Edit docstrings attached to free resolutions

src/sage/homology/free_resolution.py

@@ -7,8 +7,8 @@
 
 .. MATH::
 
-    R^{n_0} \xleftarrow{d_1}  R^{n_1} \xleftarrow{d_2}
-    \cdots \xleftarrow{d_k} R^{n_k} \xleftarrow{d_{k+1}} 0
+    0 \rightarrow R^{n_k} \xrightarrow{d_k}
+    \cdots \xrightarrow{d_2} R^{n_1} \xrightarrow{d_1} R^{n_0}
 
 terminating with a zero module at the end that is exact (all homology groups
 are zero) such that the image of `d_1` is `M`.
@@ -87,8 +87,8 @@ class FreeResolution(SageObject, metaclass=ClasscallMetaclass):
 
     .. MATH::
 
-        R^{n_1} \xleftarrow{d_1}  R^{n_1} \xleftarrow{d_2}
-        \cdots \xleftarrow{d_k} R^{n_k} \xleftarrow{d_{k+1}} \cdots
+        \cdots \rightarrow R^{n_k} \xrightarrow{d_k}
+        \cdots \xrightarrow{d_2} R^{n_1} \xrightarrow{d_1} R^{n_0}
 
     that is exact (all homology groups are zero) such that the image
     of `d_1` is `M`.
@@ -792,14 +792,14 @@ class FiniteFreeResolution_singular(FiniteFreeResolution):
     The available algorithms and the corresponding Singular commands
     are shown below:
 
-        ============= ============================
-        algorithm     Singular commands
-        ============= ============================
-        ``minimal``   ``mres(ideal)``
-        ``shreyer``   ``minres(sres(std(ideal)))``
-        ``standard``  ``minres(nres(std(ideal)))``
-        ``heuristic`` ``minres(res(std(ideal)))``
-        ============= ============================
+    ============= ============================
+    algorithm     Singular commands
+    ============= ============================
+    ``minimal``   ``mres(ideal)``
+    ``shreyer``   ``minres(sres(std(ideal)))``
+    ``standard``  ``minres(nres(std(ideal)))``
+    ``heuristic`` ``minres(res(std(ideal)))``
+    ============= ============================
 
     EXAMPLES::
 

src/sage/homology/graded_resolution.py

@@ -97,16 +97,16 @@ class GradedFiniteFreeResolution(FiniteFreeResolution):
 
     - ``module`` -- a homogeneous submodule of a free module `M` of rank `n`
       over `S` or a homogeneous ideal of a multivariate polynomial ring `S`
+
     - ``degrees`` -- (default: a list with all entries `1`) a list of integers
       or integer vectors giving degrees of variables of `S`
+
     - ``shifts`` -- a list of integers or integer vectors giving shifts of
       degrees of `n` summands of the free module `M`; this is a list of zero
       degrees of length `n` by default
-    - ``name`` -- a string; name of the base ring
 
-    .. WARNING::
+    - ``name`` -- a string; name of the base ring
 
-        This does not check that the module is homogeneous.
     """
     def __init__(self, module, degrees=None, shifts=None, name='S', **kwds):
         r"""
@@ -317,10 +317,6 @@ class GradedFiniteFreeResolution_free_module(GradedFiniteFreeResolution, FiniteF
     r"""
     Graded free resolution of free modules.
 
-    .. WARNING::
-
-        This does not check that the module is homogeneous.
-
     EXAMPLES::
 
         sage: from sage.homology.free_resolution import FreeResolution
@@ -434,8 +430,10 @@ class GradedFiniteFreeResolution_singular(GradedFiniteFreeResolution, FiniteFree
 
     - ``algorithm`` -- Singular algorithm to compute a resolution of ``ideal``
 
-    If ``module`` is an ideal of `S`, it is considered as a submodule of a
-    free module of rank `1` over `S`.
+    OUTPUT: a graded minimal free resolution of ``ideal``
+
+    If ``module`` is an ideal of `S`, it is considered as a submodule of a free
+    module of rank `1` over `S`.
 
     The degrees given to the variables of `S` are integers or integer vectors of
     the same length. In the latter case, `S` is said to be multigraded, and the
@@ -446,25 +444,18 @@ class GradedFiniteFreeResolution_singular(GradedFiniteFreeResolution, FiniteFree
     rank one over `S`, denoted `S(-d)` with shift `d`.
 
     The computation of the resolution is done by using ``libSingular``.
-    Different Singular algorithms can be chosen for best performance.
-
-    OUTPUT: a graded minimal free resolution of ``ideal``
-
-    The available algorithms and the corresponding Singular commands are shown
+    Different Singular algorithms can be chosen for best performance. The
+    available algorithms and the corresponding Singular commands are shown
     below:
 
-        ============= ============================
-        algorithm     Singular commands
-        ============= ============================
-        ``minimal``   ``mres(ideal)``
-        ``shreyer``   ``minres(sres(std(ideal)))``
-        ``standard``  ``minres(nres(std(ideal)))``
-        ``heuristic`` ``minres(res(std(ideal)))``
-        ============= ============================
-
-    .. WARNING::
-
-        This does not check that the module is homogeneous.
+    ============= ============================
+    algorithm     Singular commands
+    ============= ============================
+    ``minimal``   ``mres(ideal)``
+    ``shreyer``   ``minres(sres(std(ideal)))``
+    ``standard``  ``minres(nres(std(ideal)))``
+    ``heuristic`` ``minres(res(std(ideal)))``
+    ============= ============================
 
     EXAMPLES::