# Download PDF by D. G. Northcott: A first course of homological algebra

By D. G. Northcott

ISBN-10: 0521299764

ISBN-13: 9780521299763

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Extra info for A first course of homological algebra

Example text

11. The symmetries of a geometric object often form a group. We look at some special cases. Consider an equilateral triangle in the plane with a side on the X-axis labeling the ordered vertices A, B, C, with line segment AB the base. Let r be a counterclockwise rotation of 120o . The triangle looks the same but the vertices are now ordered 50 III. GROUPS C, A, B, with base CA. , we have the relation r3 is the identity. So r is a cyclic group of three elements. ) Now, viewing the plane in R3 let f denote the flip along the line of the apex of the triangle perpendicular to the base.

S This is only useful if we can compute the size of the equivalence class x. 14. (1) A partition of a set A is a collection C of subsets of A such that A = C B. Let R be an equivalence relation on A. Then A partitions A. Conversely, let C partition A. Define a relation ∼ on A by a ∼ b if a and b belong to the same set in C. Show ∼ is an equivalence relation on A. So an equivalence relation and a partition of a set are essentially the same. (2) Draw lines in R2 perpendicular to the X-axis through all integer points and the analogous lines parallel to the Y -axis.

This means that 1 ≡ bi mod mi and 0 ≡ bi mod mj if i = j. Consequently, x0 := c1 b1 + · · · + cr br ≡ ci bi ≡ ci mod mi , with i = 1, . . , r and x0 works. Uniqueness. , mi | x0 − y0 for 1 ≤ i ≤ r. 8 (2), we have x0 ≡ y0 mod m. We want to interpret what we did above in the language of “rings” and “ring homomorphisms”. Let m > 1 in Z and (a, m) = 1. 13), there are integers x and y satisfying 1 = ax + my. , a has a multiplicative inverse in Z/mZ. An element a having a multiplicative inverse is called a unit.