Error in tensor(Ring, Ring, Join => false)

[mahrud]

Here's an unexpected error:

i1 : S1 = QQ[x,y, DegreeRank => 1];

i2 : S2 = QQ[x,y, DegreeRank => 2];

i3 : tensor(S1, S2, Join => false)
stdio:3:1:(3): error: with Join => false, expected degree map to return a list of length at most 1

i4 : tensor(S2, S1, Join => false)

o4 = QQ[x , y , x , y ]
         0   0   1   1

o4 : PolynomialRing

Looking at the code, I can't figure out what was the intention here: https://github.com/Macaulay2/M2/blob/5572c8a245d187f6c22f8eaa0b03cac0b8d86e02/M2/Macaulay2/m2/monoids.m2#L703-L705

[mahrud]

More generally, when tensoring two (or more) rings, why are DegreeMap and DegreeLift only mapping the degrees of the first factor and the tensor ring?

[DanGrayson]

The meaning of Join => true is that the degrees from the various factors are concatenated to get the degrees in the tensor product. The meaning of Join => false is that they are not, so they all have to be the same length.

[mahrud]

I don't understand your point. In both examples above I'm using Join => false with the same rings.

[DanGrayson]

S1 has degrees of length 1 and S2 has degrees of length 2. There is no sensible way to combine these in a tensor product.

[mahrud]

Once again, that doesn't explain why switching the order of S1 and S2 changes the result. Can you clarify what is your definition of tensor product where A⊗B and B⊗A are not only non-isomorphic, but only one is defined and there is no sensible way to define the other?!

[mikestillman]

What should the degrees of the generators in the tensor product be if Join => false is used? I'm not sure...

[mahrud]

I think there's a misunderstanding about what the issue is. I'm not asking something about Join => true vs. Joint => false. I'm only switching the order of the two rings:

i3 : tensor(S1, S2, Join => false)
stdio:3:1:(3): error: with Join => false, expected degree map to return a list of length at most 1

i4 : tensor(S2, S1, Join => false)

o4 = QQ[x , y , x , y ]
         0   0   1   1

I claim the results should be the same, so either both should give an error, or the outputs should be isomorphic, except with different degrees:

i3 : degrees tensor(S2, S1, Join => false)

o3 = {{1, 0}, {0, 1}, {0, 1}, {0, 1}}

o3 : List

i4 : degrees tensor(S1, S2, Join => false)

o4 = {{1, 0}, {1, 0}, {1, 0}, {0, 1}} -- hypothetical answer

o4 : List

[DanGrayson]

Good point!

By the way, use Join=>true to get the degree group of the tensor product to be G1++G2.

[mahrud]

By the way, use Join=>true to get the degree group of the tensor product to be G1++G2.

Correct. I edited this out because it wasn't relevant.

[mahrud]

This is causing a bug in PushForward package:

Here is a basic example from docs for reesAlgebra:

needsPackage "PushForward"
needsPackage "ReesAlgebra"

kk = ZZ/101;
S = kk[x_0..x_4]
I = trim monomialCurveIdeal(S, {2,3,5,6})
R = reesAlgebra I
f = map(R, S, DegreeMap => prepend_0)
assert isHomogeneous f

It seems like pushForward from Core works fine but pushFwd loses homogeneity:

isHomogeneous(N = coker basis(2, R^1)) -- true
isHomogeneous pushForward_f N          -- true
isHomogeneous pushFwd_f N              -- false

See @devlin-mallory's comment in https://github.com/Macaulay2/M2/issues/3077#issuecomment-1901395089.