MODULES WITH EXTREMAL RESOLUTIONS

Luchezar L. Avramov

Dedicated to the memory of Hideyuki Matsumura

Introduction

Let R be a commutative noetherian local ring R with maximal ideal 𝔪 and residue field k = R/𝔪. The size of a minimal free resolution of a finite R–module M is given by its Betti numbers β_{n}^{R}(M) = rank_{k} Ext_{R}^{∗}(M, k). Dually, that of a minimal injective resolution is measured by the Bass numbers μ_{M}^{n}(M) = rank_{k} Ext_{R}^{∗}(k, M).

We estimate these sizes asymptotically on a natural, a polynomial, and an exponential scale. The first yields the classical homological dimensions. The second produces notions of complexity, which distinguish between modules of infinite homological dimensions. The third leads to the new concept of homological curvatures, which discriminate among modules of infinite complexities.

It is well known that k has maximal homological dimensions among all finite R–modules. An elementary computation shows that its complexities and curvatures are maximal as well. By analyzing the representations on Ext_{R}^{∗}(M, k) and Ext_{R}^{∗}(k, M) of the homotopy Lie algebra π^{∗} of the local ring R, we prove that modules with extremal homological invariants occur with unexpected frequency.

1. Asymptotic invariants

The projective complexity proj cx_{R} M is equal to d if d − 1 is the smallest degree of a polynomial p(t) such that β_{n}^{R}(M) ≤ p(n) for all n; if no such d exists one sets proj cx_{R} M = ∞. (Under the name of complexity, this concept was introduced in representation theory in [1], and transplanted to local algebra in [4], [5].) Replacing Betti numbers by Bass numbers, one gets a notion of injective complexity inj cx_{R} M.

We define the projective curvature and the injective curvature of M by

proj curv_{R} M = lim sup_{n→∞} \sqrt[n]{β_{R}^{n}(M)} and inj curv_{R} M = lim sup_{n→∞} \sqrt[n]{μ_{R}^{n}(M)} .

The names of the last two numbers reflect the fact that they are reciprocal values of the radii of convergence of the Poincaré series P_{M}^{R}(t) = ∑_{n} β_{n}^{R}(M)t^{n} and of the Bass series I_{R}^{M}(t) = ∑_{n} μ_{R}^{n}(M)t^{n}, respectively.

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 13D02, 13D05, 13D40, 13H05, 13H10.

Research partially supported by NSF Grant No. DMS-9102951. .

For background and further terminology from commutative algebra we refer to the beautiful exposition of Matsumura [12]. In describing relations among the various invariants we let the prefix ‘hom’ stand for either ‘proj’ or ‘inj’.

(1) hom dim_{R} M < ∞ ⟺ hom cx_{R} M = 0 ⟺ hom curv_{R} M = 0 .

(2) hom dim_{R} M = ∞ ⟺ 1 ≤ hom curv_{R} M.

(3) hom cx_{R} M < ∞ ⟹ hom curv_{R} M ≤ 1 ; the converse is an open problem, discussed for projective invariants in [6].

(4) It is unknown if the finitistic complexity f-hom cx R = sup_{M}{hom cx_{R} M < ∞} is finite. By [7; (3.10)] for each finite R–module M there is an R̂–module N such that μ_{n}^{R}(M) = βR̂ ; n−d(N) for n > d = dim M, hence f-inj cx R ≤ f-proj cx R; furthermore, in all known cases f-proj cx R ≤ rank_{k} 𝔪/𝔪^{2} − depth R, cf. [6].

(5) proj curv_{R} M < ∞, cf. e.g. [4; (2.5)], hence inj curv_{R} N < ∞ by Lemma 1.

We use ≼ and ≽ to denote coefficientwise inequalities of formal power series.

