Basic Commutative Algebra by Balwant Singh

By Balwant Singh

This textbook, set for a one or semester path in commutative algebra, offers an creation to commutative algebra on the postgraduate and learn degrees. the most necessities are familiarity with teams, jewelry and fields. Proofs are self-contained.

The publication might be helpful to rookies and skilled researchers alike. the fabric is so prepared that the newbie can examine via self-study or by way of attending a path. For the skilled researcher, the booklet may well serve to offer new views on a few recognized effects, or as a reference.

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Then ei = 1 ei = n≥0 en ei . Comparing homogeneous components of degree i we get ei = e0 ei for every i. Now, e0 = e0 1 = i≥0 e0 ei = i≥0 ei = 1, showing that 1 ∈ A0 . This proves that A0 is a subring of A. Since A0 An ⊆ An , each An is an A0 -submodule of A. (2) Clear. Note that A0 being a subring of A makes A a commutative A0 -algebra. We usually identify A0 with A/A+ as in the above proposition. Let A and B be graded rings. A ring homomorphism A → B is said to be graded if f (An ) ⊆ Bn for every n.

3) a is generated (as an ideal) by homogeneous elements. Proof. (1) ⇒ (2). By (1), every element a of a has an expression of the form a = n an (finite sum) with an ∈ a ∩ An for every n, and the uniqueness of this expression is clear. So we have (2). (2) ⇒ (3). By (2), a is generated by homogeneous elements. n≥0 (a ∩ An ), which is a set of (3) ⇒ (1). Let a be generated by a set S of homogeneous elements. Let a ∈ a, let n ≥ 0, and let an be the homogeneous component of a of degree n. We have to show that an ∈ a.

2) ⇒ (3). Given t as in (2), consider 1M − tg ∈ HomA (M, M ). We have g(1M − tg) = g − g = 0. Therefore im (1M − tg) ⊆ ker g = f (M ). Let s ∈ HomA (M, M ) be the composite of 1M −tg with the inverse of the isomorphism f : M → f (M ). e. f s + tg = 1M . This implies that f sf = f − tgf = f − 0 = f 1M . Now, since f is injective, we get sf = 1M . (3) ⇒ (1). Trivial. In order to prove the last assertion, given t as in (2), construct s as in the proof of (2) ⇒ (3). Then f s + tg = 1M . It follows that M = f (M ) + t(M ).

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