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# RingMap ** CellComplex -- tensors labels via a ring map

- Operator: **
- Usage:
    `f ** C`
- Inputs:
  - f, a ring map,
  - C, a cell complex,
- Outputs:
  - a cell complex, whose cells labels are given by taking the tensor product by the ring map f

## Description

Given a ring map f and a cell complex C, then for each label, viewed as a module M, this function constructs a cell complex whose new labels are f ** M

    i1 : S = QQ[x,y,z];

    i2 : R = QQ[a,b,c];

    i3 : f = map(R,S,matrix{{a,b,c^2}});

    o3 : RingMap R <-- S

    i4 : v1 = newCell({},x);

    i5 : v2 = newCell({},y);

    i6 : v3 = newCell({},z);

    i7 : e12 = newCell({v1,v2});

    i8 : e23 = newCell({v2,v3});

    i9 : C = cellComplex(S,{e12,e23});

    i10 : cells(1,C)/cellLabel

    o10 = {y*z, x*y}

    o10 : List

    i11 : D = f ** C;

    i12 : cells(1,D)/cellLabel

                   2
    o12 = {a*b, b*c }

    o12 : List

## See also

- relabelCellComplex -- relabels a cell complex

## Ways to use this method:

- RingMap ** CellComplex -- tensors labels via a ring map

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CellularResolutions/doc.m2:1111:0.