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trimmed the documentation of SpectralSequences further
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‎M2/Macaulay2/packages/SpectralSequences/docs.m2‎

Lines changed: 78 additions & 84 deletions
Original file line numberDiff line numberDiff line change
@@ -308,24 +308,21 @@ Node
308308
Text
309309
By multiplying a chain complex by successive powers of an ideal we obtain a filtered complex.
310310
Example
311-
B = QQ[a..d]
312-
J = ideal vars B
313-
C = res monomialCurveIdeal(B,{1,3,4})
314-
K = filteredComplex(J,C,4)
311+
S = QQ[a..d]
312+
J = ideal vars S
313+
C = res monomialCurveIdeal(S,{1,3,4})
314+
K = filteredComplex(J,C,2)
315315
Text
316316
Here are higher some pages of the associated spectral sequence:
317317
Example
318-
e = prune spectralSequence K
319-
e^2
320-
-- e^3
321-
-- e^3 .dd
322-
-- e^4
323-
-- e^4 .dd
324-
assert(all(keys support e^0, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,0)))
325-
assert(all(keys support e^1, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,1)))
326-
assert(all(keys support e^2, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,2)))
327-
assert(all(keys support e^3, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,3)))
328-
assert(all(keys support e^4, j -> isIsomorphism homologyIsomorphism(e,j#0,j#1,4)))
318+
E = prune spectralSequence K
319+
E^2
320+
E^3
321+
assert all(keys support E^0, j -> isIsomorphism homologyIsomorphism(E, j#0, j#1, 0))
322+
assert all(keys support E^1, j -> isIsomorphism homologyIsomorphism(E, j#0, j#1, 1))
323+
assert all(keys support E^2, j -> isIsomorphism homologyIsomorphism(E, j#0, j#1, 2))
324+
assert all(keys support E^3, j -> isIsomorphism homologyIsomorphism(E, j#0, j#1, 3))
325+
assert all(keys support E^4, j -> isIsomorphism homologyIsomorphism(E, j#0, j#1, 4))
329326

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Node
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Key
@@ -627,13 +624,14 @@ doc ///
627624
B = QQ[a..d];
628625
J = ideal vars B;
629626
C = res monomialCurveIdeal(B,{1,3,4});
630-
K = filteredComplex(J,C,4);
627+
K = filteredComplex(J,C,2);
631628
E = spectralSequence K
632629
Text
633-
To view pages or maps we proceed as follows (note we suppress the output of the E^0.dd command to prevent excessive output).
630+
To view pages or maps we proceed as follows.
634631
Example
635-
E^0
636-
E^0 .dd;
632+
E^2
633+
support E^2 .dd
634+
E^2 .dd _{0,1}
637635
E^infinity
638636
SeeAlso
639637
SpectralSequence
@@ -661,14 +659,10 @@ doc ///
661659
Description
662660
Text
663661
Returns the k-th page of the spectral sequence.
664-
665-
Consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
666-
the rational quartic space curve by successive powers of the irrelevant ideal.
667662
Example
668-
B = QQ[a..d];
669-
J = ideal vars B;
670-
C = res monomialCurveIdeal(B,{1,3,4});
671-
K = filteredComplex(J,C,4);
663+
S = QQ[a..d];
664+
C = koszulComplex vars S
665+
K = filteredComplex C
672666
Text
673667
Let $E$ be the spectral sequence determined by $K$.
674668
Example
@@ -767,10 +761,10 @@ doc ///
767761
obtained by multiplying the minimal free resolution of the rational
768762
quartic space curve by successive powers of the irrelevant ideal.
769763
Example
770-
B = QQ[a..d];
771-
J = ideal vars B;
772-
C = res monomialCurveIdeal(B,{1,3,4});
773-
K = filteredComplex(J,C,4);
764+
S = QQ[a..d];
765+
J = ideal vars S;
766+
C = res monomialCurveIdeal(S, {1,3,4});
767+
K = filteredComplex(J,C,2);
774768
Text
775769
Compare some pages of the non-pruned version of the spectral sequence with that of the pruned version.
776770
Example
@@ -832,10 +826,10 @@ doc ///
832826
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
833827
the rational quartic space curve by successive powers of the irrelevant ideal.
834828
Example
835-
B = QQ[a..d];
836-
J = ideal vars B;
837-
C = res monomialCurveIdeal(B,{1,3,4});
838-
K = filteredComplex(J,C,4);
829+
S = QQ[a..d];
830+
J = ideal vars S;
831+
C = res monomialCurveIdeal(S,{1,3,4});
832+
K = filteredComplex(J,C,2);
839833
Text
840834
The infinity page of the resulting spectral sequence is computed below.
841835
Example
@@ -929,8 +923,8 @@ doc ///
929923
> "hilbertPolynomial(Page)"
930924
> "SpectralSequencePage ^ List"
931925
Caveat
932-
The isomorphisms $4$ and $4$' are not explicitly
933-
part of the data type, although they can be obtained by using the command @TO"homologyIsomorphism"@.
926+
The isomorphisms above are not explicitly part of the data type,
927+
but they can be obtained by using the command @TO"homologyIsomorphism"@.
934928
SeeAlso
935929
SpectralSequence
936930
SpectralSequencePageMap
@@ -955,10 +949,10 @@ doc ///
955949
Consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
956950
the rational quartic space curve by successive powers of the irrelevant ideal.
957951
Example
958-
B = QQ[a..d];
959-
J = ideal vars B;
960-
C = res monomialCurveIdeal(B,{1,3,4});
961-
K = filteredComplex(J,C,4);
952+
S = QQ[a..d];
953+
J = ideal vars S;
954+
C = res monomialCurveIdeal(S, {1,3,4});
955+
K = filteredComplex(J,C,2);
962956
Text
963957
Let $E$ be the spectral sequence determined by $K$.
964958
Example
@@ -967,7 +961,7 @@ doc ///
967961
We now compute some pages.
968962
Example
969963
E^0
970-
E^1
964+
E^2
971965
E^infinity
972966
///
973967

