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 matric ...
, 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 ...
or a module 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 (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 ...
(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 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. 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 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 variable ...
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 ...
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 princip ...
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 resolutions, 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 , 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, and be a left - module. 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 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 variable ...
K _1, \dots, x_n/math> over a field, then any th syzygy module is a free module.


Stable properties

Generally speaking, in the language of K-theory, a property is ''stable'' if it becomes true by making a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
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 (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 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 conte ...
:L\longrightarrow M \longrightarrow 0, where the left arrow is the linear map 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 lea ...
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 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 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 variable ...
in indeterminates over a field , then every free resolution is finite of length at most . The global dimension 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-Noethe ...
is either infinite, or the minimal such that every free resolution is finite of length at most . A commutative Noetherian ring is
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
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 generall ...
. 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 resultants 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 in his 1890 article, which contains three fundamental theorems on polynomials, Hilbert's syzygy theorem, Hilbert's basis theorem and Hilbert's Nullstellensatz. 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.


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