Algebra of chain complexes I

Many algebraic concepts are applicable to chain complexes.

$\blacktriangleright$ Subcomplex: $$(S,\partial ^S)<(T,\partial ^T),$$ provided $$S_k<T_k, \forall k,$$ $$\partial _k ^S = \partial _k^T |_{S_k}, \forall k.$$

$\blacktriangleright$ Inclusion chain map for subcomplex: $$(S,\partial ^S)<(T,\partial ^T),$$ $$i: (S,\partial ^S) \hookrightarrow (T,\partial ^T),$$ defined by $$i_k = \left(i_k: S_k \hookrightarrow T_k\right), \forall k,$$ $$i_{k-1}\partial _k^S = \partial _k^T i_k, \forall k.$$

$\blacktriangleright$ Restriction chain map for subcomplex and chain map: $$(S,\partial ^S)<(T,\partial ^T),$$ $$f: (T,\partial ^T) \rightarrow (Q,\partial ^Q),$$ $$f|_{(S,\partial ^S)}: (S,\partial ^S) \rightarrow (Q,\partial ^Q),$$ defined by $$(f|_{(S,\partial ^S)})_k = \left(f_k|_{S_k}: S_k \rightarrow T_k\right), \forall k,$$ $$f_{k-1}|_{S_{k-1}}\partial _k^S = \partial _k^T f_k|_{S_{k}}, \forall k.$$

$\blacktriangleright$ Quotient complex for subcomplex: $$(S,\partial ^S)<(T,\partial ^T),$$ $$(Q, \partial ^Q) = (T,\partial ^T) / (S,\partial ^S),$$ defined by $$Q_k = T_k / S_k, \forall k,$$ $$\partial _k^Q (s_k+S_k)=\partial _k^T (s_k) +S_{k-1}, \forall k.$$

$\blacktriangleright$ Kernel of chain map: $$f: (S,\partial ^S) \rightarrow (T,\partial ^T),$$ $$(Q, \partial ^Q) = \ker f,$$ defined by $$Q_k = \ker f_k,$$ $$\partial _k^{Q} =\partial _k^S |_{Q_k}, \forall k.$$

$\blacktriangleright$ Image of chain map: $$f: (S,\partial ^S) \rightarrow (T,\partial ^T),$$ $$(Q, \partial ^Q) =\text{im } f,$$ defined by $$Q_k = \text{im } f_k, \forall k,$$ $$\partial _k^{Q} =\partial _k^T |_{Q_k}, \forall k.$$

$\blacktriangleright$ Exact sequence of chain complexes and chain maps: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \ra{} & S^{m+1} & \ra{f^{m+1}} & S^{m} & \ra{f^{m}} & S^{m-1} & \ra{f^{m-1}} & S^{m-2} & \ra{} & ...,\\ \end{array} $$ provided $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccc} ... & \ra{} & S^{m+1}_k & \ra{f^{m+1}_k} & S^{m}_k & \ra{f^{m}_k} & S^{m-1}_k & \ra{f^{m-1}_k} & S^{m-2}_k & \ra{} & ...\\ \end{array} $$ is exact for all $k$.

$\blacktriangleright$ Intersection of subcomplexes: $$(S_i,\partial ^{S_i})<(Q,\partial ^Q),$$ $$(R,\partial ^R) = {\LARGE \cap } _i (S^i, \partial ^{S_i}),$$ defined by $$R_k = {\LARGE \cap } _i S_k^i, \forall k,$$ $$\partial _k^{R} =\partial _k^Q |_{R_k}, \forall k.$$

$\blacktriangleright$ Sum of subcomplexes: $$(S_i,\partial ^{S_i})<(Q,\partial ^Q),$$ $$(R,\partial ^R) = \sum _i (S^i, \partial ^{S_i}),$$ defined by $$R_k = \sum _i S_k^i, \forall k,$$ $$\partial _k^{R} =\partial _k^Q |_{R_k}, \forall k.$$

$\blacktriangleright$ Direct sum of complexes: $$(R,\partial ^R) = \bigoplus _i (S^i, \partial ^{S_i}),$$ defined by $$R_k =\bigoplus _i S_k^i, \forall k,$$ $$\partial _k^{R} =\bigoplus _i\partial ^i_k, \forall k.$$

$\blacktriangleright$ Tensor product of complexes: $$Q=S\otimes T$$ defined by $$Q_k = \bigoplus_{i+j=k} S_i \otimes T_j, \forall k,$$ $$\partial ^Q(a\otimes b)=\partial ^S a \otimes b + (-1)^{|a|} a \otimes \partial ^T b.$$