@@ -988,15 +982,18 @@ doc ///
988982
Text
989983
This is the primitive spectral sequence page constructor.
990984
Example
991-
B = QQ[a..d];
992-
J = ideal vars B;
993-
C = res monomialCurveIdeal(B,{1,3,4});
994-
K = filteredComplex(J,C,4);
985+
S = QQ[a..d];
986+
J = ideal vars S;
987+
C = res monomialCurveIdeal(S,{1,3,4});
988+
K = filteredComplex(J,C,2);
995989
E = spectralSequence K
996990
Text
997-
To view pages and or maps we proceed, for example, as follows
991+
To view pages or maps we proceed as follows.
998992
Example
999-
E^0
993+
E^2
994+
support E^2 .dd
995+
E^2 .dd _{0,1}
996+
E^infinity
1000997
SeeAlso
1001998
spectralSequence
1002999
(spectralSequence, FilteredComplex)
@@ -1053,18 +1050,11 @@ doc ///
10531050
Description
10541051
Text
10551052
Returns the pruning maps which are cached in the process of pruning the spectral sequence page.
1056-
1057-
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
1058-
the rational quartic space curve by successive powers of the irrelevant ideal.
10591053
Example
1060-
B = QQ[a..d];
1061-
J = ideal vars B;
1062-
C = res monomialCurveIdeal(B,{1,3,4});
1063-
K = filteredComplex(J,C,4);
1064-
Text
1065-
We compute an example of a pruning map below.
1066-
Example
1067-
E = prune spectralSequence K;
1054+
S = QQ[a..d];
1055+
C = koszulComplex vars S
1056+
K = filteredComplex C
1057+
E = prune spectralSequence K
10681058
pruningMaps E^2
10691059
SeeAlso
10701060
(prune, SpectralSequence)
@@ -1084,25 +1074,25 @@ doc ///
10841074
L:List
10851075
E:SpectralSequencePage
10861076
Outputs
1087-
B:Matrix --Note!! The output should actually be a page!!
1077+
B:Page
10881078
Description
10891079
Text
1090-
Returns generators for the requested (multi)degree of the spectral sequence page. It is designed to extend
1091-
the function @TO"basis"@ which can be applied to modules, for instance.
1080+
Returns generators for the requested (multi)degree of the spectral sequence page.
1081+
It is designed to extend the function @TO"basis"@ which can be applied to modules, for instance.
10921082

