ringFiberProduct -- constructs the fiber product of several rings

Usage:

ringFiberProduct(I, J)

ringFiberProduct(S, T)
Inputs:
- I, an ideal, a homogeneous ideal of a PolynomialRing.
- J, an ideal, a homogeneous ideal of a PolynomialRing.
- S, a quotient ring, a quotient of a polynomial ring by a homogeneous ideal
- T, a quotient ring, a quotient of a polynomial ring by a homogeneous ideal
Outputs:
- an ideal,
- a quotient ring,

Description

Suppose we have rings $S_1 = R_1/I_1$ and $S_2 = R_2/I_2$ for some homogeneous ideals $I_1$ of $R_1 = K[x_1, \dots, x_n]$ and $I_2$ of $R_2 = K[y_1, \dots, y_m]$, where $K$ is a field. Their fiber product over $K$ is the ring: \[ S_1 \times_K S_2 = K[x_1, \dots, x_n, y_1, \dots, y_m] / (I_1 + I_2 + (x_i y_j : 1 \leq i \leq n, 1 \leq j \leq m )) \]

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2, y^2)

             2   2
o2 = ideal (x , y )

o2 : Ideal of R

i3 : S = QQ[a,b]

o3 = S

o3 : PolynomialRing

i4 : J = ideal(a^4, b^4)

             4   4
o4 = ideal (a , b )

o4 : Ideal of S

i5 : ringFiberProduct(R/I, S/J)

                QQ[x..y, a..b]
o5 = ------------------------------------
       2   2   4   4
     (x , y , a , b , x*a, x*b, y*a, y*b)

o5 : QuotientRing

i6 : R = (ZZ/7)[x,y,z]

o6 = R

o6 : PolynomialRing

i7 : I = ideal(x^2, y^3, z^4)

             2   3   4
o7 = ideal (x , y , z )

o7 : Ideal of R

i8 : S = (ZZ/7)[a,b,c]

o8 = S

o8 : PolynomialRing

i9 : J = ideal(a, b^3, c^3)

                3   3
o9 = ideal (a, b , c )

o9 : Ideal of S

i10 : ringFiberProduct(R/I, S/J)

                                 ZZ
                                 --[x..z, a..c]
                                  7
o10 = --------------------------------------------------------------------
        2   3   4      3   3
      (x , y , z , a, b , c , x*a, x*b, x*c, y*a, y*b, y*c, z*a, z*b, z*c)

o10 : QuotientRing

Ways to use ringFiberProduct:

ringFiberProduct(Ideal,Ideal)
ringFiberProduct(List)
ringFiberProduct(QuotientRing,QuotientRing)
ringFiberProduct(Sequence)

For the programmer

The object ringFiberProduct is an associative binary method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/MacaulayPosets.m2:1624:0.

ringFiberProduct -- constructs the fiber product of several rings

Description

See also

Ways to use ringFiberProduct:

For the programmer