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Linear relation

In , particularly in linear algebra and , a linear , or simply a , among a finite set of elements f_1, \dots, f_t in a M over a R is a (a_1, \dots, a_t) \in R^t such that a_1 f_1 + \dots + a_t f_t = 0. These relations capture dependencies among the elements and form the syzygy module \mathrm{Syz}(f_1, \dots, f_t), which is the kernel of the surjection from the free module R^t to the submodule generated by the f_i. Linear relations are fundamental in the study of module presentations and free resolutions, where higher-order relations (syzygies on syzygies) build the resolution chain. They play a key role in , , and computational methods for determining properties like projective dimension.

Fundamentals

Definition

In module theory, a linear relation on elements e_1, \dots, e_n in an R-module M, where R is , is a tuple (f_1, \dots, f_n) \in R^n such that f_1 e_1 + \dots + f_n e_n = 0 in M. The set of all such linear relations forms the module of relations, which is a submodule of the free R^n. When the elements \{e_1, \dots, e_n\} generate a submodule of M, each linear relation is a , and the module of relations is denoted \mathrm{Syz}(e_1, \dots, e_n), the first syzygy module of these generators. This arises as the kernel of the surjective R- homomorphism R^n \to \langle e_1, \dots, e_n \rangle sending the standard basis vectors to e_1, \dots, e_n. Over a , where modules are vector spaces, linear relations reduce to the standard notion of linear dependence among . However, the general setting over rings allows for more complex structures, as modules need not be and relations may involve non-invertible coefficients.

Examples

A linear relation among elements of a provides intuition for the concept in more general modules. Consider the two-dimensional real \mathbb{R}^2 equipped with the vectors e_1 = (1,0) and e_2 = (0,1). These vectors are linearly independent, so the only linear relation they satisfy is the trivial one (0,0), yielding $0 \cdot e_1 + 0 \cdot e_2 = 0. In contrast, the set \{e_1, 2e_1\} is linearly dependent, with the non-trivial relation (2, -1) satisfying $2 \cdot e_1 + (-1) \cdot (2e_1) = 2e_1 - 2e_1 = 0. In the category of abelian groups, which are \mathbb{Z}-modules, linear relations appear as syzygies in module presentations. For the cyclic module \mathbb{Z}/6\mathbb{Z}, one presentation uses two generators corresponding to the images of 2 and 3 under the surjection \mathbb{Z}^2 \to \mathbb{Z}/6\mathbb{Z} given by the matrix (2 \ 3). The syzygy module, or module of relations, is generated by (3, -2), since $3 \cdot 2 + (-2) \cdot 3 = 6 - 6 = 0 \equiv 0 \pmod{6}. To verify step-by-step: multiply the first generator by 3 to get $3 \cdot 2 = 6 \equiv 0 \pmod{6}; multiply the second by -2 to get -2 \cdot 3 = -6 \equiv 0 \pmod{6}; their sum is zero in the module, confirming the relation holds in the matrix representation of the syzygy module as the kernel. Trivial relations like (0,0) always exist but do not contribute to the presentation. For modules over polynomial rings, relations often reflect ring structure. In the polynomial ring k[x,y] over a field k, the ideal I = (x,y) is generated by x and y. The first syzygy module \mathrm{Syz}(x,y) is the kernel of the surjection k[x,y]^2 \to I sending the standard basis to x and y. This kernel is free of rank 1, generated by the relation (y, -x), since y \cdot x + (-x) \cdot y = xy - xy = 0. This relation can be represented in matrix form as the column vector \begin{pmatrix} y \\ -x \end{pmatrix}, whose image under the presentation map is zero.

