Ufyukyu

1 $begingroup$ The exercise 5.10 of Atiyah's Commutative Algebra gives the definition of $going-up$ map: A ring homomorphism $f:Arightarrow B$ is said to have the $going-up$ (resp. the $going-down$ property) if the conclusion of going-up theorem (resp. the going-down theorem) holds for $B$ and its subring $f(A)$ . I think this definition is unnatural and I think the definition should be ....... if the conclusion of going-up theorem (resp. the going-down theorem) holds for $B$ and $A$ . It seems these two definitions are different. Why the book chooses the "unnatural" definition? Also one part of exercise is i) $f$ has the going-down property ii) For any prime ideal $q$ of $B$ , if $p=q^{c}$ , then $f^{*}: Spec(B_{q})to Spec(A_{p})$ is surjective prove that i)

1 $begingroup$ $ (E,S)$ is an independence system with $ w: E rightarrow mathbb{R_+}$ . $E= { e_1,...,e_m} $ is the set of edges with $ w(e_1) geq....geq w(e_m), w(e_{m+1}):=0$ . Define the set: $ E_i:= { e_1,...,e_i}$ and $ T subset E$ with $ w'(T) = sum_{ e in T} w(e)$ . I will show: $$ w'(T)= sum_{ i=1}^m |T cap E_i|big(w(e_i)-w(e_{i+1})big)$$ Can somebody give me a hint to do that? combinatorics graph-theory independence matroids share | cite | improve this question edited Jan 12 at 16:01 greedoid 42k 11 52 105