FINITE REGULARITY AND KOSZUL ALGEBRAS

LUCHEZAR L. AVRAMOV AND IRENA PEEVA

Abstract. We determine the positively graded commutative algebras over which the residue field modulo the homogeneous maximal ideal has finite Castelnuovo-Mumford regularity: they are the polynomial rings in finitely many indeterminates over Koszul algebras; this proves a conjecture in [3]. We also show that if the residue field of a finitely generated graded algebras has finite regularity, then so do all finitely generated graded modules.

Introduction

The graded Betti numbers β_{i,j}^{R}(M) are among the most important numerical invariants of a finitely generated graded module M over a positively graded commutative algebra R that is finitely generated over a field R_{0}.

The supremum of those i for which β_{i,j}^{R}(M) ≠ 0 (for some j) is equal to the projective dimension of M. It measures the number of systems of linear equations over R that have to be solved in order to build a minimal free resolution of M. Algebras over which all modules have finite projective dimension are characterized by the Hilbert Syzygy Theorem and the Auslander-Buchsbaum-Serre Theorem, which establish the equivalence of the following conditions:

(i) R is a polynomial ring in finitely many indeterminates over a field.

(ii) Every finitely generated graded R-module has finite projective dimension. (iii) The residue field R/R_{⩾ 1} has finite projective dimension. The supremum of the differences (j −i) occurring when β_{i,j}^{R}(M) ≠ 0 is called the

Castelnuovo-Mumford regularity of M. It estimates the range of degrees involved in solving any of the systems of equations arising in the construction of a minimal free resolution. We characterize those algebras over which all modules have finite regularity, proving that the following conditions are equivalent:

(i) R is a polynomial ring in finitely many indeterminates over a Koszul algebra. (ii) Every finitely generated graded R-module has finite regularity. (iii) The residue field R/R_{⩾ 1} has finite regularity.

Koszul algebras, appearing in (i), are defined by the vanishing of the regularity of the residue field. These algebras have received considerable attention due to their extraordinary homological properties and to their appearance in many cases of interest in algebra, algebraic geometry, topology, and combinatorics.

Our main result, Theorem 2, proves that (i) and (iii) are equivalent, even for algebras that are not necessarily finitely generated. This establishes a generalized version of a conjecture of Avramov and Eisenbud [3]. The fact that (iii) implies (ii) is contained in Theorem 1 and extends a result from [3].

Date: December 6, 2000.

Both authors were partly supported by grants from the National Science Foundation. The second author was also supported by a Sloan Research Fellowship.

Results

In this paper k denotes a field, R = ⨁_{n∈ℕ} R_{n} a commutative graded algebra with R_{0} = k, and M = ⨁_{p∈ℤ} M_{p} a graded R-module. For each d ∈ ℤ we denote M(d) the graded R-module with M(d)_{p} = M_{d+p}. By a customary abuse of notation, we let k denote also the residue field R/R_{⩾ 1}.

Every graded R-module M has a graded free resolution

F = … ⟶ F_{i} ⟶^{∂_{i}} F_{i−1} ⟶ … ⟶ F_{1} ⟶^{∂_{1}} F_{0} ⟶ 0

where for each i the module F_{i} is isomorphic to a direct sum of copies of R(−j) and the homomorphism ∂_{i} preserves the grading. If M_{p} = 0 for p ≪ 0, then F can be chosen minimal, in the sense that ∂_{i}(F_{i}) ⊆ (R_{⩾ 1})F_{i−1} for all i. Any two minimal resolutions are isomorphic as complexes of graded R-modules, so the number of summands of F_{i} isomorphic to R(−j) is an invariant of M, called the ij’th graded Betti number β_{ij}^{R}(M). The Castelnuovo-Mumford regularity of M is the number

reg_{R}(M) = sup {n ∈ ℤ | β_{i, i+n}^{R}(M) ≠ 0 for some i ∈ ℕ} .

