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By Bandini A.

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Also, Definition Let A be a finite-dimensional algebra over the field k. If Vand Ware finitely generated A-modules, we define the intertwining number of Vand W. 2. Let P be a jinitely generated projective A-module, V ajinitely generated R-free A-module. Then: (i) HomA(P, V) is ajinitely generated R-free R-module. 5, we have P ~ Ae l EB ... EB Ae" where the Ae, are principal indecomposable A-modules and the ei are ryrimitive idempotents. Correspondingly, and Hence in proving (i) and (ii), we may assume P = Ae.

There are 2P such monomials, and a permutes them in cycles of length p except for the two fixed points a(XP) = XP, a( YP) = P. Therefore (X + YY = sum of all such monomials p-1 = XP + P +I t I ai(fiX, Y), i= 0 j= 1 where f1 (X, Y), ... , fr(X, Y) are monomials and t 294 = (l/p)(2 P - 2). In 47. 295 Number of Irreducible Modules particular, we have p-l + by = aP + (a bP t L L + i~O j~ If W l , ... , wp are each either a or b, then we have W W ••• W 1 2 P = (Ji(fia, b». 1 (W l . , . (Ji(W W . . 1 2 w;)(w i + 1 w) P ...

J(d» an idempotent in A. Setting f j +1(t) = f(fi t», this means that (iv) holds for j + I. f and fj have no constant term, so neither does fj + 1 and (i) holds for j + I. j+ 1 (a) = f(fj(a» == f(a) (mod I), by (ii) for j. ii is an idempotent in A/I, so a k == a (mod I) for any integer k > 0, and we see that f j +1 (a) == f(a) == f(1)a = a (mod I), proving (ii) for j + I. j(a) = fia) (mod Ij), proving (iii) for j Step 2 + I. The sequence {fit)} satisfies (1), (2), (3). Proof (1) is (i). j(an is a Cauchy sequence in (A, dJ(A»' A is complete, so (2) holds.

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3-Selmer groups for curves y^2 = x^3 + a by Bandini A.

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