Finite representation $k$-algebras and $C^\infty$-generated finite rings

Document Type : Research Paper

Authors

Semnan University

Abstract

In this article, we begin by outlining the historical development of Zariski topology on rings of multivariate polynomials with complex coefficients. Subsequently, for the presumed commutative and unitary ring $k$, we aim to introduce one of the existing structures on the dual side of finite representation k-algebras, which can be extended in a similar manner to the  category L, i.e., the dual category of all -generated finite rings. It is known as Zariski site.

Keywords


[1] M. F. Atiyeh and I. G. MacDonald, Introduction to Commutative Algebra, Addision-Wesley Publishing Company, 1969.
[2] F. Borceux, Handbook of Categorical Algebras 1 and 3, Cambridge University Press, 1994.
[3] D. Cox, J. Little and D. Ośhea, Ideals, Varieties and Algoritms, Third eddition, Springer-Verlag, New York, 2007.
[4] A. Grothendieck, The cohomology theory of abstract algebraic varieties, Proc. Internat. Congress Math., Cambridge Univ. Press, New York, 1960 103–118.
[5] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974.
[6] T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
[7] R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer, Kluwer Academic Publishers Group, Dordrecht, 1996.
[8] S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic, Universitext. Springer-Verlag, New York, 1994.
[9] I. Moerdijk and G. E. Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, New York, 1991.
[10] D. Mumford, The Red Book of Varieties and Schemes, 1358, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.
[11] M. Reid, Undergraduate Algebraic Geometry, New York, Cambridge University Press, 1988.
[12] B. R. Tennison, Sheaf Theory, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.
[13] O. Zariski, The Fundamental Ideas of Abstract Algebraic Geometry, Proceedings of the International Congress of Mathemati-cians, Cambridge, Mass., 2 1950 77–89.