Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-18T04:49:28.449Z Has data issue: false hasContentIssue false

Grothendieck groups for categories of complexes

Published online by Cambridge University Press:  09 January 2008

Hans-Bjørn Foxby
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark, foxby@math.ku.dk.
Esben Bistrup Halvorsen
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark, esben@math.ku.dk.
Get access

Abstract

The new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have nd = dim R.

One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

Type
Research Article
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bruns, Winfried and Herzog, Jürgen, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 95h:13020Google Scholar
2.Foxby, Hans-Bjørn, K-theory for complexes with finite length homology, Copenhagen University Preprint Series no. 1, 1982.Google Scholar
3.Foxby, Hans-Bjørn, The Macrae invariant, Commutative algebra: Durham 1981 (Durham, 1981), London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge, 1982, pp. 121128. MR 85e:13033Google Scholar
4.Halvorsen, Esben Bistrup, Algebraic K-theory and local Chern characters applied to Serre's conjectures on intersection multiplicity, Master's thesis, University of Copenhagen, 2004.Google Scholar
5.Hilton, Peter J. and Stammbach, Urs, A course in homological algebra, second ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 97k:18001Google Scholar
6.Roberts, Paul C., Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics, vol. 133, Cambridge University Press, Cambridge, 1998. MR 2001a:13029Google Scholar
7.Roberts, Paul C. and Srinivas, V., Modules of finite length and finite projective dimension, Invent. Math. 151 (2003), no. 1, 127. MR MR1943740 (2003j:13017)CrossRefGoogle Scholar