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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, a linear relation, or simply relation, between elements of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
or a module is a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficie ...
that has these elements as a solution. More precisely, if $e_1,\dots,e_n$ are elements of a (left) module over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
(the case of a vector space over a field is a special case), a relation between $e_1,\dots,e_n$ is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
$\left(f_1,\dots, f_n\right)$ of elements of such that :$f_1e_1+\dots+f_ne_n=0.$ The relations between $e_1,\dots,e_n$ form a module. One is generally interested in the case where $e_1,\dots,e_n$ is a generating set of a finitely generated module , in which case the module of the relations is often called a syzygy module of . 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 $S_1$ and $S_2$ 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 $L_1$ and $L_2$ such that $S_1\oplus L_1$ and $S_2\oplus L_2$ are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
. Higher order syzygy modules are defined recursively: a first syzygy module of a module is simply its syzygy module. For , a th syzygy module of is a syzygy module of a -th syzygy module. Hilbert's syzygy theorem states that, if
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
in indeterminates over a field, then every th syzygy module is free. The case is the fact that every finite dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
has a basis, and the case is the fact that is a principal ideal domain and that every submodule of a finitely generated free module is also free. The construction of higher order syzygy modules is generalized as the definition of
free resolution 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 objects of an abelian category), which is used to ...
s, which allows restating Hilbert's syzygy theorem as ''a polynomial ring in indeterminates over a field has global homological dimension . If and are two elements of the
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not s ...
, then 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. The concept of trivial relations can be generalized to higher order syzygy modules, and this leads to the concept of the Koszul complex of an ideal, which provides information on the non-trivial relations between the generators of an ideal.

# Basic definitions

Let be a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, and be a left - module. A '' linear relation'', or simply a ''relation'' between elements $x_1, \dots, x_k$ of is a sequence $\left(a_1, \dots, a_k\right)$ of elements of such that :$a_1x_1+\dots+ a_kx_k=0.$ If $x_1, \dots, x_k$ is a generating set of , the relation is often called a ''syzygy'' of . It makes sense to call it a syzygy of $M$ without regard to $x_1,..,x_k$ because, although the syzygy module depends on the chosen generating set, most of its properties are independent; see , below. If the ring is Noetherian, or, at least coherent, and if is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a ''second syzygy module'' of . Continuing this way one can define a th syzygy module for every positive integer . Hilbert's syzygy theorem asserts that, if is a finitely generated module over a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...

# Stable properties

Generally speaking, in the language of
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geome ...
, a property is ''stable'' if it becomes true by making a direct sum with a sufficiently large free module. A fundamental property of syzygies modules is that there are "stably independent" on choices of generating sets for involved modules. The following result is the basis of these stable properties. ''Proof.'' As $\$ is a generating set, each $y_i$ can be written $\textstyle y_i=\sum \alpha_x_j.$ This provides a relation $r_i$ between $x_1,\dots, x_m, y_1,\dots, y_n.$ Now, if $\left(a_1, \dots,a_m, b_1,\dots,b_n\right)$ is any relation, then $\textstyle r-\sum b_ir_i$ is a relation between the $x_1,\dots, x_m$ only. In other words, every relation between $x_1,\dots, x_m, y_1,\dots, y_n$ is a sum of a relation between $x_1,\dots, x_m,$ and a linear combination of the $r_i$s. It is straightforward to prove that this decomposition is unique, and this proves the result. $\blacksquare$ This proves that the first syzygy module is "stably unique". More precisely, given two generating sets $S_1$ and $S_2$ of a module , if $S_1$ and $S_2$ are the corresponding modules of relations, then there exist two free modules $L_1$ and $L_2$ such that $S_1\oplus L_1$ and $S_2\oplus L_2$ are isomorphic. For proving this, it suffices to apply twice the preceding proposition for getting two decompositions of the module of the relations between the union of the two generating sets. For obtaining a similar result for higher syzygy modules, it remains to prove that, if is any module, and is a free module, then and have isomorphic syzygy modules. It suffices to consider a generating set of that consists of a generating set of and a basis of . For every relation between the elements of this generating set, the coefficients of the basis elements of are all zero, and the syzygies of are exactly the syzygies of extended with zero coefficients. This completes the proof to the following theorem.

