Syzygy (mathematics)
Encyclopedia
In mathematics
, a syzygy is a relation between the generators of a module
M. The set of all such relations is called the "first syzygy module of M". A relation between generators of the first syzygy module is called a "second syzygy" of M, and the set of all such relations is called the "second syzygy module of M". Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)th syzygy module of M. If M is finitely generated over a polynomial ring
over a field
, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert's syzygy theorem
). The syzygy modules of M are not unique, for they depend on the choice of generators at each step.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a syzygy is a relation between the generators of a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
M. The set of all such relations is called the "first syzygy module of M". A relation between generators of the first syzygy module is called a "second syzygy" of M, and the set of all such relations is called the "second syzygy module of M". Continuing in this way, we derive the nth syzygy module of M by taking the set of all relations between generators of the (n − 1)th syzygy module of M. If M is finitely generated over a polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert's syzygy theorem
Hilbert's syzygy theorem
In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it...
). The syzygy modules of M are not unique, for they depend on the choice of generators at each step.