Types of Relations

Trivial relations

In commutative algebra, a linear relation among elements f_1, \dots, f_n generating a module M over a commutative ring R is an element (g_1, \dots, g_n) \in R^n such that \sum g_i f_i = 0 in M. Such a relation is called trivial if it arises from a direct dependency between a pair of generators; for two generators f and g, the prototypical trivial relation is (g, -f), since g \cdot f - f \cdot g = 0 holds identically due to the commutativity of R. More generally, for n > 2, trivial relations are those generated by such pairwise dependencies, forming the submodule of the syzygy module \mathrm{Syz}(f_1, \dots, f_n) spanned by elements like (f_j, -f_i, 0, \dots, 0) and (0, \dots, f_k, \dots, -f_i, \dots, 0) for distinct indices i, j, k. The submodule of trivial syzygies, often denoted \mathrm{TrivSyz}(f_1, \dots, f_n), is a submodule of the full module \mathrm{Syz}(f_1, \dots, f_n) = \ker(R^n \to M), generated explicitly by these obvious pairwise dependencies among the generators. This submodule captures the "expected" relations imposed solely by the ring's commutativity and the choice of generators, without reflecting deeper structural properties of M. In minimal presentations, distinguishing trivial syzygies allows for the isolation of non-trivial ones that reveal additional . Trivial relations are fully characterized in specific module types. For a free module M \cong R^k, the syzygy module vanishes entirely, so all (absent) relations are vacuously trivial, as there are no non-zero dependencies. In general modules, trivial relations correspond precisely to the syzygies induced by the presentation map to the quotient module M / N, where N is the submodule generated by higher-order relations; thus, they embed the relations of the ambient quotient without introducing new ones. This correspondence highlights how trivial syzygies preserve the dependency structure from the free cover. A concrete example arises in cyclic modules presented non-minimally with two generators. Consider M = R / (d) for some d \in R, redundantly presented as M \cong R^2 / \langle (r, -s) \rangle, where r e_1 - s e_2 = 0 in M (i.e., r and s satisfy r \cdot 1 = s \cdot 1 modulo d). Here, the relation (r, -s) is trivial, as it directly enforces the dependency between the two generators mapping to the single generator of the cyclic module, generating the entire syzygy module without non-trivial extensions. This case illustrates how trivial relations account for overgeneration in presentations of cyclic structures.

Non-trivial relations

Non-trivial linear relations in the context of module presentations are those dependencies among a generating set that lie outside the submodule generated by trivial relations. These relations indicate hidden structural dependencies that cannot be explained by basic actions alone. To detect minimal non-trivial syzygies, Gröbner bases provide an effective computational by yielding a for the syzygy through the reduction of S-polynomials and linear algebra over the . Complementarily, the resolves the and identifies non-trivial syzygies via its groups, where non-vanishing homology in relevant degrees signals additional relations beyond the acyclic trivial part. The significance of non-trivial relations lies in their revelation of module complexity, particularly for non-free modules over polynomial rings, where they demonstrate that a minimal generating set does not form a free basis and necessitate extended free resolutions to capture the full structure. In such cases, these relations quantify deviations from freeness, influencing homological invariants like Betti numbers and projective dimension. A representative example occurs in the R = k[x,y,z]/(xy - z^2), where the images \bar{x}, \bar{y}, \bar{z} generate the m = (\bar{x}, \bar{y}, \bar{z}). Beyond trivial Koszul relations, the defining induces a non-trivial ( \bar{y}, \bar{x}, -2\bar{z} ) in the first of m, satisfying \bar{y} \cdot \bar{x} + \bar{x} \cdot \bar{y} - 2\bar{z} \cdot \bar{z} = 2(\bar{x}\bar{y} - \bar{z}^2) = 0 in R (assuming not 2), highlighting the dependency in this generating set.

Properties

Stable properties

Stable properties of linear relations refer to those characteristics of the modules that remain invariant under changes in the generating set or certain ring operations. The module associated to a generating set of a M is unique up to stable , meaning that for two generating sets e1 and e2, the corresponding modules S1 and S2 satisfy S1 ⊕ F1 ≅ S2 ⊕ F2 for some free modules F1 and F2. This stability ensures that the essential relations are preserved regardless of the choice of generators. A key states that for a module M with a minimal generating set of n elements, the relation module Rel(M) decomposes as the of the syzygy module Syz() and a of n-1. This splitting highlights the of the relations, where the component accounts for trivial dependencies among the generators, while Syz() captures the non-trivial linear relations. The explicit is given by \text{Rel}(M) \cong \text{Syz}(e) \oplus R^{n-1}, where R is the underlying and is the minimal generating set. This is fundamental for understanding the homological of M. These properties exhibit invariance under localization of the . Specifically, localizing at a multiplicative set S yields a relation module for the localized module M_S that corresponds to the localization of the original relation module, preserving the stable isomorphism and splitting up to the localized free modules. This invariance allows the study of linear relations to be reduced to local cases without loss of essential information.

