By Gregor Kemper
This textbook deals a radical, glossy advent into commutative algebra. it really is intented mostly to function a consultant for a process one or semesters, or for self-study. The rigorously chosen subject material concentrates at the strategies and effects on the middle of the sphere. The ebook continues a relentless view at the typical geometric context, permitting the reader to realize a deeper figuring out of the cloth. even though it emphasizes conception, 3 chapters are dedicated to computational features. Many illustrative examples and routines increase the textual content.
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Additional resources for A Course in Commutative Algebra (Graduate Texts in Mathematics, Volume 256)
As a consequence of some general topological considerations, aﬃne varieties can be decomposed into irreducible components. Another consequence is that a Noetherian ring contains only ﬁnitely many minimal prime ideals. Readers who are unfamiliar with the language of topology can ﬁnd all that is needed for this book in any textbook on topology (for example Bourbaki ), usually on the ﬁrst few pages. 1 Aﬃne Varieties In this section we deﬁne the Zariski topology on K n and on its subsets. We ﬁrst need a proposition.
By the deﬁnition of n, all a ∈ S are algebraic over L := Quot (K[a1 , . . , an ]), so Quot(A) is algebraic over L, too. There exists a nonzero element a ∈ P1 . We have a nonzero k i polynomial G = i=0 gi x ∈ L[x] with G(a) = 0. Since a = 0, we may assume g0 = 0. Furthermore, we may assume gi ∈ K[a1 , . . , an ]. Then k g0 = − gi ai ∈ P1 , i=1 so viewing g0 as a polynomial in n indeterminates over K, we obtain g0 (a1 + P1 , . . , an +P1 ) = 0, contradicting the algebraic independence of the ai +P1 ∈ A/P1 .
Xn ] a subset. , we may assume S to be an ideal, and in fact even a radical ideal. (b) For a subset X ⊆ K n , the topological closure (also called the Zariski closure) is X = V (I(X)) . (c) If Y ⊆ K n is an aﬃne variety, then by deﬁnition the Zariski topology on Y has the subvarieties of Y as closed sets. (d) On Rn and Cn , the Zariski topology is coarser than the usual Euclidean topology. (e) Every ﬁnite subset of K n is Zariski closed. In other words, K n is a T1 space. This also applies to every subset Y ⊆ K n .
A Course in Commutative Algebra (Graduate Texts in Mathematics, Volume 256) by Gregor Kemper