⊿ Point. {R} Glossary. ◢ Keyword: L. ◥ University. {q} PhD. {tr} Training. ⚫ UK. ↂ EndNote. ☝️ Weerakkody. Linear relation. Type of mathematical equation More precisely, if is a sequence of elements of R such that The relations between form a module. One is generally interested in the case where is a generating set of a finitely generated module M, in which case the module of the relations is often called a syzygy module of M. And are syzygy modules corresponding to two generating sets of the same module, then they are stably isomorphic, which means that there exist two free modules such that are isomorphic. Higher order syzygy modules are defined recursively: a first syzygy module of a module M is simply its syzygy module. Is a polynomial ring in n indeterminates over a field, then every nth syzygy module is free.
The construction of higher order syzygy modules is generalized as the definition of free resolutions, which allows restating Hilbert's syzygy theorem as a polynomial ring in n indeterminates over a field has global homological dimension n. of M is a sequence If without regard to Proposition—Let Proof. Between.