Lemma 1. For each R–module M there exists a number ℓ ≥ 1 such that

P_{M}^{R}(t) ≼ ℓ · P_{k}^{R}(t) = ℓ · I_{R}^{k}(t) ≽ I_{R}^{M}(t)

Deferring the proof, we note an immediate consequence:

Proposition 2. If R is a local ring with residue field k, then

proj dim_{R} k = sup_{M}{proj dim_{R} M} = sup_{M}{inj dim_{R} M} = inj dim_{R} k;
proj cx_{R} k = sup_{M}{proj cx_{R} M} = sup_{M}{inj cx_{R} M} = inj cx_{R} k;
proj curv_{R} k = sup_{M}{proj curv_{R} M} = sup_{M}{inj curv_{R} M} = inj curv_{R} k .

when M ranges over all finite R–modules. □

The first row of equalities is classical, and defines the global dimension gl dim R. Accordingly, we use the second row to define the global complexity gl cx R of R, and the third row to introduce its global curvature gl curv R.

By the Auslander-Buchsbaum-Serre theorem, gl dim R = d < ∞ precisely when R is regular of dimension d. To formulate similar statements for the asymptotic global invariants, we recall some earlier results. By Tate [17; Theorem 6], if R is a complete intersection, then P_{k}^{R}(t) = (1 + t)^{e}/(1 − t^{2})^{c} with e = rank_{k} 𝔪/𝔪^{2} and c = e − dim R, hence cx_{R} k is equal to the codimension c of R. On the other hand, if cx_{R} k < ∞, then R is a complete intersection by Gulliksen [8; (2.3)]. The same conclusion is derived in [3; (6.2)] from the much weaker condition curv_{R} k ≤ 1.

In the language introduced here, these results may be restated as:

Theorem 3. The following conditions on a local ring R are equivalent. (i) R is a complete intersection (respectively, of codimension c). (ii) gl cx R is finite (respectively, and equal to c). (iii) gl curv R is at most 1. □

Proof of Lemma 1. The proof is by induction on the Krull dimension dim M. When dim M = 0 a secondary induction on the length of M, using the cohomology exact sequence associated with a proper submodule, yields P_{M}^{R}(t) ≼ (length_{R} M) P_{k}^{R}(t).

If dim M > 0 and L = ⋃_{i⩾0}(0 :_{M} 𝔪^{i}), then the cohomology exact sequence associated with 0 ⟶ L ⟶ M ⟶ N ⟶ 0 yields P_{M}^{R}(t) ≼ P_{L}^{R}(t) + P_{N}^{R}(t), hence it suffices to prove the inequality for N. As depth_{R} N > 0, there is an exact sequence 0 ⟶ N ⟶^{x} N ⟶ P ⟶ 0 for some x ∈ 𝔪. The first homomorphism induces the zero map in cohomology, so P_{N}^{R}(t) ≼ P_{P}^{R}(t). Since dim P < dim N = dim M, the induction hypothesis applies to P, and we are done.

A parallel argument deals with I_{M}^{R}(t). Finally, note that P_{k}^{R}(t) = I_{k}^{R}(t). □

2. Extremal submodules

The main result of this section implies that each non-zero R–module may be obtained as an extension of a module of maximal complexity and curvature by a second module with the same characteristics. In particular, the Grothendieck group of finite R–modules is generated by modules with extremal behavior.

Theorem 4. If M is a finite module and L is a submodule such that L ⊇ 𝔪M ≠ 𝔪L, then the following hold:

proj cx_{R} L = inj cx_{R} L = gl cx R;
proj curv_{R} L = inj curv_{R} L = gl curv R .

The proof is immediate from Proposition 2 and Lemma 6 below. □

Corollary 5. For each i ≥ 1, the R–module 𝔪^{i}M is either zero, or has extremal complexities and curvatures. □

Remark. The corollary implies that if L = 𝔪M ≠ 0 and proj dim_{R} L or inj dim_{R} L is finite, then R is regular: the first result is due to Levin [9], cf. also [10; (1.1)]; the second is a new characterization of regularity. In conjunction with Theorem 3, the corollary also produces several criteria for complete intersections.

The asymptotic equalities in Theorem 4 follow from the more precise inequalities of power series established below, by using cohomology representation of the homotopy Lie algebra π^{∗} of the local ring R, discussed in more detail in [3].

Lemma 6. If e = rank_{k} 𝔪/𝔪^{2}, then for L as in Theorem 4 there are inequalities

P_{L}^{R}(t) · (1 + t)^{e} ≽ b · P_{k}^{R}(t)
(I_{R}^{L}(t) − s) · (1 + t)^{e} ≽ m · P_{k}^{R}(t) · t

with b = rank_{k} M/L > 0, s = rank_{k}(0 :_{L} 𝔪), and m = rank_{k} M/(𝔪L :_{M} 𝔪) > 0.

