Fixes sheaf cohomology printing, adds check that coefficient ring is …

…a field (oscar-system#3087) * docu sheaf cohomology * correction * fixes sheaf coh table printing * adjusts docu, checks that coefficient ring is a field * fix getindex and docu * fixes tests --------- Co-authored-by: Wolfram Decker <[email protected]> Co-authored-by: Rafael Mohr <[email protected]>
JohnAAbbott · Dec 11, 2023 · 6d999d6 · 6d999d6
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diff --git a/docs/doc.main b/docs/doc.main
@@ -168,6 +168,9 @@
         "AlgebraicGeometry/Schemes/ProjectiveSchemes.md",
         "AlgebraicGeometry/Schemes/MorphismsOfProjectiveSchemes.md",
      ],
+     "Sheaf Cohomology" => [
+        "AlgebraicGeometry/SheafCohomology/sheaf_cohomology.md",
+     ],
      "Algebraic Sets" => [
         "AlgebraicGeometry/AlgebraicSets/AffineAlgebraicSet.md",
         "AlgebraicGeometry/AlgebraicSets/ProjectiveAlgebraicSet.md",

diff --git a/docs/oscar_references.bib b/docs/oscar_references.bib
@@ -461,6 +461,18 @@ @Article{Cor21
   doi           = {10.1007/s00029-021-00679-6}
 }
 
+@InCollection{DE02,
+  author        = {Decker, Wolfram and Eisenbud, David},
+  title         = {Sheaf algorithms using the exterior algebra},
+  booktitle     = {Computations in algebraic geometry with Macaulay 2},
+  zbl           = {0994.14010},
+  publisher     = {Berlin: Springer},
+  pages         = {215--249},
+  year          = {2002},
+  language      = {English},
+  zbmath        = {1693054}
+}
+
 @Article{DES93,
   author        = {Decker, Wolfram and Ein, Lawrence and Schreyer, Frank-Olaf},
   title         = {Construction of surfaces in ${\mathbb P}^4$},
@@ -631,6 +643,21 @@ @Article{Der99
   doi           = {10.1006/aima.1998.1787}
 }
 
+@Article{EFS03,
+  author        = {Eisenbud, David and Fl{\o}ystad, Gunnar and Schreyer, Frank-Olaf},
+  title         = {Sheaf cohomology and free resolutions over exterior algebras},
+  zbl           = {1063.14021},
+  journal       = {Trans. Am. Math. Soc.},
+  fjournal      = {Transactions of the American Mathematical Society},
+  volume        = {355},
+  number        = {11},
+  pages         = {4397--4426},
+  year          = {2003},
+  doi           = {10.1090/S0002-9947-03-03291-4},
+  language      = {English},
+  zbmath        = {1963988}
+}
+
 @Article{EHU03,
   author        = {Eisenbud, David and Huneke, Craig and Ulrich, Bernd},
   title         = {What is the {Rees} algebra of a module?},
@@ -704,6 +731,16 @@ @Book{Eis95
   year          = {1995}
 }
 
+@Book{Eis98,
+  author        = {Eisenbud, David},
+  title         = {Computing cohomology. A chapter in W. Vasconcelos, Computational methods in commutative algebra and
+                  algebraic geometry},
+  publisher     = {Berlin: Springer},
+  pages         = {209--216},
+  year          = {1998},
+  language      = {English}
+}
+
 @Article{FGLM93,
   author        = {Faugère, J.C. and Gianni, P. and Lazard, D. and Mora, T.},
   title         = {Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering},

diff --git a/docs/src/AlgebraicGeometry/SheafCohomology/sheaf_cohomology.md b/docs/src/AlgebraicGeometry/SheafCohomology/sheaf_cohomology.md
@@ -0,0 +1,22 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = quote
+  using Oscar
+end
+```
+
+# Sheaves on Projective Space
+
+We present two algorithms for computing sheaf cohomology over projective $n$-space.
+The algorithms are based on Tate resolutions via the **B**ernstein-**G**elfand-**G**elfand-correspondence
+as introduced in [EFS03](@cite) and on local cohomology (see [Eis98](@cite)), respectively. While the first
+algorithm makes use of syzygy computations over the exterior algebra, the second algorithm is based on
+syzygy computations over the symmetric algebra (see [DE02](@cite) for a tutorial). Thus, in most examples,
+the first algorithm is much faster.
+
+```@docs
+sheaf_cohomology(M::ModuleFP{T}, l::Int, h::Int; algorithm::Symbol = :bgg) where {T <: MPolyDecRingElem}
+```
+
+
+
diff --git a/src/Modules/ModulesGraded.jl b/src/Modules/ModulesGraded.jl
@@ -1484,7 +1484,7 @@ mutable struct sheafCohTable
 end
 
