13 <tensor product>
A module M, N
Tensor product M ⊗A N
Given conditions
(1) M ⊗A N is A module that is generated by { x ⊗ y | x ∈M, y ∈ N }
(2) ( x + x’ ) ⊗ y = x ⊗ y + x’ ⊗ y, x ⊗ (y + y’ ) = x ⊗ y + x ⊗ y’ a ∈ A a ( x ⊗ y ) = ax ⊗y = x ⊗ ay
14 <multiplicative set>
Ring A
Subset of A S
1 ∈ S, 0 ∉ S
S is closed by multiplication.
S is Multiplicative set.
15 <localization>
Ring A
S is A multiplicative set,
S-1A = { a/s | a ∈ A, s ∈ S }
S-1A is ring.
16 <total quotient ring, canonical map>
Ring A
Ring S = { s ∈ A | s is non-zero divisor }
S-1A is total quotient ring.
A is subring of total quotient ring S-1A.
Canonical is next..
(1) Map iA : A → S-1A, iA (a ）= a/1
(2) Map iM : M → S-1M, I (x ) = x/1
Tokyo September 24, 2007
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