ringConnectedSum -- constructs the connected sum of several rings

Usage:

ringConnectedSum(I, J, ...)

ringConnectedSum(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 Gorenstein 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. Let $m_1, m_2$ be the maximal elements of the monomial posets of $S_1, S_2$, respectively. The connected sum of $S_1$ and $S_2$ is their fiber product mod $m_1-m_2$. In symbols, $S_1 \# S_2 = \frac{S_1 \times_K S_2}{(m_1-m_2)}$.

i1 : R = (ZZ/2)[u,v,w]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(u^4, v^2-w^2, w^2-v*u)

             4   2    2         2
o2 = ideal (u , v  + w , u*v + w )

o2 : Ideal of R

i3 : S = R/I

o3 = S

o3 : QuotientRing

i4 : T = newRing(S, Variables=>{x,y,z})

o4 = T

o4 : QuotientRing

i5 : U = ringConnectedSum(S, T)

o5 = U

o5 : QuotientRing

i6 : ideal U

             4   2    2         2   4   2    2         2                     
o6 = ideal (u , v  + w , u*v + w , x , y  + z , x*y + z , u*x, u*y, u*z, v*x,
     ------------------------------------------------------------------------
                               5    5
     v*y, v*z, w*x, w*y, w*z, w  + z )

              ZZ
o6 : Ideal of --[u..z]
               2

i7 : isMacaulay S

o7 = true

i8 : isMacaulay U

o8 = false

Ways to use ringConnectedSum:

ringConnectedSum(Ideal,Ideal)
ringConnectedSum(List)
ringConnectedSum(QuotientRing,QuotientRing)
ringConnectedSum(Sequence)

For the programmer

The object ringConnectedSum 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.

ringConnectedSum -- constructs the connected sum of several rings

Description

See also

Ways to use ringConnectedSum:

For the programmer