In mathematics, rigidity of ''K''-theory encompasses results relating
algebraic ''K''-theory of different rings.
Suslin rigidity
''Suslin rigidity'', named after
Andrei Suslin
Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee ...
, refers to the invariance of mod-''n'' algebraic ''K''-theory under the base change between two
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s: showed that for an
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
:
of algebraically closed fields, and an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
''X'' / ''F'', there is an isomorphism
:
between the mod-''n'' ''K''-theory of coherent sheaves on ''X'', respectively its base change to ''E''. A textbook account of this fact in the case ''X'' = ''F'', including the resulting computation of ''K''-theory of algebraically closed fields in characteristic ''p'', is in .
This result has stimulated various other papers. For example show that the base change functor for the mod-''n''
stable A1-homotopy category
:
is fully faithful. A similar statement for non-commutative motives has been established by .
Gabber rigidity
Another type of rigidity relates the mod-''n'' K-theory of an
henselian ring In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now rest ...
''A'' to the one of its
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
''A''/''m''. This rigidity result is referred to as ''Gabber rigidity'', in view of the work of who showed that there is an isomorphism
:
provided that ''n''≥1 is an integer which is invertible in ''A''.
If ''n'' is not invertible in ''A'', the result as above still holds, provided that K-theory is replaced by the fiber of the
trace map In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...
between K-theory and
topological cyclic homology. This was shown by .
Applications
used Gabber's and Suslin's rigidity result to reprove Quillen's computation of
K-theory of finite fields.
References
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* {{Citation, last=Weibel, first=Charles A., title=The ''K''-book, series=Graduate Studies in Mathematics, volume=145, publisher=American Mathematical Society, Providence, RI, year=2013, isbn=978-0-8218-9132-2, mr=3076731, url=http://www.math.rutgers.edu/~weibel/Kbook.html
Algebraic K-theory