Seminar Algebra and Geometry: Danny de Jesus Gomez (Universität Osnabrück)

On the Connectedness of Forcing Schemes

Let R be a commutative ring with unity; f,f₁,...,f_n elements of R (data) and I=(f₁,...,f_n), the corresponding ideal of R. The Forcing Algebra A:=R[T₁,...,T_n]/(f₁T₁+...+f_nT_n-f) for the data f,f₁,...,f_n is the most "natural" R-algebra such that f belongs to the expansion of I in A. On this talk we present criteria and properties over the base ring R and the data in order to guarantee the connectedness of the corresponding forcing scheme Spec A. In particular, we present an arithmetical Criteria for Principal Ideals Domains. Besides, we rewrite the condition of belonging to the integral closure of I by means of the universal connectedness of the canonical morphism between the forcing scheme and the base space Spec R. Finally, we study the local nature (over the base) of the connectedness in a very general setting.

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