In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graph...In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graphs. The organized reduction pairs lead to sequences of projection maps that reduce the number of generators while preserving the homology groups of the original chain complex. This sequencing of reduction pairs allows updating the boundary information in a single step for a whole set of reductions, which shows impressive gains in computational performance compared to existing methods. In addition, our method gives the homology generators for a small additional cost.展开更多
For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology func...For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology functor Tor^X,W, relative homology functors Tor^W and Tor^X. Many results in Iacob [Comm. Algebra, 35, 1589-1606 （2007）] and Liang [Algebr. Represent. Theory, 16, 1541-1560 （2013）] are generalized and improved.展开更多
文摘In this paper, we propose a new algorithm to compute the homology of a finitely generated chain complex. Our method is based on grouping several reductions into structures that can be encoded as directed acyclic graphs. The organized reduction pairs lead to sequences of projection maps that reduce the number of generators while preserving the homology groups of the original chain complex. This sequencing of reduction pairs allows updating the boundary information in a single step for a whole set of reductions, which shows impressive gains in computational performance compared to existing methods. In addition, our method gives the homology generators for a small additional cost.
基金Supported by National Natural Science Foundation of China(Grant Nos.11301240,11401475)Natural Science Foundation of Chongqing(cstc2017jcyjAX0298)
文摘For two classes of right R-modules W, X such that P W X, where P is the class of projective right R-modules, we show that there is an Avramov-Martsinkovsky type exact sequence with generalized Tate homology functor Tor^X,W, relative homology functors Tor^W and Tor^X. Many results in Iacob [Comm. Algebra, 35, 1589-1606 （2007）] and Liang [Algebr. Represent. Theory, 16, 1541-1560 （2013）] are generalized and improved.
