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ChainComplex ** FilteredComplex -- filtered tensor product of complexes

Description

Returns the two filtrations of the tensor product complex determined by the double complex. The following example illustrates the syntax.

i1 : A = QQ[x,y];
 
i2 : B = koszul vars A;
 
i3 : C = koszul vars A;
 
i4 : F' = (filteredComplex B) ** C

o4 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                           
          0           1           2           3           4

     0 : image | 1 | <-- image {1} | 1 0 0 0 | <-- image {2} | 1 0 0 0 0 0 | <-- image 0 <-- image 0
                               {1} | 0 1 0 0 |           {2} | 0 0 0 0 0 0 |                  
         0                     {1} | 0 0 0 0 |           {2} | 0 0 0 0 0 0 |     3           4
                               {1} | 0 0 0 0 |           {2} | 0 0 0 0 0 0 |
                                                         {2} | 0 0 0 0 0 0 |
                         1                               {2} | 0 0 0 0 0 0 |
                                                    
                                                   2

     1 : image | 1 | <-- image {1} | 1 0 0 0 | <-- image {2} | 1 0 0 0 0 0 | <-- image {3} | 1 0 0 0 | <-- image 0
                               {1} | 0 1 0 0 |           {2} | 0 1 0 0 0 0 |           {3} | 0 1 0 0 |      
         0                     {1} | 0 0 1 0 |           {2} | 0 0 1 0 0 0 |           {3} | 0 0 0 0 |     4
                               {1} | 0 0 0 1 |           {2} | 0 0 0 1 0 0 |           {3} | 0 0 0 0 |
                                                         {2} | 0 0 0 0 1 0 |      
                         1                               {2} | 0 0 0 0 0 0 |     3
                                                    
                                                   2

          1      4      6      4      1
     2 : A  <-- A  <-- A  <-- A  <-- A
                                      
         0      1      2      3      4

o4 : FilteredComplex
 
i5 : F'' = B ** (filteredComplex C)

o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 <-- image 0
                                                           
          0           1           2           3           4

     0 : image | 1 | <-- image {1} | 0 0 0 0 | <-- image {2} | 0 0 0 0 0 0 | <-- image 0 <-- image 0
                               {1} | 0 0 0 0 |           {2} | 0 0 0 0 0 0 |                  
         0                     {1} | 0 0 1 0 |           {2} | 0 0 0 0 0 0 |     3           4
                               {1} | 0 0 0 1 |           {2} | 0 0 0 0 0 0 |
                                                         {2} | 0 0 0 0 0 0 |
                         1                               {2} | 0 0 0 0 0 1 |
                                                    
                                                   2

     1 : image | 1 | <-- image {1} | 1 0 0 0 | <-- image {2} | 0 0 0 0 0 0 | <-- image {3} | 0 0 0 0 | <-- image 0
                               {1} | 0 1 0 0 |           {2} | 0 1 0 0 0 0 |           {3} | 0 0 0 0 |      
         0                     {1} | 0 0 1 0 |           {2} | 0 0 1 0 0 0 |           {3} | 0 0 1 0 |     4
                               {1} | 0 0 0 1 |           {2} | 0 0 0 1 0 0 |           {3} | 0 0 0 1 |
                                                         {2} | 0 0 0 0 1 0 |      
                         1                               {2} | 0 0 0 0 0 1 |     3
                                                    
                                                   2

          1      4      6      4      1
     2 : A  <-- A  <-- A  <-- A  <-- A
                                      
         0      1      2      3      4

o5 : FilteredComplex

See also

Ways to use this method:

The source of this document is in SpectralSequences.m2:3587:0.