Asia Mathematika

An International Journal. ISSN: 2457-0834 (online)

Volume 2, Issue 3, December 2018, Pages: 8-13

Quasi-coherent modules over disjoint union of affine schemes

Mohamed Aqalmoun

Department of Mathematics, Faculty of Science, University Moulay Ismail, B.P. 11201, Zitoune, Mekn`es 50000, Morocco.

Abstract

In this paper, we study the category of quasi-coherent modules over disjoint union of affine schemes. We show that is has enough projectives and we give example of non affine scheme with trivial quasi-coherent cohomology.

Keywords

Projective object, quasi-coherent module, cohomology, scheme, affine scheme.

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Reference

1. M.F. Atiyah, I.G. MacDonald, An introduction to commutative algebra, Addison- Wesley, 1969.

2. D. Eisenbud, Commutative algebra with a View Toward Algebraic Geometry,Springer, 2004.

3. R. Godement, Th´eorie des faisceaux, Actualit´e scientifiques et industrielles, 1252, Hermann, Paris (1964).

4. U. G¨ortz, T. Wedhorn, Algebraic Geometry I, Vieweg + Teubner, Wiesbaden, 2010.

5. A. Grothendieck, Sur quelques points d’alg`ebre homologique, Tohoku mathematical journal,9,(1955), pp. 119-221.

6. A. Grothendieck, J. Dieudonn´e, El´ements de g´eom´etrie alg´ebrique., ´ Inst. des Hautes Etudes de Science ´

7. J. Harris, D. Eisenbud, The Geometry of schemes, Springer, 2000.

8. R. Hartshorne, Algebraic Geometry,Springer, 1977.

9. Q. Liu, Algebraic Geometry and Arithmetic Curves,6th Edition, Oxfrord Gradu- ate Texts in Mathematics, 2002.

10. D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer, New York.

11. A. Neeman, Steins, affines and Hilbert’s fourtheenth problem, Annals of Mathematics, 127(1988),229-244.

12. Serre, J.-P., Faisceaux alg´ebriques coherent, Ann. of Maths, 61,(1955).

13. I. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, Springer-Verlag, 1994.

14. I. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Springer-Verlag, 1996

15. C. A. Weibel, An Antroduction to Homological Algebra,Vol. 38, Cambridge University Press, Cambridge, 1994.

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