Minimal relations

In commutative algebra, a minimal relation refers to an element of a minimal generating set for the syzygy module of a module over a polynomial ring, where the syzygy module consists of relations among the generators of the original module. Specifically, in the context of a minimal free resolution of a finitely generated graded module M over a polynomial ring S = k[x_1, \dots, x_n], the minimal syzygies are the generators of the kernel of the differential map from the i-th free module F_i to F_{i-1}, chosen such that the resolution is minimal. A free resolution is minimal if the image of each differential lies in the homogeneous maximal ideal \mathfrak{m} = (x_1, \dots, x_n) times the target free module, ensuring that the matrices representing the differentials have no entries in k (i.e., no unit coefficients). These minimal relations form a basis without redundant elements and are typically obtained through algorithms that select a minimal set of generators for each syzygy module at successive steps of the resolution. Minimal relations exhibit key properties that make them fundamental to understanding module structure. The rank of the free module generated by minimal syzygies at each step equals the minimal number of generators required for that syzygy module, known as the Betti number \beta_i(M), which is an invariant of M. Since the differentials have entries in \mathfrak{m}, the coefficients of minimal relations contain no units from the base field k, preventing trivial or invertible adjustments that would not affect the kernel. This minimality ensures that the entire free resolution is unique up to isomorphism of complexes, providing a canonical way to study the homological properties of M. Computationally, minimal relations play a central role in algebraic software systems designed for . In Macaulay2, the syz function computes a matrix whose columns form a minimal generating set for the of a given of generators, automatically trimming redundant relations to produce the minimal basis. Similarly, in Singular, the syz command generates the , with options to compute minimal presentations via algorithms, facilitating the construction of minimal free resolutions for ideals and . These tools rely on methods or Schreyer frames to efficiently identify and reduce to minimal syzygies, enabling practical computations even for high-dimensional rings. A example illustrates minimal relations for the I = (x^2, xy, y^2) in the k[x,y] over a k. The first syzygy module is generated by the minimal relations corresponding to the columns of the matrix \begin{pmatrix} y & 0 \\ -x & y \\ 0 & -x \end{pmatrix}, where the first relation (y, -x, 0) satisfies y \cdot x^2 - x \cdot xy + 0 \cdot y^2 = 0, and the second (0, y, -x) satisfies $0 \cdot x^2 + y \cdot xy - x \cdot y^2 = 0. These two generators form a minimal basis for the syzygy , as neither is redundant and their rank matches the \beta_1(I) = 2.

Homological Connections

Relationship with free resolutions

In commutative algebra, linear relations among the generators of a finitely generated module M over a commutative ring R play a fundamental role in constructing free resolutions by forming the first syzygy module \mathrm{Syz}_1(M). Specifically, if M is presented by a surjective homomorphism \phi_0: F_0 \to M where F_0 = R^n is a free module with basis corresponding to the generators of M, then \mathrm{Syz}_1(M) = \ker(\phi_0) consists precisely of the linear relations, which are elements (a_1, \dots, a_n) \in R^n satisfying \sum a_i z_i = 0 for generators z_i of M. These relations generate \mathrm{Syz}_1(M) as an R-module, providing the structural kernel that encodes the dependencies among the generators. The construction of a free resolution begins with this presentation and extends it iteratively: the resolution takes the form \dots \to F_1 \xrightarrow{\phi_1} F_0 \xrightarrow{\phi_0} M \to 0, where F_1 is a whose basis is chosen to generate \mathrm{Syz}_1(M), typically minimally to ensure uniqueness up to . The map \phi_1: F_1 \to F_0 is defined such that its image is exactly \mathrm{Syz}_1(M), making the sequence exact at F_0. This continues by resolving the higher syzygy modules, with linear relations initiating the chain by determining the rank and structure of F_1. For minimal resolutions, the differentials satisfy \mathrm{im}(\phi_{i+1}) \subseteq \mathfrak{m} F_i for a \mathfrak{m}, ensuring the resolution captures the essential homological information without redundancy. Linear relations induce higher syzygies through the exactness of the , as the of each subsequent map \phi_i yields \mathrm{Syz}_{i+1}(M), propagating the relational structure outward in the . This iterative formation reflects how initial linear dependencies among generators evolve into relations among relations, forming the backbone of the entire . The integration of linear relations is captured succinctly in the short exact sequence $0 \to \mathrm{Syz}_1(M) \to F_0 \to M \to 0, which embeds the relations as a submodule of the free module F_0 and serves as the starting point for extending to a full resolution.

