diff --git a/M2/Macaulay2/packages/AInfinity.m2 b/M2/Macaulay2/packages/AInfinity.m2
index 0fe2366f48..9cc12c749f 100644
--- a/M2/Macaulay2/packages/AInfinity.m2
+++ b/M2/Macaulay2/packages/AInfinity.m2
@@ -126,7 +126,7 @@ d1d1 := hashTable for i from min B to max B2 -2 list
 
 D := map(labeledTensorComplex{A0,B},B2,d1d1, Degree => -2);
 assert (isComplexMap D);
-m0 := nullHomotopy D;
+m0 := nullHomotopy(D, FreeToExact => true);
 for i from 4 to limit do(
     (C,K) := componentsAndIndices B2_i;
     for k in K do (
@@ -212,7 +212,7 @@ for i from lo+1 to hi list
 	 (A0**G).dd_(i-2)*D_i - D_(i-1)*BG.dd_i
 	 );
 
-m0 := nullHomotopy D;
+m0 := nullHomotopy(D, FreeToExact => true);
 for i from 2 to min(limit, 1+(concentration G)_1) do( --was just 1+(concentration G)_1
     (C,K) := componentsAndIndices BG_i;
     for k in K do (
@@ -386,7 +386,7 @@ p = new MutableHashTable from
 for i from 2 to length G list i=>(presentation R ** G_(i-2))*(BG_i)^[{2,i-2}]
 p#2
 p#3
-nullHomotopy map(G[-2],BG,p)
+nullHomotopy(map(G[-2],BG,p), FreeToExact => true)
 G[-2]
 BG
 
@@ -646,7 +646,7 @@ if n >= 2 then (
     d1d1 := hashTable for i from min B to max B list 
        i+2 => (d1**id_(B_i))*(B2_(i+2))^[{2,i}] - (id_(B_i)**d1)*(B2_(i+2))^[{i,2}];
     D := map(A0**B,B2,d1d1, Degree => -2);
-    m0 := nullHomotopy D;
+    m0 := nullHomotopy(D, FreeToExact => true);
 
     tlist := flatten for i from 4 to 1+(concentration B)_1 list(
         (C,K) := componentsAndIndices B2_i;
diff --git a/M2/Macaulay2/packages/Complexes.m2 b/M2/Macaulay2/packages/Complexes.m2
index 1bdbd03347..f8f6e8bc95 100644
--- a/M2/Macaulay2/packages/Complexes.m2
+++ b/M2/Macaulay2/packages/Complexes.m2
@@ -63,6 +63,7 @@ export {
     "yonedaMap'",
     "yonedaProduct",
     -- Option names
+    "FreeToExact", -- used in nullHomotopy
     "OverField",
     "OverZZ",
     "Homogenization",
diff --git a/M2/Macaulay2/packages/Complexes/ChainComplexMap.m2 b/M2/Macaulay2/packages/Complexes/ChainComplexMap.m2
index 2329e95257..478f2457e4 100644
--- a/M2/Macaulay2/packages/Complexes/ChainComplexMap.m2
+++ b/M2/Macaulay2/packages/Complexes/ChainComplexMap.m2
@@ -1068,9 +1068,6 @@ coimage ComplexMap := Complex => f -> (
 -- homotopy --------------------------------------------------------
 --------------------------------------------------------------------
 isNullHomotopyOf = method()
-isNullHomotopic = method()
-nullHomotopy = method() -- this function attempts to construct one, might fail
-
 isNullHomotopyOf(ComplexMap, ComplexMap) := (h, f) -> (
     -- returns true if h is a null homotopy for f : C --> D.
     -- if debugLevel > 0, then more info as to where it is not, is given
@@ -1104,12 +1101,20 @@ isNullHomotopyOf(ComplexMap, ComplexMap) := (h, f) -> (
         )
     )
 
