ringMap -- The degree-zero ring map underlying a DGAlgebraMap

Usage:

phi.ringMap

Description

Every DGAlgebraMap phi from A to B includes a key ringMap: a RingMap from A.ring to B.ring giving the restriction of phi to degree zero. When the two DG algebras live over the same ring, phi.ringMap is the identity; in general it is the degree-zero ring map used to lift phi.

i1 : R = ZZ/101[a,b,c]/ideal{a^3, b^3, c^3}

o1 = R

o1 : QuotientRing

i2 : S = R / ideal{a^2*b^2*c^2}

o2 = S

o2 : QuotientRing

i3 : A = acyclicClosure(R, EndDegree => 2)

o3 = {Ring => R                                        }
      Underlying algebra => R[T   ..T   ]
                               1,1   2,3
                                 2       2       2
      Differential => {a, b, c, a T   , b T   , c T   }
                                   1,1     1,2     1,3

o3 : DGAlgebra

i4 : B = acyclicClosure(S, EndDegree => 2)

o4 = {Ring => S                                                                                    }
      Underlying algebra => S[T   ..T   , T   ..T   , T   ..T   ]
                               1,1   1,3   2,1   2,4   3,1   3,3
                                 2       2       2         2 2       2 2        2 2        2 2
      Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   }
                                   1,1     1,2     1,3         1,1       2,1        2,3        2,2

o4 : DGAlgebra

i5 : phi = liftToDGMap(B, A, map(S, R))

o5 = map (S[T   ..T   , T   ..T   , T   ..T   ], R[T   ..T   ], {T   , T   , T   , T   , T   , T   , a, b, c})
             1,1   1,3   2,1   2,4   3,1   3,3      1,1   2,3     1,1   1,2   1,3   2,1   2,2   2,3

o5 : DGAlgebraMap

i6 : phi.ringMap

o6 = map (S, R, {a, b, c})

o6 : RingMap S <-- R

The ringMap is the bridge used internally by toComplexMap to pushforward complexes between different rings.

For the programmer

The object ringMap is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8989:0.

ringMap -- The degree-zero ring map underlying a DGAlgebraMap

Description

See also

For the programmer