 function Base.getindex(st::sheafCohTable, ind...)
-  row_ind = ind[1] + 1
+  row_ind = size(st.values, 1) - ind[1]
   col_ind = ind[2] - first(st.twist_range) + 1
   return st.values[row_ind, col_ind]
 end
@@ -1503,7 +1503,7 @@ function Base.show(io::IO, table::sheafCohTable)
   # row labels
   row_label_length = max(_ndigits(nrows - 1), 3) + 3
   for i in 1:nrows
-    pushfirst!(print_rows[i], rpad("$(i-1): ", row_label_length, " "))
+    pushfirst!(print_rows[i], rpad("$(nrows-i): ", row_label_length, " "))
   end
   pushfirst!(chi_print, rpad("chi: ", row_label_length, " "))  
 
@@ -1524,18 +1524,22 @@ end
 @doc raw"""
     sheaf_cohomology(M::ModuleFP{T}, l::Int, h::Int; algorithm::Symbol = :bgg) where {T <: MPolyDecRingElem}
 
-Compute the cohomology of twists of of the coherent sheaf on projective
-space associated to `M`. The range of twists is between `l` and `h`.
-In the displayed result, '-' refers to a zero enty and '*' refers to a
-negative entry (= dimension not yet determined). To determine all values
-in the desired range between `l` and `h` use `sheafCoh_BGG_regul(M, l-ngens(base_ring(M)), h+ngens(base_ring(M)))`.
-The values of the returned table can be accessed by indexing it
-with a cohomological index and a value between `l` and `h` as shown
-in the example below.
+If `M` is a graded module over a standard graded multivariate polynomial ring with coefficients in a field `K`, 
+say, and $\mathcal F = \widetilde{M}$ is the coherent sheaf associated to `M` on the corresponding projective 
+space $\mathbb P^n(K)$, consider the cohomology groups $H^i(\mathbb P^n(K), \mathcal F(d))$ as vector spaces 
+over $K$, and return their dimensions $h^i(\mathbb P^n(K), \mathcal F(d))$ in the range of twists $d$ 
+indicated by `l` and `h`. The result is presented as a table, where '-' indicates that
+$h^i(\mathbb P^n(K), \mathcal F(d)) = 0$. The line starting  with `chi` lists the Euler characteristic 
+of each twist under consideration. The values in the table can be accessed as shown in the 
+first example below. Note that this example addresses the cotangent bundle on projective 3-space, while the 
+second example is concerned with the structure sheaf of projective 4-space.
 
 The keyword `algorithm` can be set to
-- `:bgg` (uses the Bernstein-Gelfand-Gelfand correspondence),
-- `:loccoh` (uses local duality),
+- `:bgg` (use the Tate resolution via the Bernstein-Gelfand-Gelfand correspondence),
+- `:loccoh` (use local cohomology).
+
+!!! note 
+    Due to the shape of the Tate resolution, the algorithm addressed by `bgg` does not compute all values in the given range `l` $<$ `h`. The missing values are indicated by a `*`. To determine all values in the range `l` $<$ `h`, enter `sheaf_cohomology(M, l-ngens(base_ring(M)), h+ngens(base_ring(M)))`.
 
 ```jldoctest
 julia> R, x = polynomial_ring(QQ, "x" => 1:4);
@@ -1555,35 +1559,47 @@ julia> M = cokernel(map(FI, 2));
 julia> tbl = sheaf_cohomology(M, -6, 2)
 twist:  -6  -5  -4  -3  -2  -1   0   1   2
 ------------------------------------------
-0:      70  36  15   4   -   -   -   -   *
-1:       *   -   -   -   -   -   -   -   -
-2:       *   *   -   -   -   -   1   -   -
-3:       *   *   *   -   -   -   -   -   6
+3:      70  36  15   4   -   -   -   -   *
+2:       *   -   -   -   -   -   -   -   -
+1:       *   *   -   -   -   -   1   -   -
+0:       *   *   *   -   -   -   -   -   6
 ------------------------------------------
 chi:     *   *   *   4   -   -   1   -   *
 
-julia> tbl[0, -6]
-70
+julia> tbl[0, 2]
+6
 
-julia> tbl[2, 0]
+julia> tbl[1, 0]
 1
 
+julia> sheaf_cohomology(M, -9, 5)
+twist:   -9   -8   -7   -6   -5   -4   -3   -2   -1    0    1    2    3    4    5
+---------------------------------------------------------------------------------
+3:      280  189  120   70   36   15    4    -    -    -    -    -    *    *    *
+2:        *    -    -    -    -    -    -    -    -    -    -    -    -    *    *
+1:        *    *    -    -    -    -    -    -    -    1    -    -    -    -    *
+0:        *    *    *    -    -    -    -    -    -    -    -    6   20   45   84
+---------------------------------------------------------------------------------
+chi:      *    *    *   70   36   15    4    -    -    1    -    6    *    *    *
+```
+
+```jldoctest
 julia> R, x = polynomial_ring(QQ, "x" => 1:5);
 
-julia> R, x = grade(R);
+julia> S, _  = grade(R);
 
-julia> F = graded_free_module(R, 1);
+julia> F = graded_free_module(S, 1);
 
-julia> sheaf_cohomology(F, -7, 2, algorithm = :bgg)
-twist:  -7  -6  -5  -4  -3  -2  -1   0   1   2
-----------------------------------------------
-0:      15   5   1   -   -   -   *   *   *   *
-1:       *   -   -   -   -   -   -   *   *   *
-2:       *   *   -   -   -   -   -   -   *   *
-3:       *   *   *   -   -   -   -   -   -   *
-4:       *   *   *   *   -   -   -   1   5  15
-----------------------------------------------
-chi:     *   *   *   *   -   -   *   *   *   *
+julia> sheaf_cohomology(F, -8, 3, algorithm = :loccoh)
+twist:  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3
+------------------------------------------------------
+4:      35  15   5   1   -   -   -   -   -   -   -   -
+3:       -   -   -   -   -   -   -   -   -   -   -   -
+2:       -   -   -   -   -   -   -   -   -   -   -   -
+1:       -   -   -   -   -   -   -   -   -   -   -   -
+0:       -   -   -   -   -   -   -   -   1   5  15  35
+------------------------------------------------------
+chi:    35  15   5   1   -   -   -   -   1   5  15  35
 ```
 """
 function sheaf_cohomology(M::ModuleFP{T},
@@ -1611,6 +1627,36 @@ The values of the returned table can be accessed by indexing it
 with a cohomological index and a value between `l` and `h` as shown
 in the example below.
 
