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# concentration(ComplexMap) -- indices on which a complex map may be non-zero

- Function: concentration
- Usage:
    `(lo,hi) = concentration f`
- Inputs:
  - f, a map of complexes,
- Outputs:
  - a sequence, a pair of integers lo, hi such that f_i = 0 for i < lo or i > hi.

## Description

In this package, each map of complexes has a concentration (lo, hi) such that lo <= hi. When lo <= i <= hi, the map f_i might be zero.

This function is mainly used in programming, to loop over all non-zero maps. This should not be confused with the support of the source or target.

    i1 : S = ZZ/101[a..c]

    o1 = S

    o1 : PolynomialRing

    i2 : C = freeResolution coker vars S

          1      3      3      1
    o2 = S  <-- S  <-- S  <-- S

         0      1      2      3

    o2 : Complex

    i3 : concentration id_C

    o3 = (0, 3)

    o3 : Sequence

    i4 : D = C ++ C[5]

          1      3      3      1             1      3      3      1
    o4 = S  <-- S  <-- S  <-- S  <-- 0  <-- S  <-- S  <-- S  <-- S

         -5     -4     -3     -2     -1     0      1      2      3

    o4 : Complex

    i5 : concentration id_D

    o5 = (-5, 3)

    o5 : Sequence

    i6 : f = randomComplexMap(D, C)

              1              1
    o6 = 0 : S  <---------- S  : 0
                   | 24 |

              3                           3
         1 : S  <----------------------- S  : 1
                   {1} | -36 19  -29 |
                   {1} | -30 19  -8  |
                   {1} | -29 -10 -22 |

              3                           3
         2 : S  <----------------------- S  : 2
                   {2} | -29 -16 34  |
                   {2} | -24 39  19  |
                   {2} | -38 21  -47 |

              1                   1
         3 : S  <--------------- S  : 3
                   {3} | -39 |

    o6 : ComplexMap

    i7 : concentration f

    o7 = (0, 3)

    o7 : Sequence

Indices that are outside of the concentration automatically return the zero object.

    i8 : f_-1

    o8 = 0

    o8 : Matrix 0 <-- 0

    i9 : (id_D)_4

    o9 = 0

    o9 : Matrix 0 <-- 0

Maps inside the concentration may nevertheless be zero.

    i10 : (id_D)_-1

    o10 = 0

    o10 : Matrix 0 <-- 0

The concentration of a zero complex can be arbitrary, however, after pruning, its concentration will be (0,0).

    i11 : C0 = (complex S^0)[4]

    o11 = 0

          -4

    o11 : Complex

    i12 : g = id_C0

    o12 = -4 : 0 <----- 0 : -4
                    0

    o12 : ComplexMap

    i13 : concentration g

    o13 = (-4, -4)

    o13 : Sequence

    i14 : prune g

    o14 = 0

    o14 : ComplexMap

    i15 : concentration oo

    o15 = (0, 0)

    o15 : Sequence

## See also

- concentration(Complex) -- indices on which a complex may be non-zero
- ComplexMap _ ZZ -- access individual matrices in a complex map

## Ways to use this method:

- concentration(ComplexMap) -- indices on which a complex map may be non-zero

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Complexes/ChainComplexMapDoc.m2:768:0.