Syzygies and homological dimension

In homological algebra, higher syzygies arise from iterating the construction of linear relations on modules. Given a module M over a ring R with a presentation F_1 \to F_0 \to M \to 0, the first syzygy module \mathrm{Syz}_1(M) is the kernel of the map F_1 \to F_0. The second syzygy \mathrm{Syz}_2(M) consists of the linear relations among the generators of \mathrm{Syz}_1(M), obtained from a free presentation of \mathrm{Syz}_1(M), and this process continues to define \mathrm{Syz}_i(M) for i \geq 1. These higher syzygies form the successive kernels in a free resolution of M, determining the resolution length. The projective dimension of a M, denoted \mathrm{pd}_R(M), is the minimal length n of a projective of M, which equals the largest integer i such that \mathrm{Syz}_i(M) \neq 0 in a minimal free (assuming projective modules are free in this context). Equivalently, \mathrm{pd}_R(M) = \sup \{ i \mid \mathrm{Ext}^i_R(M, N) \neq 0 \text{ for some } N \}. For rings like rings R = k[x_1, \dots, x_n] over a k, guarantees that every finitely generated graded has \mathrm{pd}_R(M) \leq n, ensuring the chain of higher syzygies terminates after at most n steps. The global dimension of a ring R, denoted \mathrm{gl.dim}(R), is the supremum of \mathrm{pd}_R(M) over all R-modules M, reflecting the maximal length of syzygy chains across all modules. For polynomial rings in n variables, \mathrm{gl.dim}(R) = n, a finite value due to the bounded nature of higher syzygies, as established by Hilbert's theorem; this contrasts with rings of infinite global dimension where syzygy chains can be arbitrarily long. These structures enable key applications in , particularly in computing derived functors. A projective derived from the syzygy chain allows the computation of \mathrm{Ext}^i_R(M, N) as the of the \mathrm{Hom}_R(P_\bullet, N), where P_\bullet \to M \to 0 is the resolution, vanishing for i > \mathrm{pd}_R(M). Similarly, tensoring the resolution with another N yields a for \mathrm{Tor}^R_i(M, N), providing invariants of module interactions essential for studying extensions and geometric properties.

Historical Development

Origins and early concepts

The concept of linear relations emerged in the mid-19th century within the framework of , where mathematicians sought to identify polynomials unchanged under linear transformations of variables. Arthur laid foundational groundwork in 1847 by exploring dependencies among forms and their transformations, particularly through the study of involutions in , which implicitly involved linear relations among algebraic expressions. These ideas extended to matrices, where Cayley examined how transformations preserved certain structural properties, setting the stage for viewing relations as constraints in algebraic systems. In the context of determinants, linear relations were understood as linear dependencies among the rows or columns of matrices whose entries belonged to rings, ensuring consistency in systems of equations. This perspective arose from efforts to compute via determinants of symbolic matrices, where such dependencies captured the of linear maps between modules. Cayley's work highlighted how these relations manifested in the minors of matrices, providing a mechanism to eliminate variables and derive invariant quantities without explicit computation. Early examples of linear relations appeared in the study of binary forms and their covariants, where they ensured the algebraic consistency of expressions under group actions. For instance, in analyzing quadratic binary forms, relations among covariants of specific degrees and orders maintained the , allowing mathematicians to construct complete systems of invariants. These examples illustrated how linear relations acted as "yokings" between generators, preventing overcounting in the invariant ring. James Joseph Sylvester advanced these ideas pre-Hilbert by incorporating linear relations into the theory of resultants, where syzygies—relations among —facilitated the elimination of variables in systems defining common roots. In his 1853 work, Sylvester formalized such relations as syzygies, applying them to resultants of binary forms to derive conditions for simultaneous zeros, thus linking dependencies in polynomial ideals to geometric intersections. This approach extended Cayley's matrix-based insights, emphasizing syzygies in the construction of eliminants over rings.

Key advancements and theorems

Hilbert's syzygy theorem, established in 1890, asserts that for a field k and the polynomial ring R = k[x_1, \dots, x_n], every finitely generated module over R admits a free resolution of length at most n, with the nth syzygy module being free. A modern proof proceeds by induction on the number of variables n. For the base case n=1, the Koszul complex provides an exact resolution. Assuming the result for n-1 variables, one adjoins the nth variable and uses the Koszul complex on that variable to extend the resolution, ensuring termination at length n. This theorem implies that the global dimension of R is exactly n, meaning the longest possible projective dimension of any finitely generated is n, which underpins much of modern by guaranteeing finite homological dimensions in settings. In , Auslander and Buchsbaum extended these ideas with their , which relates the projective dimension of a finitely generated M over a commutative Noetherian local ring R to depths: \mathrm{pd}_R(M) + \mathrm{depth}(M) = \mathrm{depth}(R). This connects syzygies directly to ring invariants like depth and , generalizing Hilbert's bounds to broader classes of rings.