D. Eisenbud's Commutative Algebra, with a View Toward Algebraic Geometry PDF

By D. Eisenbud

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Example text

2, are valid in the EEP. 6 In the EEP, we have 1. Two distinct points lie on a unique line. 2. Two distinct lines intersect in a unique point. parallel lines non-parallel lines Proof. 5, a line ∈ R2 when extended is denoted by ∗ . 1. Let A and B be two distinct points of the EEP. (a) If A and B are two distinct points of R2 , then (AB)∗ is the unique line of the EEP which contains them. (b) If A and B are two distinct points at ∞, then ∞ is the unique line of the EEP which contains them. (c) Suppose A ∈ R2 , and B ∈ ∞ .

O B B1 O C1 A1 B C C X A A Proof. Let π be a plane (of order greater than two) of a projective space Sn , of dimension n ≥ 3. Let ABC, A B C be two triangles of π, in perspective from a point X. Take any line ∈ / π, through X. There are at least two other points 42 Introduction to axiomatic geometry (distinct from X) on . Let O, O be two such points. Now A O, O A intersect in a point since they are distinct lines of the plane OXA. Let A O ∩ O A = A1 . / π, Similarly, define the points B1 and C1 .

3. Consider the following diagram and suppose the points and lines are those of a Pappian plane π. By considering the two triads of points {A2 , D, B1 } and {B2 , C, A1 }, prove that the line QR passes though the point a ∩ b. a A1 C Q A2 R D b B2 B1 4. Let π be a Desarguesian plane. In π, let A, B, C lie on a line m and A , B , C lie on a line n such that AA , BB , and CC are concurrent in a point V . Prove that the points AB ∩ A B, AC ∩ A C, and BC ∩ B C lie on a line passing through the point of intersection of m and n.

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Commutative Algebra, with a View Toward Algebraic Geometry [bad OCR] by D. Eisenbud


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