Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
K-theory
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Examples and properties == === K<sub>0</sub> of a field === The easiest example of the Grothendieck group is the Grothendieck group of a point <math>\text{Spec}(\mathbb{F})</math> for a field <math>\mathbb{F}</math>. Since a vector bundle over this space is just a finite dimensional vector space, which is a free object in the category of coherent sheaves, hence projective, the monoid of isomorphism classes is <math>\N</math> corresponding to the dimension of the vector space. It is an easy exercise to show that the Grothendieck group is then <math>\Z</math>. === K<sub>0</sub> of an Artinian algebra over a field === One important property of the Grothendieck group of a [[Noetherian scheme]] <math>X</math> is that it is invariant under reduction, hence <math>K(X) = K(X_{\text{red}})</math>.<ref>{{Cite web|url=https://mathoverflow.net/questions/77089/grothendieck-group-for-projective-space-over-the-dual-numbers|title=Grothendieck group for projective space over the dual numbers|website=mathoverflow.net|access-date=2017-04-16}}</ref> Hence the Grothendieck group of any [[Artinian ring|Artinian]] <math>\mathbb{F}</math>-algebra is a direct sum of copies of <math>\Z</math>, one for each connected component of its spectrum. For example, <math display="block">K_0 \left(\text{Spec}\left(\frac{\mathbb{F}[x]}{(x^9)}\times\mathbb{F}\right)\right) = \mathbb{Z}\oplus\mathbb{Z}</math> === K<sub>0</sub> of projective space === One of the most commonly used computations of the Grothendieck group is with the computation of <math>K(\mathbb{P}^n)</math> for projective space over a field. This is because the intersection numbers of a projective <math>X</math> can be computed by embedding <math>i:X \hookrightarrow \mathbb{P}^n </math> and using the push pull formula <math>i^*([i_*\mathcal{E}]\cdot [i_*\mathcal{F}])</math>. This makes it possible to do concrete calculations with elements in <math>K(X)</math> without having to explicitly know its structure since<ref>{{Cite web|title=kt.k theory and homology - Grothendieck group for projective space over the dual numbers|url=https://mathoverflow.net/questions/77089/grothendieck-group-for-projective-space-over-the-dual-numbers|access-date=2020-10-20|website=MathOverflow}}</ref> <math display="block">K(\mathbb{P}^n) = \frac{\mathbb{Z}[T]}{(T^{n+1})}</math> One technique for determining the Grothendieck group of <math>\mathbb{P}^n</math> comes from its stratification as <math display="block">\mathbb{P}^n = \mathbb{A}^n \coprod \mathbb{A}^{n-1} \coprod \cdots \coprod \mathbb{A}^0</math> since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to <math>\mathbb{Z}</math>, and the intersection of <math>\mathbb{A}^{n-k_1},\mathbb{A}^{n-k_2}</math> is generically <math display="block">\mathbb{A}^{n-k_1} \cap \mathbb{A}^{n-k_2} = \mathbb{A}^{n-k_1-k_2}</math> for <math>k_1 + k_2 \leq n</math>. === K<sub>0</sub> of a projective bundle === Another important formula for the Grothendieck group is the projective bundle formula:<ref>{{Cite journal|last=Manin|first=Yuri I|author-link=Yuri Manin|date=1969-01-01|title=Lectures on the K-functor in algebraic geometry|journal=Russian Mathematical Surveys|language=en|volume=24|issue=5|pages=1–89|doi=10.1070/rm1969v024n05abeh001357|issn=0036-0279|bibcode=1969RuMaS..24....1M}}</ref> given a rank r vector bundle <math>\mathcal{E}</math> over a Noetherian scheme <math>X</math>, the Grothendieck group of the projective bundle <math>\mathbb{P}(\mathcal{E})=\operatorname{Proj}(\operatorname{Sym}^\bullet(\mathcal{E}^\vee))</math> is a free <math>K(X)</math>-module of rank ''r'' with basis <math>1,\xi,\dots,\xi^{n-1}</math>. This formula allows one to compute the Grothendieck group of <math>\mathbb{P}^n_\mathbb{F}</math>. This make it possible to compute the <math>K_0</math> or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group <math>K(\mathbb{P}^n)</math> by observing it is a projective bundle over the field <math>\mathbb{F}</math>. === K<sub>0</sub> of singular spaces and spaces with isolated quotient singularities === One recent technique for computing the Grothendieck group of spaces with minor singularities comes from evaluating the difference between <math>K^0(X)</math> and <math>K_0(X)</math>, which comes from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of the [[Singularity category]] <math>D_{sg}(X)</math><ref>{{Cite web|title=ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?