Algebraic Geometry and Commutative Algebra (Universitext) by Siegfried Bosch

By Siegfried Bosch

Algebraic geometry is an engaging department of arithmetic that mixes tools from either, algebra and geometry. It transcends the restricted scope of natural algebra via geometric building ideas. furthermore, Grothendieck’s schemes invented within the past due Nineteen Fifties allowed the appliance of algebraic-geometric tools in fields that previously far-off from geometry, like algebraic quantity concept. the recent innovations lead the way to fantastic development comparable to the evidence of Fermat’s final Theorem through Wiles and Taylor.

The scheme-theoretic method of algebraic geometry is defined for non-experts. extra complicated readers can use the ebook to develop their view at the topic. A separate half bargains with the required necessities from commutative algebra. On a complete, the e-book presents a truly available and self-contained creation to algebraic geometry, as much as a fairly complicated level.

Every bankruptcy of the publication is preceded through a motivating advent with a casual dialogue of the contents. usual examples and an abundance of workouts illustrate each one part. this fashion the publication is a wonderful resolution for studying on your own or for complementing wisdom that's already current. it could actually both be used as a handy resource for classes and seminars or as supplemental literature.

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Example text

Let M be an R-module. A subgroup N ⊂ M is called a submodule or, in more precise terms, an R-submodule of M if rx ∈ N for all r ∈ R and x ∈ N . In particular, N is then an R-module itself, using the addition and scalar multiplication inherited from M . For example, the R-submodules of any ring R consist precisely of the ideals in R. Remark 5. Let ϕ : M ✲ N be a morphism of R-modules. 4 Modules 33 the kernel of ϕ, is an R-submodule of M , and im ϕ := ϕ(M ), the image of ϕ, is an R-submodule of N .

Therefore y 1 ∈ im d and, hence, ker g 1 ⊂ im d. Let us mention a special case of the Snake Lemma, which is quite neat to state. 5 Finiteness Conditions and the Snake Lemma Corollary 2. Let ✲ 0 f1 M1 ✲ u1 ✲ 0 f2 M2 ✲ u2 ❄ N1 g1 ✲ M3 ✲ 0 ✲ 0 u3 ❄ N2 g2 ✲ ❄ N3 be a commutative diagram of R-module homomorphisms with exact rows. Then, using the notation of Lemma 1, there is a corresponding exact sequence 0 ✲ ker u1 f1 ✲ f2 ✲ ker u2 d ✲ ker u3 coker u1 g1 ✲ coker u2 g2 ✲ coker u3 ✲ 0. Proof. Apply the Snake Lemma.

Remark 6. Let R be a ring and a ⊂ R an ideal. Let π : R canonical residue homomorphism. Then: (i) j(a) = π −1 j(R/a) . (ii) rad(a) = π −1 rad(R/a) . (iii) rad(a) = p∈Spec R, a⊂p p. 3 Radicals 29 ✲ π −1 (n) Proof. To justify assertions (i) and (iii) observe that the map n defines a bijection between all maximal (resp. prime) ideals of R/a and the maximal (resp. prime) ideals in R that contain a. Therefore (i) and (iii) follow from Definition 1 and Proposition 4; just use the fact that the formation of inverse images with respect to π commutes with intersections.

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