+```jldoctest
+julia> R, x = polynomial_ring(QQ, "x" => 1:5);
+
+julia> R, x = grade(R);
+
+julia> F = graded_free_module(R, 1);
+
+julia> Oscar._sheaf_cohomology_bgg(F, -7, 2)
+twist:  -7  -6  -5  -4  -3  -2  -1   0   1   2
+----------------------------------------------
+4:      15   5   1   -   -   -   *   *   *   *
+3:       *   -   -   -   -   -   -   *   *   *
+2:       *   *   -   -   -   -   -   -   *   *
+1:       *   *   *   -   -   -   -   -   -   *
+0:       *   *   *   *   -   -   -   1   5  15
+----------------------------------------------
+chi:     *   *   *   *   -   -   *   *   *   *
+
+julia> sheaf_cohomology(F, -11, 6)
+twist:  -11  -10   -9   -8   -7   -6   -5   -4   -3   -2   -1    0    1    2    3    4    5    6
+------------------------------------------------------------------------------------------------
+4:      210  126   70   35   15    5    1    -    -    -    -    -    -    -    *    *    *    *
+3:        *    -    -    -    -    -    -    -    -    -    -    -    -    -    -    *    *    *
+2:        *    *    -    -    -    -    -    -    -    -    -    -    -    -    -    -    *    *
+1:        *    *    *    -    -    -    -    -    -    -    -    -    -    -    -    -    -    *
+0:        *    *    *    *    -    -    -    -    -    -    -    1    5   15   35   70  126  210
+------------------------------------------------------------------------------------------------
+chi:      *    *    *    *   15    5    1    -    -    -    -    1    5   15    *    *    *    *
+```
+
 ```jldoctest
 julia> R, x = polynomial_ring(QQ, "x" => 1:4);
 
@@ -1626,38 +1672,21 @@ S^4 <---- S^6 <---- S^4 <---- S^1 <---- 0
 
 julia> M = cokernel(map(FI, 2));
 
-julia> tbl = Oscar._sheaf_cohomology_bgg(M, -6, 2)
+julia> tbl = sheaf_cohomology(M, -6, 2, algorithm = :loccoh)
 twist:  -6  -5  -4  -3  -2  -1   0   1   2
 ------------------------------------------
-0:      70  36  15   4   -   -   -   -   *
-1:       *   -   -   -   -   -   -   -   -
-2:       *   *   -   -   -   -   1   -   -
-3:       *   *   *   -   -   -   -   -   6
+3:      70  36  15   4   -   -   -   -   -
+2:       -   -   -   -   -   -   -   -   -
+1:       -   -   -   -   -   -   1   -   -
+0:       -   -   -   -   -   -   -   -   6
 ------------------------------------------
-chi:     *   *   *   4   -   -   1   -   *
+chi:    70  36  15   4   -   -   1   -   6
 
-julia> tbl[0, -6]
+julia> tbl[3, -6]
 70
 
-julia> tbl[2, 0]
+julia> tbl[1, 0]
 1
-
-julia> R, x = polynomial_ring(QQ, "x" => 1:5);
-
-julia> R, x = grade(R);
-
-julia> F = graded_free_module(R, 1);
-
-julia> Oscar._sheaf_cohomology_bgg(F, -7, 2)
-twist:  -7  -6  -5  -4  -3  -2  -1   0   1   2
-----------------------------------------------
-0:      15   5   1   -   -   -   *   *   *   *
-1:       *   -   -   -   -   -   -   *   *   *
-2:       *   *   -   -   -   -   -   -   *   *
-3:       *   *   *   -   -   -   -   -   -   *
-4:       *   *   *   *   -   -   -   1   5  15
-----------------------------------------------
-chi:     *   *   *   *   -   -   *   *   *   *
 ```
 """
 function _sheaf_cohomology_bgg(M::ModuleFP{T},
@@ -1703,17 +1732,17 @@ julia> M = cokernel(map(FI, 2));
 julia> tbl = Oscar._sheaf_cohomology_loccoh(M, -6, 2)
 twist:  -6  -5  -4  -3  -2  -1   0   1   2
 ------------------------------------------
-0:      70  36  15   4   -   -   -   -   -
-1:       -   -   -   -   -   -   -   -   -
-2:       -   -   -   -   -   -   1   -   -
-3:       -   -   -   -   -   -   -   -   6
+3:      70  36  15   4   -   -   -   -   -
+2:       -   -   -   -   -   -   -   -   -
+1:       -   -   -   -   -   -   1   -   -
+0:       -   -   -   -   -   -   -   -   6
 ------------------------------------------
 chi:    70  36  15   4   -   -   1   -   6
 
