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# isWellDefined -- whether a map is well defined

## Synopsis

• Usage:
isWellDefined f
• Inputs:
• f, or
• Outputs:
• , whether f is a well-defined map

## Description

In order to check whether a matrix, whose source module is not free, is well defined, then a Gröbner basis computation will probably be required.
 i1 : R = QQ[a..d]; i2 : f = map(R^1,coker vars R,{{1_R}}) o2 = | 1 | o2 : Matrix i3 : isWellDefined f o3 = false i4 : isWellDefined map(coker vars R, R^1, {{1_R}}) o4 = true
In order to check whether a ring map is well defined, it is often necessary to check that the image of an ideal under a related ring map is zero. This often requires a Gröbner basis as well.
 i5 : A = ZZ/5[a] o5 = A o5 : PolynomialRing i6 : factor(a^3-a-2) 3 o6 = (a - a - 2) o6 : Expression of class Product i7 : B = A/(a^3-a-2); i8 : isWellDefined map(A,B) o8 = false i9 : isWellDefined map(B,A) o9 = true

• map -- make a map

## Ways to use isWellDefined :

• isWellDefined(Matrix)
• isWellDefined(RingMap)
• isWellDefined(CoherentSheaf) (missing documentation)
• isWellDefined(SheafMap) (missing documentation)
• isWellDefined(Variety) (missing documentation)

## For the programmer

The object isWellDefined is .