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Krull Dimension Krull dimension of a ring What does it mean for a ring to be 0 dimensional? For all x we have a relation x n (1 − ux) = 0 If the ring is reduced (a nilpotent element is 0) this simplifies to ∀x.∃u.x(1 − xu) = 0 von Neumann regular ring 9
Krull Dimension Krull dimension of a ring Lemma: If R is a k-algebra and a1, . . . , al are algebraically dependent, i.e. there exists a relation P (a1, . . . , al) = 0 with P in k[X1, . . . , Xl] then a1, . . . , al has a complementary sequence Example for n = 2: we can assume the relation to be of the form a k2 2 (ak1 1 (1 + a1u1) + a2u2) = 0 and we can take b2 = a k1 1 (1 + a1u1) + a2u2 and b1 = 1 + a1u1 10
Page 1 and 2: Krull Dimension Thierry Coquand Tri
Page 3 and 4: Krull Dimension Zariski spectrum An
Page 5 and 6: Krull Dimension Krull dimension of
Page 7 and 8: Krull Dimension Logical complexity
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Page 15 and 16: Krull Dimension Kronecker’s Theor
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Page 23 and 24: Krull Dimension Forster’s Theorem
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