The concept was introduced in the classical context of a polynomial ring on finitely many variables, where Hilbert’s Basis Theorem and Syzygy Theorem imply that each finitely generated graded module has finite regularity. We prove that all finitely generated graded modules over an arbitrary finitely generated algebra have finite regularity whenever one of them—the residue field—has this property.

Theorem 1. Let R be a finitely generated positively graded commutative algebra over a field k.

If reg_{R}(k) < ∞ then reg_{R}(M) < ∞ for each graded finitely generated module M.

For algebras generated in degree 1 the theorem was proved in [3]; the general case follows immediately from a relation between the graded Betti numbers of M and those of k, established in Lemma 4.

Following Priddy [7], we say that R is Koszul if reg_{R}(k) = 0. Each Koszul algebra is generated in degree 1 and its relations follow from those of degree 2. However, the Koszul property is much subtler and cannot be inferred from the knowledge of any finite number of Betti numbers of k. In fact, Roos [8] constructed for each integer n ≥ 3 a commutative algebra S^{n}, generated by 6 linear forms subject to 11 quadratic relations, with β_{ij}^{S^{n}}(k) = 0 for j ≠ i < n and β_{n,n+1}^{S^{n}}(k) ≠ 0.

Our main result pinpoints the relationship of the Koszul property and the finiteness of the regularity of the residue field.

Theorem 2. Let R be a positively graded commutative algebra over a field k.

If reg_{R}(k) < ∞, then the algebra S = k[R_{1}] is Koszul and R is a polynomial ring on finitely many indeterminates over S.

Conversely, if R is a polynomial ring on indeterminates t_{1}, …, t_{ℓ} over a Koszul algebra, then reg_{R}(k) = ∑_{p=1}^{ℓ}(deg(t_{p}) − 1) .

A special case of the theorem establishes a conjecture from [3] and yields a new characterization of Koszul algebras.

Corollary 3. A graded commutative k-algebra R is Koszul if and only if it is generated by elements of degree 1 and reg_{R}(k) is finite.

Proofs

If F is a minimal free resolution of M, then H_{i}(k ⊗_{R} F) = k ⊗_{R} F, hence

β_{ij}^{R}(M) = rank_{k} Tor_{i}^{R}(k, M)_{j} .

The isomorphisms of graded vector spaces Tor_{i}^{R}(k, M) ≅ H_{i}(A ⊗_{R} M) for all i ∈ ℕ show that the graded Betti numbers of M can be computed from any graded resolution A of the R-module k.

The following graded version of [2, (4.1.9)] implies Theorem 1.

Lemma 4. If M is a finitely generated graded module over a finitely generated positively graded commutative k-algebra R, then there exists an integer s (depending on M) such that for all (i, j) ∈ ℕ × ℤ the following inequality holds:

β_{ij}^{R}(M) ≤ ∑_{p ⩽ s}(rank_{k} M_{p}) · β_{i,j−p}^{R}(k) .

Proof. For each r ≥ 1 the R-module M/(R_{⩾ 1})^{r}M is graded and has finite nonzero length, so s(r) = sup{j ∈ ℤ|(M/(R_{⩾ 1})^{r}M)_{j} ≠ 0} is finite. The inclusions M_{> s(r)} ⊆ (R_{⩾ 1})^{r}M ⊆ M define homomorphisms of graded vector spaces

Tor_{i}^{R}(k, M) ⟶ Tor_{i}^{R}(k, M/M_{> s(r)}) ⟶ Tor_{i}^{R}(k, M/(R_{⩾ 1})^{r}M) .

By [1, (A.4)] or [2, (4.1.8)] we may choose r so that the composition above is injective for each i ∈ ℕ. It follows that the first map is injective, hence we may assume that M_{j} = 0 for j > s = s(r). If A is a minimal resolution of k, then

rank_{k} ( H_{i}(A ⊗_{R} M)_{j}) ≤ rank_{k} ((A_{i} ⊗_{R} M)_{j}) = rank_{k} ( ⨁_{p+q=j} M_{p}^{b_{iq}})

where b_{iq} = β_{iq}^{R}(k). The desired inequality follows.

