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Add mres and nres Singular Calls for Free resolutions #2500

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Sep 18, 2023
Merged

Add mres and nres Singular Calls for Free resolutions #2500

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5c04f88
add `mres` with fix
Sep 7, 2023
fdff1c7
fixes no relation edge case
ederc Sep 7, 2023
c9c0d6c
adds `nres`, updated doc, adds tests
Sep 7, 2023
87d2059
fix typo `mres` -> `nres`
Sep 7, 2023
347349c
compatibility changes to other `free_resolution` methods
Sep 13, 2023
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71 changes: 36 additions & 35 deletions src/Modules/UngradedModules.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -6215,11 +6215,13 @@
@doc raw"""
free_resolution(F::FreeMod)

Return a free resolution of `F`.
Return a free resolution of `F`. The `length` and `algorithm`
keywords are here only for compatibility reasons with the other `free_resolution`
methods and have no effect on the computation.

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# Examples
"""
function free_resolution(F::FreeMod)
function free_resolution(F::FreeMod; length::Int=0, algorithm::Symbol=:fres)
res = presentation(F)
set_attribute!(res, :show => free_show, :free_res => F)
return FreeResolution(res)
Expand Down Expand Up @@ -6371,7 +6373,8 @@
Return a free resolution of `M`.

If `length != 0`, the free resolution is only computed up to the `length`-th free module.
At the moment, `algorithm` only allows the option `:fres`.
At the moment, options for `algorithm` are `:fres`, `:mres` and `:nres`. With `:mres` or `:nres`,
minimal free resolutions are returned.