|url=https://mathoverflow.net/questions/133383/is-the-algebraic-grothendieck-group-of-a-weighted-projective-space-finitely-gene|access-date=2020-10-20|website=MathOverflow}}</ref><ref name=":0">{{cite journal|last1=Pavic|first1=Nebojsa|last2=Shinder|first2=Evgeny|title=K-theory and the singularity category of quotient singularities|journal=Annals of K-Theory|year=2021|volume=6|issue=3|pages=381–424|doi=10.2140/akt.2021.6.381|arxiv=1809.10919|s2cid=85502709}}</ref> from [[derived noncommutative algebraic geometry]]. It gives a long exact sequence starting with <math display="block">\cdots \to K^0(X) \to K_0(X) \to K_{sg}(X) \to 0</math> where the higher terms come from [[Algebraic K-theory|higher K-theory]]. Note that vector bundles on a singular <math>X</math> are given by vector bundles <math>E \to X_{sm}</math> on the smooth locus <math>X_{sm} \hookrightarrow X</math>. This makes it possible to compute the Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities. In particular, if these singularities have isotropy groups <math>G_i</math> then the map <math display="block">K^0(X) \to K_0(X)</math> is injective and the cokernel is annihilated by <math>\text{lcm}(|G_1|,\ldots, |G_k|)^{n-1}</math> for <math>n = \dim X</math>.<ref name=":0" /><sup>pg 3</sup> === K<sub>0</sub> of a smooth projective curve === For a smooth projective curve <math>C</math> the Grothendieck group is <math display="block">K_0(C) = \mathbb{Z}\oplus\text{Pic}(C)</math> for [[Picard group]] of <math>C</math>. This follows from the [[Quillen spectral sequence|Brown-Gersten-Quillen spectral sequence]]<ref name=":1">{{Cite book|last=Srinivas, V.|url=https://www.worldcat.org/oclc/624583210|title=Algebraic K-theory|date=1991|publisher=Birkhäuser|isbn=978-1-4899-6735-0|location=Boston|oclc=624583210}}</ref><sup>pg 72</sup> of [[algebraic K-theory]]. For a [[regular scheme]] of finite type over a field, there is a convergent spectral sequence <math display="block">E_1^{p,q} = \coprod_{x \in X^{(p)}}K^{-p-q}(k(x)) \Rightarrow K_{-p-q}(X)</math> for <math>X^{(p)}</math> the set of codimension <math>p</math> points, meaning the set of subschemes <math>x: Y \to X</math> of codimension <math>p</math>, and <math>k(x)</math> the algebraic function field of the subscheme. This spectral sequence has the property<ref name=":1" /><sup>pg 80</sup> <math display="block">E_2^{p,-p} \cong \text{CH}^p(X)</math> for the Chow ring of <math>X</math>, essentially giving the computation of <math>K_0(C)</math>. Note that because <math>C</math> has no codimension <math>2</math> points, the only nontrivial parts of the spectral sequence are <math>E_1^{0,q},E_1^{1,q}</math>, hence <math display="block">\begin{align} E_\infty^{1,-1}\cong E_2^{1,-1} &\cong \text{CH}^1(C) \\ E_\infty^{0,0} \cong E_2^{0,0} &\cong \text{CH}^0(C) \end{align}</math> The [[coniveau filtration]] can then be used to determine <math>K_0(C)</math> as the desired explicit direct sum since it gives an exact sequence <math display="block">0 \to F^1(K_0(X)) \to K_0(X) \to K_0(X)/F^1(K_0(X)) \to 0</math> where the left hand term is isomorphic to <math> \text{CH}^1 (C) \cong \text{Pic}(C)</math> and the right hand term is isomorphic to <math>CH^0(C) \cong \mathbb{Z}</math>. Since <math>\text{Ext}^1_{\text{Ab}}(\mathbb{Z},G) = 0</math>, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if <math>C</math> is a smooth projective curve of genus <math>g</math> over <math>\mathbb{C}</math>, then <math display="block">K_0(C) \cong \mathbb{Z}\oplus(\mathbb{C}^g/\mathbb{Z}^{2g})</math> Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolated [[Cohen–Macaulay ring|Cohen-Macaulay]] singularities, giving techniques for computing the Grothendieck group of any singular algebraic curve. This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)