We start the proof of Theorem 2 with a well known computation.

Lemma 5. If R = S[t_{1}, …, t_{ℓ}] for a positively graded k-algebra S and indeterminates t_{1}, …, t_{ℓ} of positive degree, then

reg_{R}(k) = reg_{S}(k) + ∑_{p=1}^{ℓ}(deg(t_{p}) − 1) .

Proof. By induction, it suffices to deal with a single variable t. The complex

C = 0 ⟶ k[t](−d) ⟶^{t} k[t] ⟶ 0 ,

where d = deg(t), is a minimal free resolution of k = k[t]/(t) over k[t]. If B is a minimal free resolution of k = S/S_{⩾ 1} over S, then B ⊗_{k} C is a minimal free resolution of k ⊗_{k} k = k over S ⊗_{k} k[t] ≅ R. For all i, j this yields equalities β_{ij}^{R}(k) = β_{ij}^{S}(k) + β_{i−1 , j−d}^{S}(k). Thus, reg_{R}(k) = reg_{S}(k) + (d − 1).

We fix some terminology and notation. A DG algebra is a complex of graded R-modules B with B_{0} = R, equipped with a morphism of complexes of graded Rmodules B ⊗_{R} B → B that yields an associative and unitary product on ⨁_{i∈ℕ} B_{i}. An element b ∈ (B_{i})_{j} is said to be homogeneous of homological degree i and internal degree j; this is denoted |b| = i and deg(b) = j. We require that b b′ = (−1)^{|b||b′|}b′ b, and b^{2} = 0 when |b| is odd. If B is a DG algebra, then B⟨W ⟩ denotes a DG algebra obtained by adjunction of sets W_{i} of homogeneous exterior variables in odd homological degrees i ≥ 1 and sets W_{i} of homogeneous divided powers variables in even homological degrees i ≥ 2, cf. [10], [6, §1.1], or [2, §6.1] for details.

In the proof of the next result we play variations on arguments of Gulliksen [5].

Lemma 6. Let B be a DG algebra. If A = B⟨w⟩ for some exterior variable w and A is a minimal free resolution of k, then |w| = 1.

Proof. Set |w| = 2h+1 and consider the exact sequence of complexes of R-modules

0 ⟶ B ⟶ A ⟶^{ϑ} B[2h + 1] ⟶ 0

where ϑ(a + wb) = b, and B[2h + 1] is obtained from B by shifting it 2h + 1 steps to the left and changing the signs of the differentials.

We assume that h > 0 and get a contradiction. In the homology exact sequence

H_{i+2h+1}(A) ⟶ H_{i}(B) ⟶^{ð_{i}} H_{i+2h}(B) ⟶ H_{i+2h}(A)

the connecting homomorphism ð_{i} is bijective because H_{q}(A) = 0 for q ≥ 1. In particular, for each integer n ≥ 0 the composition δ_{n} = ð_{2hn} ∘⋯∘ð_{0} is an isomorphism of k = H_{0}(B) with H_{2h(n+1)}(B). It remains to show that δ_{n}(1) = 0 for some n.

For z = ∂(w) we have z = ∑_{p=1}^{n} r_{p}a_{p} with r_{p} ∈ R_{⩾ 1} = ∂(A_{1}) and a_{p} ∈ A_{2h}. As A_{i} = B_{i} for i ≤ 2h, we see that a_{p} ∈ B_{2h} and that r_{p} = ∂(b_{p}) for appropriate b_{1}, …, b_{n} ∈ B_{1}. Thus, we have z = u + ∂(∑_{p=1}^{n} b_{p}a_{p}), and u = ∑_{p=1}^{n} b_{p}∂(a_{p}) is a cycle. Recalling that b_{p}^{2} = 0 for p = 1, …, n, we get u^{n+1} = 0. A direct computation shows that ð_{i}(cls(c)) = cls(zc) = cls(uc) for any cycle c ∈ B_{i}. Thus, we obtain δ_{n}(1) = cls(u^{n+1}) = 0, as desired.