--- TODO: we are keeping this version so that we may compare the 
---   more general version with this version, at a later date.
-nullHomotopyFreeSource = f -> (
-    -- f:ComplexMap
-    -- key assumption: 'source f' is a complex of free modules
+isNullHomotopic = method()
+isNullHomotopic ComplexMap := Boolean => f -> (
+    g := homomorphism' f;
+    H := target g; 
+    d := degree f;
+    g1 := g_0 // dd^H_(d+1); 
+    g_0 == dd^H_(d+1) * g1
+    )
+
+nullHomotopyFreeToExact = f -> (
+    -- key assumption: 'source f' is a complex of free modules AND the target is exact
     -- result is a ComplexMap h : C --> D, of degree degree(f)+1
+    if not isFree source f then error "expected source of complex map to be free";
+    -- Note: we do not check that the target is exact!
     C := source f;
     D := target f;
     deg := degree f + 1;
@@ -1130,21 +1135,9 @@ nullHomotopyFreeSource = f -> (
     map(D, C, new HashTable from hs, Degree => deg)
     )
 
-isNullHomotopic ComplexMap := Boolean => f -> (
-    g := homomorphism' f;
-    H := target g; 
-    d := degree f;
-    g1 := g_0 // dd^H_(d+1); 
-    g_0 == dd^H_(d+1) * g1
-    )
-
-nullHomotopy ComplexMap := ComplexMap => f -> (
-    -- we check that the source is free, as that can be much faster
-    -- TODO: nullHomotopy should perhaps be hook-ified.
-    -- The following code might require that the source is free
-    --  and the target is exact?
-    --result := if isFree source f then nullHomotopyFreeSource f;
-    --if result =!= null then return result;
+nullHomotopy = method(Options => true) -- this function attempts to construct one, might fail
+nullHomotopy ComplexMap := ComplexMap => {FreeToExact => false} >> opts -> f -> (
+    if opts.FreeToExact then return nullHomotopyFreeToExact f;
     g := homomorphism' f;
     H := target g; 
     d := degree f;
diff --git a/M2/Macaulay2/packages/Depth.m2 b/M2/Macaulay2/packages/Depth.m2
index a9a0ef251c..6beb0255d5 100644
--- a/M2/Macaulay2/packages/Depth.m2
+++ b/M2/Macaulay2/packages/Depth.m2
@@ -648,7 +648,7 @@ doc///
 	  (depth, Ideal, Module)
 	  (depth, Module)
           (depth, Ideal, Ideal)
-     Headline 
+     Headline
           computes the depth of a ring
      Usage
           d = depth(I,M)
diff --git a/M2/Macaulay2/packages/Macaulay2Doc/shared.m2 b/M2/Macaulay2/packages/Macaulay2Doc/shared.m2
index a31095f933..3d777937ea 100644
--- a/M2/Macaulay2/packages/Macaulay2Doc/shared.m2
+++ b/M2/Macaulay2/packages/Macaulay2Doc/shared.m2
@@ -15,6 +15,9 @@ document { Key => isVeryAmple,  methodstr, SeeAlso => {
 	"Divisor::isVeryAmple(WeilDivisor)", "Polyhedra::isVeryAmple(Polyhedron)",
 	"PositivityToricBundles::isVeryAmple(ToricVectorBundleKlyachko)",
 	"NormalToricVarieties::isVeryAmple(ToricDivisor)" } }
+document { Key => isNormal,      methodstr, SeeAlso => { "Polyhedra::isNormal(Polyhedron)","IntegralClosure::isNormal(Ring)"} }
+document { Key => normalCone,      methodstr, SeeAlso => { "Polyhedra::normalCone(Polyhedron,Polyhedron)","ReesAlgebra::normalCone(Ideal)"} }
+
 
 document { Key => cone,
     Headline => "mapping cone or polyhedral cone",
diff --git a/M2/Macaulay2/packages/ReesAlgebra.m2 b/M2/Macaulay2/packages/ReesAlgebra.m2
index d6b3b58f61..79f1cbc254 100644
--- a/M2/Macaulay2/packages/ReesAlgebra.m2
+++ b/M2/Macaulay2/packages/ReesAlgebra.m2
@@ -57,7 +57,8 @@ check "ReesAlgebra"
 *-
 
 export{
-  "analyticSpread", 
+  "analyticSpread",
+  "associatedGradedRing",
   "distinguished",
   "intersectInP",
   "isLinearType", 
@@ -78,7 +79,6 @@ export{
   "expectedReesIdeal",
   "PlaneCurveSingularities",
   --synonyms
-  "associatedGradedRing", -- => "normalCone",
   "reesAlgebraIdeal" => "reesIdeal",
   "Trim" -- option in reesIdeal
   }
@@ -207,26 +207,6 @@ isLinearType(Module, RingElement):= o-> (N,a)->(
      J := ideal((vars S) * P);
      ((gens I) % J) == 0)
 