10931083
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
10941084
the rational quartic space curve by successive powers of the irrelevant ideal.
10951085
Example
1096-
B = QQ[a..d];
1097-
J = ideal vars B;
1098-
C = res monomialCurveIdeal(B,{1,3,4});
1099-
K = filteredComplex(J,C,4);
1086+
S = QQ[a..d];
1087+
J = ideal vars S;
1088+
C = res monomialCurveIdeal(S, {1,3,4});
1089+
K = filteredComplex(J,C,2);
11001090
Text
11011091
We compute the degree $0$ piece of the $E^3$ page below.
11021092
Example
11031093
E = prune spectralSequence K;
11041094
E^3
1105-
basis(0,E^3)
1095+
basis(0, E^3)
11061096
SeeAlso
11071097
basis
11081098
///
@@ -1125,15 +1115,17 @@ doc ///
11251115
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
11261116
the rational quartic space curve by successive powers of the irrelevant ideal.
11271117
Example
1128-
B = QQ[a..d];
1129-
J = ideal vars B;
1130-
C = res monomialCurveIdeal(B,{1,3,4});
1131-
K = filteredComplex(J,C,4);
1118+
S = QQ[a..d];
1119+
J = ideal vars S;
1120+
C = res monomialCurveIdeal(S, {1,3,4});
1121+
K = filteredComplex(J,C,2);
11321122
Text
11331123
We compute the degree $0$ piece of the $E^3$ page below.
11341124
Example
11351125
E = prune spectralSequence K;
1136-
hilbertPolynomial(E^3)
1126+
hilbertPolynomial(E^0)
1127+
hilbertPolynomial(E^1)
1128+
hilbertPolynomial(E^infinity)
11371129
///
11381130

11391131
doc ///
@@ -1158,10 +1150,10 @@ doc ///
11581150
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
11591151
the rational quartic space curve by successive powers of the irrelevant ideal.
11601152
Example
1161-
B = QQ[a..d];
1162-
J = ideal vars B;
1163-
C = res monomialCurveIdeal(B,{1,3,4});
1164-
K = filteredComplex(J,C,4);
1153+
S = QQ[a..d];
1154+
J = ideal vars S;
1155+
C = res monomialCurveIdeal(S, {1,3,4});
1156+
K = filteredComplex(J,C,2);
11651157
Text
11661158
Compare some pruned and non-prunded pages the spectral sequence $E$ below.
11671159
Example
@@ -1243,15 +1235,17 @@ doc ///
12431235
As a specific example consider the filtered complex $K$ below, obtained by multiplying the minimal free resolution of
12441236
the rational quartic space curve by successive powers of the irrelevant ideal.
12451237
Example
1246-
B = QQ[a..d];
1247-
J = ideal vars B;
1248-
C = res monomialCurveIdeal(B,{1,3,4});
1249-
K = filteredComplex(J,C,4);
1238+
S = QQ[a..d];
1239+
J = ideal vars S;
1240+
C = res monomialCurveIdeal(S, {1,3,4});
1241+
K = filteredComplex(J,C,2);
12501242
Text
12511243
We compute an example of a pruning map below.
12521244
Example
12531245
E = prune spectralSequence K;
1254-
pruningMaps E^2
1246+
D = E^2 .dd
1247+
support D
1248+
D_{0,1}
12551249
SeeAlso
12561250
Page
12571251
SpectralSequencePageMap
@@ -1326,10 +1320,10 @@ doc ///
13261320
Text
13271321
The notation @TT "P.dd_{p,q}"@ returns the map with source the homological {p,q} term on a spectral sequence page.
13281322
Example
1329-
B = QQ[a..d];
1330-
J = ideal vars B;
1331-
C = res monomialCurveIdeal(B,{1,2,3});
1332-
K = filteredComplex(J,C,4);
1323+
S = QQ[a..d];
1324+
J = ideal vars S;
1325+
C = res monomialCurveIdeal(S, {1,2,3});
1326+
K = filteredComplex(J,C,2);
13331327
Text
13341328
We compute a map on the third page of the spectral sequence associated to $K$.
13351329
Example

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