Hilbert's syzygy theorem

The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890). The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem (Theorem III), which is used in part IV to discuss the Hilbert polynomial. The last part, part V, proves finite generation of certain rings of invariants. Incidentally part III also contains a special case of the Hilbert–Burch theorem. ==Syzygies (relations)==

Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring. Given a generating set g_1, \ldots, g_k of a module over a ring , a relation or first syzygy between the generators is a -tuple (a_1, \ldots, a_k) of elements of such that :a_1g_1 + \cdots + a_kg_k =0. Let L_0 be a free module with basis (G_1, \ldots, G_k). The -tuple (a_1, \ldots, a_k) may be identified with the element :a_1G_1 + \cdots + a_kG_k, and the relations form the kernel R_1 of the linear map L_0 \to M defined by G_i \mapsto g_i. In other words, one has an exact sequence :0 \to R_1 \to L_0 \to M \to 0. This first syzygy module R_1 depends on the choice of a generating set, but, if S_1 is the module that is obtained with another generating set, there exist two free modules F_1 and F_2 such that :R_1 \oplus F_1 \cong S_1 \oplus F_2 where \oplus denote the direct sum of modules. The second syzygy module is the module of the relations between generators of the first syzygy module. By continuing in this way, one may define the th syzygy module for every positive integer . If the th syzygy module is free for some , then by taking a basis as a generating set, the next syzygy module (and every subsequent one) is the zero module. If one does not take a basis as a generating set, then all subsequent syzygy modules are free. Let be the smallest integer, if any, such that the th syzygy module of a module is free or projective. The above property of invariance, up to the sum direct with free modules, implies that does not depend on the choice of generating sets. The projective dimension of is this integer, if it exists, or if not. This is equivalent with the existence of an exact sequence :0 \longrightarrow R_n \longrightarrow L_{n-1} \longrightarrow \cdots \longrightarrow L_0 \longrightarrow M \longrightarrow 0, where the modules L_i are free and R_n is projective. It can be shown that one may always choose the generating sets for R_n being free, that is for the above exact sequence to be a free resolution. == Statement ==

Hilbert's syzygy theorem states that, if is a finitely generated module over a polynomial ring k[x_1,\ldots,x_n] in indeterminates over a field , then the th syzygy module of is always a free module. In modern language, this implies that the projective dimension of is at most , and thus that there exists a free resolution :0 \longrightarrow L_k \longrightarrow L_{k-1} \longrightarrow \cdots \longrightarrow L_0 \longrightarrow M \longrightarrow 0 of length . This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly . The standard example is the field , which may be considered as a k[x_1,\ldots,x_n]-module by setting x_i c=0 for every and every . For this module, the th syzygy module is free, but not the th one (for a proof, see , below). The theorem is also true for modules that are not finitely generated. As the global dimension of a ring is the supremum of the projective dimensions of all modules, Hilbert's syzygy theorem may be restated as: the global dimension of k[x_1,\ldots,x_n] is . Low dimension In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every finitely generated vector space has a basis. In the case of a single indeterminate, Hilbert's syzygy theorem is an instance of the theorem asserting that over a principal ideal ring, every submodule of a free module is itself free. == Koszul complex ==

The Koszul complex, also called "complex of exterior algebra", allows, in some cases, an explicit description of all syzygy modules. Let g_1, \ldots, g_k be a generating system of an ideal in a polynomial ring R=k[x_1,\ldots,x_n], and let L_1 be a free module of basis G_1, \ldots, G_k. The exterior algebra of L_1 is the direct sum :\Lambda(L_1)=\bigoplus_{t=0}^k L_t, where L_t is the free module, which has, as a basis, the exterior products :G_{i_1} \wedge \cdots \wedge G_{i_t}, such that i_1 In particular, one has L_0=R (because of the definition of the empty product), the two definitions of L_1 coincide, and L_t=0 for . For every positive , one may define a linear map L_t\to L_{t-1} by :G_{i_1} \wedge \cdots \wedge G_{i_t} \mapsto \sum_{j=1}^t (-1)^{j+1}g_{i_j}G_{i_1}\wedge \cdots\wedge \widehat{G}_{i_j} \wedge \cdots\wedge G_{i_t}, where the hat means that the factor is omitted. A straightforward computation shows that the composition of two consecutive such maps is zero, and thus that one has a complex :0\to L_t \to L_{t-1} \to \cdots \to L_1 \to L_0 \to R/I. This is the Koszul complex. In general the Koszul complex is not an exact sequence, but it is an exact sequence if one works with a polynomial ring R=k[x_1,\ldots,x_n] and an ideal generated by a regular sequence of homogeneous polynomials. In particular, the sequence x_1,\ldots,x_n is regular, and the Koszul complex is thus a projective resolution of k=R/\langle x_1, \ldots, x_n\rangle. In this case, the th syzygy module is free of dimension one (generated by the product of all G_i); the th syzygy module is thus the quotient of a free module of dimension by the submodule generated by (x_1, -x_2, \ldots, \pm x_n). This quotient may not be a projective module, as otherwise, there would exist polynomials p_i such that p_1x_1 + \cdots +p_nx_n=1, which is impossible (substituting 0 for the x_i in the latter equality provides ). This proves that the projective dimension of k=R/\langle x_1, \ldots, x_n\rangle is exactly . The same proof applies for proving that the projective dimension of k[x_1, \ldots, x_n]/\langle g_1, \ldots, g_t\rangle is exactly if the g_i form a regular sequence of homogeneous polynomials. ==Computation==

At Hilbert's time, there was no method available for computing syzygies. It was only known that an algorithm may be deduced from any upper bound of the degree of the generators of the module of syzygies. In fact, the coefficients of the syzygies are unknown polynomials. If the degree of these polynomials is bounded, the number of their monomials is also bounded. Expressing that one has a syzygy provides a system of linear equations whose unknowns are the coefficients of these monomials. Therefore, any algorithm for linear systems implies an algorithm for syzygies, as soon as a bound of the degrees is known. The first bound for syzygies (as well as for the ideal membership problem) was given in 1926 by Grete Hermann: Let a submodule of a free module of dimension over k[x_1, \ldots, x_n]; if the coefficients over a basis of of a generating system of have a total degree at most , then there is a constant such that the degrees occurring in a generating system of the first syzygy module is at most (td)^{2^{cn}}. The same bound applies for testing the membership to of an element of . On the other hand, there are examples where a double exponential degree necessarily occurs. However such examples are extremely rare, and this sets the question of an algorithm that is efficient when the output is not too large. At the present time, the best algorithms for computing syzygies are Gröbner basis algorithms. They allow the computation of the first syzygy module, and also, with almost no extra cost, all syzygies modules. ==Syzygies and regularity==

One might wonder which ring-theoretic property of A=k[x_1,\ldots,x_n] causes the Hilbert syzygy theorem to hold. It turns out that this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let A be a Noetherian ring. Then A has finite global dimension if and only if A is regular and the Krull dimension of A is finite; in that case the global dimension of A is equal to the Krull dimension. This result may be proven using Serre's theorem on regular local rings. == See also ==