# Relationship with free resolutions

Given a generating set $g_1,\dots,g_n$ of an -module, one can consider a free module of of basis $G_1,\dots,G_n,$ where $G_1,\dots,G_n$ are new indeterminates. This defines an exact sequence :$L\longrightarrow M \longrightarrow 0,$ where the left arrow is the
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
that maps each $G_i$ to the corresponding $g_i.$ The kernel of this left arrow is a first syzygy module of . One can repeat this construction with this kernel in place of . Repeating again and again this construction, one gets a long exact sequence :$\cdots\longrightarrow L_k\longrightarrow L_ \longrightarrow \cdots\longrightarrow L_0 \longrightarrow M \longrightarrow 0,$ where all $L_i$ are free modules. By definition, such a long exact sequence is a
free resolution 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 objects of an abelian category), which is used to ...
of . For every , the kernel $S_k$ of the arrow starting from $L_$ is a th syzygy module of . It follows that the study of free resolutions is the same as the study of syzygy modules. A free resolution is ''finite'' of length if $S_n$ is free. In this case, one can take $L_n = S_n,$ and $L_k = 0$ (the zero module) for every . This allows restating Hilbert's syzygy theorem: If
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variabl ...
in indeterminates over a field , then every free resolution is finite of length at most . The
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
of a commutative
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
is either infinite, or the minimal such that every free resolution is finite of length at most . A commutative Noetherian ring is regular if its global dimension is finite. In this case, the global dimension equals its Krull dimension. So, Hilbert's syzygy theorem may be restated in a very short sentence that hides much mathematics: ''A polynomial ring over a field is a regular ring.''

# Trivial relations

In a commutative ring , one has always . This implies ''trivially'' that is a linear relation between and . Therefore, given a generating set $g_1, \dots,g_k$ of an ideal , one calls trivial relation or trivial syzygy every element of the submodule the syzygy module that is generated by these trivial relations between two generating elements. More precisely, the module of trivial syzygies is generated by the relations :$r_= \left(x_1,\dots,x_r\right)$ such that $x_i=g_j,$ $x_j=-g_i,$ and $x_h=0$ otherwise.

# History

The word ''syzygy'' came into mathematics with the work of
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
. In that paper, Cayley used it for in the theory of
resultant In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over t ...
s and discriminants. As the word syzygy was used in
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
to denote a linear relation between planets, Cayley used it to denote linear relations between minors of a matrix, such as, in the case of a 2×3 matrix: :$a\,\beginb&c\\e&f\end - b\,\begina&c\\d&f\end +c\,\begina&b\\d&e\end=0.$ Then, the word ''syzygy'' was popularized (among mathematicians) by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
in his 1890 article, which contains three fundamental theorems on polynomials, Hilbert's syzygy theorem, Hilbert's basis theorem and
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
. In his article, Cayley makes use, in a special case, of what was laterSerre, Jean-Pierre Algèbre locale. Multiplicités. (French) Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, 11 Springer-Verlag, Berlin-New York 1965 vii+188 pp.; this is the published form of mimeographed notes from Serre's lectures at the College de France in 1958. called the Koszul complex, after a similar construction in differential geometry by the mathematician Jean-Louis Koszul.

# References

* * * {{cite book, author-link=David Eisenbud, last1=Eisenbud, first1=David, title=Commutative Algebra with a View Toward Algebraic Geometry, series=Graduate Texts in Mathematics, volume=150, publisher=Springer-Verlag, year=1995, isbn=0-387-94268-8, doi=10.1007/978-1-4612-5350-1 * David Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, 2005. category:Commutative algebra category:Homological algebra category:Linear algebra category:Polynomials