sagemathgh-37132: Fix corners cases of free resolutions
    
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as seen in
```sage
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]
------------------------------------------------------------------------
---
IndexError                                Traceback (most recent call
last)
Cell In[5], line 1
----> 1 C[Integer(0)]

File ~/GitHub/sage/src/sage/homology/free_resolution.py:426, in
FiniteFreeResolution.__getitem__(self, i)
    424     F = FreeModule(self._base_ring, 0)
    425 elif i == self._length:
--> 426     F = FreeModule(self._base_ring, self._maps[i - 1].ncols())
    427 else:
    428     F = FreeModule(self._base_ring, self._maps[i].nrows())

IndexError: list index out of range
sage: C.differential(1)
------------------------------------------------------------------------
---
IndexError                                Traceback (most recent call
last)
Cell In[6], line 1
----> 1 C.differential(Integer(1))

File ~/GitHub/sage/src/sage/homology/free_resolution.py:489, in
FiniteFreeResolution.differential(self, i)
    487 elif i == self._length + 1:
    488     s = FreeModule(self._base_ring, 0)
--> 489     t = FreeModule(self._base_ring, self._maps[i - 2].ncols())
    490     m = s.hom(0, t)
    491 elif i > self._length + 1:

IndexError: list index out of range
```


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### ⌛ Dependencies

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URL: sagemath#37132
Reported by: Kwankyu Lee
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/homology/free_resolution.py

@@ -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
@@ -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:
@@ -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:
@@ -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]
@@ -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')
@@ -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())