# Examples
```jldoctest
Expand Down Expand Up @@ -6425,9 +6428,8 @@
iterative kernel computation.
"""
function free_resolution(M::SubquoModule{<:MPolyRingElem};
ordering::ModuleOrdering = default_ordering(M),
length::Int=0, algorithm::Symbol=:fres
)
ordering::ModuleOrdering = default_ordering(M),
length::Int=0, algorithm::Symbol=:fres)

coefficient_ring(base_ring(M)) isa AbstractAlgebra.Field ||
error("Must be defined over a field.")
Expand All @@ -6436,13 +6438,13 @@

#= Start with presentation =#
pm = presentation(M)
maps = [map(pm, j) for j in Hecke.map_range(pm)]
maps = [pm.maps[j] for j in 2:3]

br = base_ring(M)
kernel_entry = image(pm.maps[1])[1]

if ngens(kernel_entry) == 0
cc = Hecke.ComplexOfMorphisms(Oscar.ModuleFP, maps, check = false, seed = -2)
cc = Hecke.ComplexOfMorphisms(Oscar.ModuleFP, pushfirst!(maps, pm.maps[1]), check = false, seed = -2)
cc.fill = _extend_free_resolution
cc.complete = true
return FreeResolution(cc)
Expand All @@ -6459,6 +6461,12 @@
elseif algorithm == :lres
error("LaScala's method is not yet available in Oscar.")
gbpres = singular_kernel_entry # or as appropriate, taking into account base changes
elseif algorithm == :mres
gbpres = singular_kernel_entry
res = Singular.mres(gbpres, length)
elseif algorithm == :nres
gbpres = singular_kernel_entry
res = Singular.nres(gbpres, length)
else
error("Unsupported algorithm $algorithm")
end
Expand All @@ -6467,34 +6475,17 @@
cc_complete = true
end

codom = codomain(maps[1])

if is_graded(codom)
rk = Singular.ngens(gbpres)
SM = SubModuleOfFreeModule(codom, gbpres)
generator_matrix(SM)
ff = graded_map(codom, SM.matrix)
dom = domain(ff)
else
dom = free_module(br, Singular.ngens(gbpres))
SM = SubModuleOfFreeModule(codom, gbpres)
generator_matrix(SM)
ff = hom(dom, codom, SM.matrix)
end

maps[1] = ff

br_name = AbstractAlgebra.find_name(base_ring(M))
if br_name === nothing
br_name = "R"
end

#= Add maps from free resolution computation, start with second entry
= due to inclusion of presentation(M) at the beginning. =#
j = 2
j = 1
while j <= Singular.length(res)
if is_graded(dom)
codom = dom
if is_graded(M)
codom = domain(maps[1])
rk = Singular.ngens(res[j])
SM = SubModuleOfFreeModule(codom, res[j])
generator_matrix(SM)
Expand All @@ -6504,7 +6495,7 @@
insert!(maps, 1, ff)
j += 1
else
codom = dom
codom = domain(maps[1])
rk = Singular.ngens(res[j])
dom = free_module(br, rk)
SM = SubModuleOfFreeModule(codom, res[j])
Expand Down Expand Up @@ -6642,35 +6633,45 @@
end

@doc raw"""
free_resolution(I::MPolyIdeal)
free_resolution(I::MPolyIdeal; length::Int=0, algorithm::Symbol=:fres)

Compute a free resolution of `I`.

If `length != 0`, the free resolution is only computed up to the `length`-th free module.
At the moment, options for `algorithm` are `:fres`, `:mres` and `:nres`. With `:mres` or `:nres`,
minimal free resolutions are returned.

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# Examples
"""
function free_resolution(I::MPolyIdeal)
function free_resolution(I::MPolyIdeal;
length::Int=0, algorithm::Symbol=:fres)
S = ideal_as_module(I)
n = Hecke.find_name(I)
if n !== nothing
AbstractAlgebra.set_name!(S, string(n))
end
return free_resolution(S)
return free_resolution(S, length = length, algorithm = algorithm)
end

@doc raw"""
free_resolution(Q::MPolyQuoRing)
free_resolution(Q::MPolyQuoRing; length::Int=0, algorithm::Symbol=:fres)

Compute a free resolution of `Q`.

If `length != 0`, the free resolution is only computed up to the `length`-th free module.
At the moment, options for `algorithm` are `:fres`, `:mres` and `:nres`. With `:mres` or `:nres`,
minimal free resolutions are returned.

# Examples
"""
function free_resolution(Q::MPolyQuoRing)
function free_resolution(Q::MPolyQuoRing;
length::Int=0, algorithm::Symbol=:fres)
q = quotient_ring_as_module(Q)
n = Hecke.find_name(Q)
if n !== nothing
AbstractAlgebra.set_name!(q, String(n))
end
return free_resolution(q)
return free_resolution(q, length = length, algorithm = algorithm)
end

@doc raw"""
Expand Down
26 changes: 15 additions & 11 deletions test/Modules/UngradedModules.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -166,21 +166,25 @@ end
@test is_bijective(p)
end

R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
A = R[x; y]
B = R[x^2; x*y; y^2; z^4]
M = SubquoModule(A, B)
free_res = free_resolution(M, length=1)
R, (x, y, z) = polynomial_ring(QQ, ["x", "y", "z"])
A = R[x; y]
B = R[x^2; x*y; y^2; z^4]
M = SubquoModule(A, B)
free_res = free_resolution(M, length=1)
@test is_complete(free_res) == false
free_res[2]
@test length(free_res.C.maps) == 4
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@test free_res[3] == free_module(R, 2)
@test free_res[4] == free_module(R, 0)
@test free_res[3] == free_module(R, 2)
@test free_res[4] == free_module(R, 0)
@test is_complete(free_res) == true
free_res = free_resolution(M)
@test all(iszero, homology(free_res.C))
free_res = free_resolution_via_kernels(M)
@test all(iszero, homology(free_res))
free_res = free_resolution(M)
@test all(iszero, homology(free_res.C))
free_res = free_resolution_via_kernels(M)
@test all(iszero, homology(free_res))
free_res = free_resolution(M, algorithm = :mres)
@test all(iszero, homology(free_res.C))
free_res = free_resolution(M, algorithm = :nres)
@test all(iszero, homology(free_res.C))

N = SubquoModule(R[x+2*x^2; x+y], R[z^4;])
tensor_resolution = tensor_product(N,free_res)
Expand Down
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