nLab decomposition theory

Let $\mathcal{P}$ be a strictly full subcategory of a Grothendieck category $\mathcal{C}$ and $S$ a set. A predecomposition theory is a function which to each $M$ in $\mathcal{P}$ gives a subset $\Gamma(M)\subset S$ such that

(i) $\Gamma(M) = 0$ iff $M = 0$

(ii) For any exact sequence

$$0\to M' \to M\to M'' \to 0$$

in $\mathcal{C}$, with $M\in \mathcal{P}$, $\Gamma(M')\subset \Gamma(M) \subset \Gamma(M')\cup\Gamma(M'')$.

(iii) If $M = \sum_i M_i$ then $\Gamma(M)=\cup_i \Gamma(M_i)$.

(iv) If $M'$ is an essential subobject of $M\in Ob(\mathcal{P})$, then $\Gamma(M') = \Gamma(M)$. One often writes, by abuse of notation, $\Gamma : \mathcal{P}\to S$. For any $M\in\mathcal{P}$, the elements of $\Gamma(M)$ are called $\Gamma$-associates of $M$.

A class of examples are spectral predecomposition theories.

An object $M\in Ob(\mathcal{P})$ is said to be $\Gamma$-coirreducible if $\Gamma(M)$ contains exactly one element.

A subobject $M'\subset M$ is $\Gamma$-irreducible if $M'/M$ is $\Gamma$-coirreducible.

A predecomposition theory $\Gamma:\mathcal{P}\to M$ is caleld a decomposition theory if any subobject $M'$ of an object $M\in\mathcal{P}$ has a $\Gamma$-decomposition, that is the set of subobjects $\{M_i\}_{i\in I}$, $M_i\subset M$ such that

(D1) $\cap_{i\in I} M_i = M'$

(D2) for each $i$, $M_i$ is a $\Gamma$-irreducible subobject

(D3) $\Gamma(M/M_i) \cap \Gamma(M/M_j) = \emptyset$ if $i\neq j$

(D4) $\Gamma(M_i/M') = \Gamma(M/M')-\Gamma(M/M_i)$

(D5) $\Gamma(M/M') = \cup_i \Gamma(M/M_i)$

Examples include primary decomposition theory, Lesieur-Croisot tertiary decomposition theory, Bourbaki’s $\mathcal{P}$-primary decomposition theories…

• Nicolae Popescu, An introduction to Abelian categories with applications to rings and modules, Acad. Press 1973
• Joe W. Fisher, Harvey Wolff, Decomposition theories for abelian categories, Trans. Amer. Math. Soc. 182 (1973), 61–69 MR327870 doi

An earlier axiomatics in terms of pairs of objects is in

• John A. Riley, Axiomatic primary and tertiary decomposition theory, Trans. Amer. Math. Soc. 105 (1962), 177-201, doi