Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Hostname: page-component-559fc8cf4f-lzpzj Total loading time: 0.653 Render date: 2021-03-08T13:08:53.276Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }
EnglishFrançais
Glasgow Mathematical Journal Glasgow Mathematical Journal

Article contents

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

Published online by Cambridge University Press:  02 August 2012

SERGIO ESTRADA  and
PEDRO A. GUIL ASENSIO
SERGIO ESTRADA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia) 30100, Spain e-mail: sestrada@um.es
PEDRO A. GUIL ASENSIO
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia) 30100, Spain e-mail: paguil@um.es
Corresponding
E-mail address:
sestrada@um.es
paguil@um.es
Save pdf (0.1 mb)
Rights & Permissions[Opens in a new window]

Abstract

Finitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.

Keywords

18C35 18A35 18F20 03C98
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 59 - 68
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

1.Adámek, J., Herrlich, H. and Strecker, G. E., Abstract and concrete categories. The joy of cats, Repr. Theory Appl. Categ. 17 (2006), 1–507.Google Scholar
2.Adámek, J. and Rosický, J., Locally presentable and accessible categories, vol. 189 (Cambridge University Press, Cambridge, UK, 1994).CrossRefGoogle Scholar
3.Aldrich, S. T., Enochs, E., García-Rozas, J. R. and Oyonarte, L., Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications, J. Algebra 243 (2001), 615–630.CrossRefGoogle Scholar
4.Bican, L., El Bashir, R. and Enochs, E., All modules have flat covers, Bull. Lond. Math. Soc. 33 (4) (2001), 385–390.CrossRefGoogle Scholar
5.Crawley-Boevey, W. W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1641–1674.CrossRefGoogle Scholar
6.Dung, N. V. and García, J. L., Additive categories of locally finite representation type, J. Algebra 238 (2001), 200–238.CrossRefGoogle Scholar
7.Eklof, P. C., Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207–225CrossRefGoogle Scholar
8.Enochs, E., Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 97 (1984), 179–184.CrossRefGoogle Scholar
9.Enochs, E. and Estrada, S., Relative homological algebra in the category of quasi-coherent sheaves, Adv. Math. 194 (2005), 284–295.CrossRefGoogle Scholar
10.Enochs, E., Estrada, S. and García-Rozas, J. R., Gorenstein categories and Tate cohomology on projective schemes, Math. Nachr. 281 (2008), 525–540.CrossRefGoogle Scholar
11.Enochs, E., Estrada, S., García Rozas, J. R. and Oyonarte, L., Flat and cotorsion quasi-coherent sheaves. Applications. Algebra Represent. Theory 7 (2004), 441–456.CrossRefGoogle Scholar
12.Enochs, E., Estrada, S., García Rozas, J. R. and Oyonarte, L., Flat covers in the category of quasi-coherent sheaves over the projective line, Comm. Algebra 32 (2004), 1497–1508.CrossRefGoogle Scholar
13.Enochs, E. and Jenda, O. M. G., Relative homological algebra, GEM 30 (2000) (W. de Gruyter, Berlin, Germany).Google Scholar
14.Enochs, E. and López Ramos, J. A., Kaplansky classes, Rend. Sem. Mat. Univ. Padova 107 (2002), 67–79.Google Scholar
15.Enochs, E. and Oyonarte, L., Flat covers and cotorsion envelopes of sheaves, Proc. Amer. Math. Soc. 130 (2001), 1285–1292.CrossRefGoogle Scholar
16.Estrada, S., Guil Asensio, P., Prest, M. and Trlifaj, J., Model category structures arising from Drinfeld vector bundles, arXiv: 0906.5213v1.Google Scholar
17.Harrison, D. K., Infinite abelian groups and homological methods, Ann. Math. 69 (1959), 366–391.CrossRefGoogle Scholar
18.Murfet, D. and Salarian, S., Totally acyclic complexes over noetherian schemes, Adv. Math. 296 (2011), 1096–1133.CrossRefGoogle Scholar
19.Salce, L., Cotorsion theories for abelian groups, Symp. Math. 23 (1979), 11–32.Google Scholar
20.Šťovíček, J., Deconstructibility and the Hill lemma in Grothendieck categories, Forum Math. (to appear).Google Scholar
21.Xu, J., Flat covers of modules (Lecture Notes in Mathematics, no. 1634 (Springer-Verlag, Berlin, Germany 1996).CrossRefGoogle Scholar
22.Ziegler, M., Model theory of modules, Ann. Pure Appl. Log. 26 (2) (1984), 149–213.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 73 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 8th March 2021. This data will be updated every 24 hours.

Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *