the Grothendieckgroup of the commutative monoid of finitely generated projective R {\displaystyle...
spectral class the 1965 first model of the Honda CB450 motorbike the Grothendieckgroup in abstract algebra the Lateral earth pressure at rest the neutral...
isomorphism, one has to change the left part of the isomorphism, using the Grothendieckgroup of the category of perverse sheaves on G r {\displaystyle Gr} to replace...
structure under direct sum. One may make an abelian group out of this monoid, the Grothendieckgroup, by formally adding an additive inverse for each bundle...
k has characteristic zero. global dimension bar resolution GrothendieckgroupGrothendieck local duality Weibel (1999); Cartan & Eilenberg (1956), section...
localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary...
Global Information