Linear relation

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 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 (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[x_1,\dots,x_n] is a polynomial ring in indeterminates over a field, then every th syzygy module is free.…

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 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 (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[x_1,\dots,x_n] is a polynomial ring in indeterminates over a field, then every th syzygy module is free. The case is the fact that every finite dimensional vector space 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 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 K[x_1, \dots, x_n] 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 with a sufficiently large free module. A fundamental property of syzygies modules is that there are "stably independent" of choices of generating sets for involved modules. The following result is the basis of these stable properties. Proof. As \{x_1,\dots, x_m\} is a generating set, each y_i can be written \textstyle y_i=\sum \alpha_{i,j}x_j. This provides a relation r_i between x_1,\dots, x_m, y_1,\dots, y_n. Now, if r=(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_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 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 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_{k-1} \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_{k-1} 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[x_1, \dots, x_n] is a polynomial ring in indeterminates over a field , then every free resolution is finite of length at most . The global dimension of a commutative Noetherian ring 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_{i,j}= (x_1,\dots,x_r) such that x_i=g_j, x_j=-g_i, and x_h=0 otherwise. a\,\begin{vmatrix}b&c\\e&f\end{vmatrix} - b\,\begin{vmatrix}a&c\\d&f\end{vmatrix} +c\,\begin{vmatrix}a&b\\d&e\end{vmatrix}=0. A particular question he studied concerns a set of equations that arise in Plücker embedding, which is an embedding of the (Grassmannian) space of planes G(2, \R^n) into the projective space \mathbb P(\wedge^2 V) . Using Plücker coordinates, the space \mathbb P(\wedge^2 V) can be written as \{[p_{1,2}, p_{1,3}, \dots, p_{n-1, n}]\} , which has \tfrac 12 n(n-1)-1 dimensions. Now, since G(2, \R^n) has 2(n-2) dimensions, it should be possible to write it as the intersection of \tfrac 12 (n-2)(n-3) hypersurfaces. However, they found that G(2, \R^n) is the intersection of the \tfrac 14 n(n-1)(n-2)(n-3) equations: Q_{ijkl} := p_{i j} p_{k l}-p_{i k} p_{j l}+p_{i l} p_{j k}=0 \quad(1 \leq i Thus, these equations must be dependent on each other when n \geq 5 . For example, when n = 5 , the space G(2, \R^5) is a codimension-3 subspace of \mathbb P(\wedge^2 \R^5) defined as the intersection of 5 equations, so there are 2 redundancies, which can be exhibited by the matrix equation: \begin{pmatrix} 0 & p_{12} & p_{13} & p_{14} & p_{15}\\ -p_{12} & 0 & p_{23} & p_{24} & p_{25}\\ -p_{13} & -p_{23} & 0 & p_{34} & p_{35}\\ -p_{14} & -p_{24} & -p_{34} & 0 & p_{45}\\ -p_{15} & -p_{25} & -p_{35} & -p_{45} & 0 \end{pmatrix} \begin{pmatrix} Q_1\\ Q_2\\ Q_3\\ Q_4\\ Q_5 \end{pmatrix} Q_2 & =p_{13} p_{45}-p_{14} p_{35}+p_{15} p_{34} \\ Q_3 & =p_{12} p_{45}-p_{14} p_{25}+p_{15} p_{24} \\ Q_4 & =p_{12} p_{35}-p_{13} p_{25}+p_{15} p_{23} \\ Q_5 & =p_{12} p_{34}-p_{13} p_{24}+p_{14} p_{23} . \end{aligned} In the subspace of \mathbb A^9 \subset \mathbb P(\wedge^2 \R^5) where p_{12} = 1 , the two linear syzygies Q_1=p_{23} Q_3+p_{24} Q_4+p_{25} Q_5, \quad Q_2=-p_{13} Q_3-p_{14} Q_4-p_{15} Q_5 show that if Q_3, Q_4, Q_5 = 0 , then the other two conditions are automatically satisfied. This is the redundancy. 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 later called the Koszul complex, after a similar construction in differential geometry by the mathematician Jean-Louis Koszul. ==Notes==

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