In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between 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. The syzygy module depends on the choice of a generating set, but it is unique up to the direct sum with a free module. That is, if 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 and such that and are isomorphic.
Higher order syzygy modules are defined recursively: a first syzygy module of a module M is simply its syzygy module. For k > 1, a kth syzygy module of M is a syzygy module of a (k – 1)-th syzygy module. Hilbert's syzygy theorem states that, if is a polynomial ring in n indeterminates over a field, then every nth syzygy module is free. The case n = 0 is the fact that every finite dimensional vector space has a basis, and the case n = 1 is the fact that K[x] is a principal ideal domain and that every submodule of a finitely generated free K[x] module is also 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.
If a and b are two elements of the commutative ring R, then (b, –a) is a relation that is said trivial. The module of trivial relations of an ideal is the submodule of the first syzygy module of the ideal that is generated by the trivial relations between the elements of a generating set of an ideal.
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In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings.
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of s of an ), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions.
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A. The appellation regular is justified by the geometric meaning.