-julia> tbl[0, -6]
+julia> tbl[3, -6]
 70
 
-julia> tbl[2, 0]
+julia> tbl[1, 0]
 1
 
 julia> R, x = polynomial_ring(QQ, "x" => 1:5);
@@ -1725,11 +1754,11 @@ julia> F = graded_free_module(R, 1);
 julia> Oscar._sheaf_cohomology_loccoh(F, -7, 2)
 twist:  -7  -6  -5  -4  -3  -2  -1   0   1   2
 ----------------------------------------------
-0:      15   5   1   -   -   -   -   -   -   -
-1:       -   -   -   -   -   -   -   -   -   -
-2:       -   -   -   -   -   -   -   -   -   -
+4:      15   5   1   -   -   -   -   -   -   -
 3:       -   -   -   -   -   -   -   -   -   -
-4:       -   -   -   -   -   -   -   1   5  15
+2:       -   -   -   -   -   -   -   -   -   -
+1:       -   -   -   -   -   -   -   -   -   -
+0:       -   -   -   -   -   -   -   1   5  15
 ----------------------------------------------
 chi:    15   5   1   -   -   -   -   1   5  15
 ```
@@ -1760,9 +1789,12 @@ function _ndigits(val::Int)
 end
 
 function _weights_and_sing_mod(M::ModuleFP{T}) where {T <: MPolyDecRingElem}
+
+  CR = base_ring(base_ring(M))
+  @assert isa(CR, AbstractAlgebra.Field) "Base ring of input module must be defined over a field."
   free_mod = ambient_free_module(M)
-  @assert is_graded(M) "Module must be graded."
   @assert is_standard_graded(free_mod) "Only supported for the standard grading of polynomials."
+  @assert is_graded(M) "Module must be graded."
 
   # get a cokernel presentation of M
   p = presentation(M)
@@ -2052,6 +2084,8 @@ end
 
 elem_type(::Type{FreeMod_dec{T}}) where {T} = FreeModElem_dec{T}
 parent_type(::Type{FreeModElem_dec{T}}) where {T} = FreeMod_dec{T}
+elem_type(::FreeMod_dec{T}) where {T} = FreeModElem_dec{T}
+parent_type(::FreeModElem_dec{T}) where {T} = FreeMod_dec{T}
 
 @doc raw"""
 """

diff --git a/test/Modules/ModulesGraded.jl b/test/Modules/ModulesGraded.jl
@@ -1076,8 +1076,8 @@ end
   tbl = Oscar._sheaf_cohomology_bgg(M, -6, 2)
   lbt = sheaf_cohomology(M, -6, 2, algorithm = :bgg)
   @test tbl.values == lbt.values
-  @test tbl[0, -6] == 70
-  @test tbl[2, 0] == 1
+  @test tbl[3, -6] == 70
+  @test tbl[1, 0] == 1
   @test iszero(tbl[2, -2])
 
   F = free_module(S, 1)