Tate [10] constructs a DG algebra resolution of k in the form A = R⟨W ⟩ with ∂(W_{1}) a minimal set of homogeneous generators of R over k and {cls(z)|z ∈ ∂(W_{i+1})} a homogeneous basis of H_{i}(R⟨W_{⩽ i}⟩) for i ≥ 1. By Gulliksen [6, (1.6.4)] such a resolution is minimal, cf. also [4] or [9] for finitely generated algebras.

Lemma 7. Let A = R⟨W ⟩ be a minimal free resolution of k. The sets

X_{i} = {w ∈ W | deg(w) = |w| = i} ,  X = ⋃_{i ⩾ 1} X_{i} ,
Y_{i} = {w ∈ W | deg(w) > |w| = i} ,  Y = ⋃_{i ⩾ 1} Y_{i} ,

of homogeneous variables yield inclusions of DG algebras

R⟨X⟩ ⊆ R⟨X, Y ⟩ = A .

Proof. The minimal resolution A satisfies (A_{i})_{j} = 0 for i > j, so deg(a) ≥ |a| holds for each homogeneous a ∈ A. In particular, W = X ∪ Y, hence A = R⟨X, Y ⟩.

For x ∈ X, write ∂(x) in the form ∑_{p} r_{p}a_{p}, where the r_{p}’s are non-zero elements in R and the a_{p}’s are products of variables in W and of divided powers of variables in W. The minimality of A implies deg(r_{p}) ≥ 1. From the computation

deg(a_{p}) ≥ |a_{p}| = |x| − 1 = deg(x) − 1 = deg(a_{p}) + deg(r_{p}) − 1 ≥ deg(a_{p})

we see that |a_{p}| = deg(a_{p}) for every p. It follows that each a_{p} is in R⟨X⟩, so ∂(X) ⊆ R⟨X⟩. Thus, R⟨X⟩ is a DG subalgebra of A.

We are ready to prove the main result of this paper.

Proof of Theorem 2. We denote S the subalgebra of R generated by R_{1} over k, and let A = R⟨X, Y ⟩ be a minimal DG algebra resolution of k as in Lemma 7.

If S is Koszul and R = S[t_{1}, …, t_{ℓ}] where t_{1}, …, t_{ℓ} are indeterminates over S, then we have reg_{R}(k) = ∑_{p=1}^{ℓ}(deg(t_{p}) − 1) by Lemma 5.

For the rest of the proof we assume that reg_{R}(k) is finite. We have to show that R is generated over S by finitely many indeterminates, and that S is Koszul.

First we prove that A = R⟨X, y_{1}, …, y_{ℓ}⟩ where each |y_{p}| is an exterior variable. The minimality of A yields an isomorphisms of bigraded k-algebras

Tor^{R}(k, k) ≅ H(A ⊗_{R} k) = A ⊗_{R} k = k⟨X, Y ⟩ .

We order the variables in Y and denote P (respectively, Q) the set of finite strictly increasing sequences consisting of variables of odd (respectively, even) degree. For p = (y_{1}, …, y_{r}) ∈ P we set y_{p} = y_{1} ⋯ y_{r}. For q ∈ Q and for each map e: q → ℕ_{+} we set y_{q}^{(e)} = ∏_{y∈q} y^{(e_{y})}, where e_{y} = e(y) and y^{(n)} is the n’th divided power of y.

