Linear Relation
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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 can ...
or a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
is a linear equation 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 calle ...
(f_1,\dots, f_n) 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 In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
of a
finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts in ...
, 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 In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
, which means that there exist two
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s 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 is ...
. 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 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 ...
states that, if R=K _1,\dots,x_n/math> is 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 variables) ...
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 can ...
has a basis, and the case is the fact that is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
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 defi ...
s, which allows restating Hilbert's syzygy theorem as ''a polynomial ring in indeterminates over a field has
global homological dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring (mathematics), ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a ho ...
. 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 sp ...
, 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 In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ...
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 Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
. A '' linear relation'', or simply a ''relation'' between elements x_1, \dots, x_k of is a sequence (a_1, \dots, a_k) of elements of such that :a_1x_1+\dots+ a_kx_k=0. If x_1, \dots, x_k is a
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
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 In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
, or, at least
coherent Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deri ...
, 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 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 ...
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 variables) ...
K _1, \dots, x_n/math> over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, then any th syzygy module is a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in ...
.


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 geometry, ...
, a property is ''stable'' if it becomes true by making a direct sum with a sufficiently large
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
. 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 (a_1, \dots,a_m, b_1,\dots,b_n) 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_is. 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 In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
of of basis G_1,\dots,G_n, where G_1,\dots,G_n are new indeterminates. This defines an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
: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 Map (mathematics), mapping V \to W between two vect ...
that maps each G_i to the corresponding g_i. The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
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 defi ...
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 In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The afore ...
) for every . This allows restating
Hilbert's syzygy theorem 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 ...
: If R=K _1, \dots, x_n/math> is 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 variables) ...
in indeterminates over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, 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-Noether ...
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 In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ...
. 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_= (x_1,\dots,x_r) 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. 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 (ove ...
s and discriminants. As the word syzygy was used in
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
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 a ...
in his 1890 article, which contains three fundamental theorems on polynomials,
Hilbert's syzygy theorem 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 ...
, 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 ...
. 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 In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ...
, after a similar construction in differential geometry by the mathematician
Jean-Louis Koszul Jean-Louis Koszul (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in ...
.


Notes


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