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Fix corners cases of free resolutions #37132

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Merged
merged 2 commits into from
Feb 2, 2024
Merged

Fix corners cases of free resolutions #37132

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2c5e1da
Fix corner cases
kwankyu Jan 22, 2024
162163c
Fix a typo
kwankyu Jan 22, 2024
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66 changes: 54 additions & 12 deletions src/sage/homology/free_resolution.py
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Original file line number Diff line number Diff line change
Expand Up @@ -7,7 +7,7 @@

.. MATH::

R^{n_1} \xleftarrow{d_1} R^{n_1} \xleftarrow{d_2}
R^{n_0} \xleftarrow{d_1} R^{n_1} \xleftarrow{d_2}
\cdots \xleftarrow{d_k} R^{n_k} \xleftarrow{d_{k+1}} 0

terminating with a zero module at the end that is exact (all homology groups
Expand Down Expand Up @@ -242,9 +242,20 @@ def _repr_module(self, i):
'S^2'
sage: r # indirect doctest
S^1 <-- S^3 <-- S^2 <-- 0

TESTS::

sage: S.<x,y,z> = PolynomialRing(QQ)
sage: I = S.ideal(0)
sage: C = I.free_resolution()
sage: C
S^1 <-- 0
"""
if i == 0:
r = self._maps[0].nrows()
if self._length > 0:
r = self._maps[0].nrows()
else:
r = self._initial_differential.domain().dimension()
s = f'{self._name}^{r}'
return s
elif i > self._length:
Expand Down Expand Up @@ -422,6 +433,8 @@ def __getitem__(self, i):
raise IndexError('invalid index')
elif i > self._length:
F = FreeModule(self._base_ring, 0)
elif i == 0:
F = self.differential(0).domain()
elif i == self._length:
F = FreeModule(self._base_ring, self._maps[i - 1].ncols())
else:
Expand All @@ -444,11 +457,12 @@ def differential(self, i):
sage: r
S(0) <-- S(-2)⊕S(-2)⊕S(-2) <-- S(-3)⊕S(-3) <-- 0
sage: r.differential(3)
Free module morphism defined by the matrix []
Domain: Ambient free module of rank 0 over the integral domain
Multivariate Polynomial Ring in x, y, z, w over Rational Field
Codomain: Ambient free module of rank 2 over the integral domain
Multivariate Polynomial Ring in x, y, z, w over Rational Field
Free module morphism defined as left-multiplication by the matrix
[]
Domain: Ambient free module of rank 0 over the integral domain
Multivariate Polynomial Ring in x, y, z, w over Rational Field
Codomain: Ambient free module of rank 2 over the integral domain
Multivariate Polynomial Ring in x, y, z, w over Rational Field
sage: r.differential(2)
Free module morphism defined as left-multiplication by the matrix
[-y x]
Expand Down Expand Up @@ -476,6 +490,31 @@ def differential(self, i):
[-z^2 + y*w]
[ y*z - x*w]
[-y^2 + x*z]

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: S = P2.coordinate_ring()
sage: I = S.ideal(0)
sage: C = I.graded_free_resolution(); C
S(0) <-- 0
sage: C[1]
Ambient free module of rank 0 over the integral domain
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: C[0]
Ambient free module of rank 1 over the integral domain
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: C.differential(1)
Free module morphism defined as left-multiplication by the matrix
[]
Domain: Ambient free module of rank 0 over the integral domain
Multivariate Polynomial Ring in x, y, z over Rational Field
Codomain: Ambient free module of rank 1 over the integral domain
Multivariate Polynomial Ring in x, y, z over Rational Field
sage: C.differential(1).matrix()
[]
sage: C.differential(1).matrix().dimensions()
(1, 0)
"""
if i < 0:
raise IndexError('invalid index')
Expand All @@ -484,14 +523,17 @@ def differential(self, i):
return self._initial_differential
except AttributeError:
raise ValueError('0th differential map undefined')
elif i == self._length + 1:
s = FreeModule(self._base_ring, 0)
t = FreeModule(self._base_ring, self._maps[i - 2].ncols())
m = s.hom(0, t)
elif i > self._length + 1:
s = FreeModule(self._base_ring, 0)
t = FreeModule(self._base_ring, 0)
m = s.hom(0, t)
m = s.hom(0, t, side='right')
elif i == self._length + 1:
s = FreeModule(self._base_ring, 0)
if self._length > 0:
t = FreeModule(self._base_ring, self._maps[i - 2].ncols())
else:
t = self._initial_differential.domain()
m = s.hom(0, t, side='right')
else:
s = FreeModule(self._base_ring, self._maps[i - 1].ncols())
t = FreeModule(self._base_ring, self._maps[i - 1].nrows())
Expand Down