--- normalCone = method(TypicalValue => Ring, 
---     	    Options => {
--- 	  DegreeLimit => {},
--- 	  BasisElementLimit => infinity,
--- 	  PairLimit => infinity,
--- 	  MinimalGenerators => true,
--- 	  Strategy => null,
--- 	  Variable => "w"
--- 	  }
--- )
--- normalCone(Ideal) := o -> I -> (
---      RI := reesAlgebra(I,o);
---      RI/promote(I,RI)
---      )
-
--- normalCone(Ideal, RingElement) := o -> (I,a) -> (
---      RI := reesAlgebra(I,a,o);
---      RI/promote(I,RI)     
---      )
-
 normalConeOptions = {
 	  DegreeLimit => {},
 	  BasisElementLimit => infinity,
@@ -236,7 +216,6 @@ normalConeOptions = {
 	  Variable => "w"
 	  }
 
-
 normalCone Ideal := Ring => normalConeOptions >> o -> I -> (
      RI := reesAlgebra(I,o);
      RI/promote(I,RI)
@@ -248,8 +227,8 @@ normalCone(Ideal, RingElement) := Ring => normalConeOptions >> o  -> (I,a) -> (
      )
 
 associatedGradedRing = method(Options => normalConeOptions)
-associatedGradedRing Ideal := o -> I -> normalCone(I, o)
-associatedGradedRing(Ideal, RingElement) := o -> (I,a) -> normalCone(I, a, o)
+associatedGradedRing Ideal := Ring => o -> I -> normalCone(I, o)
+associatedGradedRing(Ideal, RingElement) := Ring => o -> (I,a) -> normalCone(I, a, o)
 
 multiplicity = method(
     	    Options => {
@@ -1280,11 +1259,16 @@ doc ///
   Key
     (normalCone, Ideal, RingElement)
     (normalCone, Ideal)
+    associatedGradedRing
+    (associatedGradedRing, Ideal)
+    (associatedGradedRing, Ideal, RingElement)
   Headline
     The normal cone of a subscheme
   Usage
     normalCone I
     normalCone(I,f)
+    associatedGradedRing I
+    associatedGradedRing(I,f)
   Inputs
     I:Ideal
     f:RingElement
@@ -2000,7 +1984,8 @@ doc ///
     [reesAlgebra,Strategy]
     [isLinearType,Strategy]
     [isReduction, Strategy]    	  
-    [multiplicity, Strategy]    	  
+    [multiplicity, Strategy]
+    [associatedGradedRing, Strategy]
     [specialFiberIdeal, Strategy]    	  
     [specialFiber, Strategy]    	  
     [analyticSpread, Strategy]    	  
@@ -2044,6 +2029,7 @@ doc ///
     [specialFiber, PairLimit]
     [specialFiberIdeal, PairLimit]
     [multiplicity, PairLimit]
+    [associatedGradedRing, PairLimit]
     [isReduction, PairLimit]
     [isLinearType,PairLimit]
     [reesAlgebra,PairLimit]
@@ -2080,6 +2066,7 @@ doc ///
     [specialFiber, MinimalGenerators]
     [specialFiberIdeal, MinimalGenerators]
     [multiplicity, MinimalGenerators]
+    [associatedGradedRing, MinimalGenerators]
     [isReduction, MinimalGenerators]
     [isLinearType,MinimalGenerators]
     [reesAlgebra,MinimalGenerators]
@@ -2116,6 +2103,7 @@ doc ///
     [analyticSpread, BasisElementLimit]
     [specialFiber, BasisElementLimit]
     [multiplicity, BasisElementLimit]
+    [associatedGradedRing, BasisElementLimit]
     [isReduction, BasisElementLimit]
     [isLinearType,BasisElementLimit]
     [reesAlgebra,BasisElementLimit]
@@ -2152,6 +2140,7 @@ doc ///
     [analyticSpread, DegreeLimit]
     [specialFiber, DegreeLimit]
     [multiplicity, DegreeLimit]
+    [associatedGradedRing, DegreeLimit]
     [isReduction, DegreeLimit]
     [isLinearType,DegreeLimit]
     [reesAlgebra,DegreeLimit]