decomposition theory

Let 𝒫\mathcal{P} be a strictly full subcategory of a Grothendieck category π’ž\mathcal{C} and SS a set. A predecomposition theory is a function which to each MM in 𝒫\mathcal{P} gives a subset Ξ“(M)βŠ‚S\Gamma(M)\subset S such that

(i) Ξ“(M)=0\Gamma(M) = 0 iff M=0M = 0

(ii) For any exact sequence

0→M′→M→M″→0 0\to M' \to M\to M'' \to 0

in π’ž\mathcal{C}, with Mβˆˆπ’«M\in \mathcal{P}, Ξ“(Mβ€²)βŠ‚Ξ“(M)βŠ‚Ξ“(Mβ€²)βˆͺΞ“(Mβ€³)\Gamma(M')\subset \Gamma(M) \subset \Gamma(M')\cup\Gamma(M'').

(iii) If M=βˆ‘ iM iM = \sum_i M_i then Ξ“(M)=βˆͺ iΞ“(M i)\Gamma(M)=\cup_i \Gamma(M_i).

(iv) If Mβ€²M' is an essential subobject of M∈Ob(𝒫)M\in Ob(\mathcal{P}), then Ξ“(Mβ€²)=Ξ“(M)\Gamma(M') = \Gamma(M). One often writes, by abuse of notation, Ξ“:𝒫→S\Gamma : \mathcal{P}\to S. For any Mβˆˆπ’«M\in\mathcal{P}, the elements of Ξ“(M)\Gamma(M) are called Ξ“\Gamma-associates of MM.

A class of examples are spectral predecomposition theories.

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

A subobject Mβ€²βŠ‚MM'\subset M is Ξ“\Gamma-irreducible if Mβ€²/MM'/M is Ξ“\Gamma-coirreducible.

A predecomposition theory Ξ“:𝒫→M\Gamma:\mathcal{P}\to M is caleld a decomposition theory if any subobject Mβ€²M' of an object Mβˆˆπ’«M\in\mathcal{P} has a Ξ“\Gamma-decomposition, that is the set of subobjects {M i} i∈I\{M_i\}_{i\in I}, M iβŠ‚MM_i\subset M such that

(D1) ∩ i∈IM i=Mβ€²\cap_{i\in I} M_i = M'

(D2) for each ii, M iM_i is a Ξ“\Gamma-irreducible subobject

(D3) Ξ“(M/M i)βˆ©Ξ“(M/M j)=βˆ…\Gamma(M/M_i) \cap \Gamma(M/M_j) = \emptyset if iβ‰ ji\neq j

(D4) Ξ“(M i/Mβ€²)=Ξ“(M/Mβ€²)βˆ’Ξ“(M/M i)\Gamma(M_i/M') = \Gamma(M/M')-\Gamma(M/M_i)

(D5) Ξ“(M/Mβ€²)=βˆͺ iΞ“(M/M i)\Gamma(M/M') = \cup_i \Gamma(M/M_i)

Examples include primary decomposition theory, Lesieur-Croisot tertiary decomposition theory, Bourbaki’s 𝒫\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

Created on February 26, 2014 at 08:33:48. See the history of this page for a list of all contributions to it.