The collection of all products y_{p}y_{q}^{(e)} forms a homogeneous basis V of k⟨X, Y ⟩ over its subalgebra k⟨X⟩. Choosing a homogeneous basis U of k⟨X⟩ over k, we get a homogeneous basis U · V of k⟨X, Y ⟩ over k. Each u ∈ U satisfies deg(u) = |u|, so

reg_{R}(k) = sup_{U,V} {deg (uy_{p}y_{q}^{(e)}) − |uy_{p}y_{q}^{(e)}| }

= sup_{U,V} {deg (y_{p}y_{q}^{(e)}) − |y_{p}y_{q}^{(e)}| + deg(u) − |u| }

= sup_{V} { ∑_{y∈p} ( deg(y) − |y|) + ∑_{y∈q} e_{y}( deg(y) − |y|)} .

Since deg(y) > |y| holds for each y ∈ Y, the finiteness of reg_{R}(k) implies that Y = {y_{1}, …, y_{ℓ}} for some ℓ, and |y_{p}| is odd for each p.

Next we prove that |y_{p}| = 1 for p = 1, …, ℓ and hence that

reg_{R}(k) = ∑_{p=1}^{ℓ} ( deg(y_{p}) − 1) .

We may assume that |y_{p}| ≤ |y_{ℓ}| for all p, so it suffices to show that |y_{ℓ}| = 1. Form the graded subalgebra B = R⟨X, y_{1}, …, y_{ℓ−1}⟩ of A. Comparison of homological degrees shows that ∂(y_{p}) ∈ B for p < ℓ. Lemma 7 yields ∂(X) ⊆ R⟨X⟩ ⊆ B. Thus, B is a DG algebra and A = B⟨y_{ℓ}⟩. From Lemma 6 we conclude that |y_{ℓ}| = 1.

Now we prove that t_{p} = ∂(y_{p}) for p = 1, …, ℓ are algebraically independent over S. Set s = ∂(X_{1}), let S′ = k[s′] be a polynomial ring on a family s′ = {s′ |s ∈ s} of indeterminates of degree 1, and let R′ = S′[t′_{1}, …, t′_{ℓ}] where t′_{1}, …, t′_{ℓ} are indeterminates with deg(t′_{p}) = deg(t_{p}). Consider the surjective homomorphism π: R′ → R of graded k-algebras defined by π(s′) = s and π(t′_{p}) = t_{p}, and set I = Ker π. By [6, (1.4.15)] or [2, (4.1.3.3)] the ideal I appears in an isomorphism

I/(R′_{⩾ 1})I ≅ H_{1}(R⟨X_{1}, Y_{1}⟩)

of graded vector spaces over k. Each element of the basis {cls(z)|z ∈ ∂(X_{2})} of H_{1}(R⟨X_{1}, Y_{1}⟩) has internal degree two, so I = R′I_{2}. The set {s, t_{1}, …, t_{ℓ}} minimally generates the k-algebra R, hence I ⊆ (R′_{⩾ 1})^{2}. Because deg(t_{p}) = deg(y_{p}) ≥ 2, we get I_{2} = I ∩ ((R′_{⩾ 1})^{2})_{2} = I ∩ ((S′_{⩾ 1})^{2})_{2} ⊆ S′_{2}. The resulting isomorphisms

S[t′_{1}, …, t′_{ℓ}] ≅ (S′/S′I_{2})[t′_{1}, …, t′_{ℓ}] ≅ R′/R′I_{2} ≅ R = S[t_{1}, …, t_{ℓ}]

of graded k-algebras show that t_{1}, …, t_{ℓ} are algebraically independent over S.

It remains to prove that S is Koszul. Since deg(y_{p}) = deg(t_{p}), comparison of the expression for reg_{R}(k) obtained above with the equality

reg_{R}(k) = reg_{S}(k) + ∑_{p=1}^{ℓ} ( deg(t_{p}) − 1)

provided by Lemma 5 shows that reg_{S}(k) = 0.

References

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[3] L. L. Avramov, D. Eisenbud, Regularity of modules over a Koszul algebra, J. Algebra 153 (1992), 85–90.

[4] T. H. Gulliksen, A proof of the existence of minimal algebra resolutions, Acta Math. 120 (1968), 53–58.

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[10] J. Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–25.

Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A.

  E-mail address: avramov@math.purdue.edu

Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.