Proof. Consider the the commutative diagram of R–modules

0 ⟶ L ⟶ M ⟶ N ⟶ 0
↓  ↓  ∥
0 ⟶ L/𝔪L ⟶ M/𝔪L ⟶ N ⟶ 0 .

It induces a commutative square

Ext_{R}^{∗} (k, N) ⟶^{ð′} Ext_{R}^{∗}(k, L)
∥  ↓
Ext_{R}^{∗} (k, N) ⟶^{ð} Ext_{R}^{∗} (k, L/𝔪L)

of homomorphisms of graded right modules over the Yoneda algebra E^{∗} = Ext_{R}^{∗}(k, k), where ð′ and ð are connecting maps of degree 1. As 𝔪 annihilates N and L/𝔪L, the modules in the lower row are isomorphic to N ⊗_{k} E^{∗} and (L/𝔪L)⊗_{k} E^{∗}.

By Levin [9] and Schoeller [15], E^{∗} is a graded Hopf algebra whose dual Tor_{∗}^{R}(k, k) is a Γ-Hopf algebra, that is, a skew-commutative graded Hopf algebra with a system of divided powers compatible with the coproduct. A theorem of Milnor and Moore [13; (5.18)] in characteristic 0 and of André [2; Theorem 17] in characteristic p > 0 (adjusted by Sjödin [16; Theorem 2] for p = 2) shows that E^{∗} is the universal enveloping algebra U(π^{∗}) of a graded Lie algebra π^{∗}.

By the Poincaré-Birkhoff-Witt theorem, E^{∗} ≅ ⋀(π^{1}) ⊗_{k} U^{∗} as graded right modules over the subalgebra U^{∗} = U(π^{>1}). Noting that π^{1} = E^{1}, we include ð in a commutative square of homomorphisms of graded U^{∗}–modules

N ⊗_{k} E^{∗} ⟶^{ð} (L/𝔪L) ⊗_{k} ⋀(E^{1}) ⊗_{k} U^{∗}
↑  ↑^{ζ⊗_{k}U^{∗}}
N ⊗_{k} U^{∗} ⟶^{ð^{0}⊗_{k}U^{∗}} Ext_{R}^{1} (k, L/𝔪L) ⊗_{k} U^{∗}

in which ζ is the composition of the canonical isomorphism Ext_{R}^{1} (k, L/𝔪L) ≅ (L/𝔪L) ⊗_{k} E^{1} with the obvious inclusion map.

By combining the two squares, we locate in Ext_{R}^{⩾1}(k, L) a copy of Im(ð^{0}) ⊗_{k} U^{∗}. As E^{1} ≅ Hom_{k} (𝔪/𝔪^{2}, k), we get (1 + t)^{e} · ∑_{i} rank_{k} U^{i}t^{i} = P_{k}^{R}(t), and thus

I_{R}^{L}(t) ≽ s + rank_{k} Im(ð^{0}) · P_{k}^{R}(t)/(1 + t)^{e} · t .

After rewriting the beginning of the cohomology exact sequence induced by the lower row of the initial diagram of R–modules in the form

0 ⟶ L/𝔪L ⟶ (𝔪L :_{M} 𝔪)/𝔪L ⟶ N ⟶^{ð^{0}} Ext_{R}^{1}(k, L/𝔪L) ,

we see by a dimension count that rank_{k} Im(ð^{0}) = rank_{k} M/(𝔪L :_{M} 𝔪) = m. To finish with Bass series, it remains to note that m > 0 because 𝔪M ≠ 𝔪L.

The proof of the inequality for Poincaré series procceeds along similar lines; it is simplified by observing that the relevant connecting homomorphisms of graded left E^{∗}–modules ð′^{0} and ð^{0} are injective. □

3. Extremal quotients

Each finite R–module M ≠ 0 has an obvious quotient with extremal invariants, namely M/𝔪M. If M has depth 0, then by dualizing the arguments of the preceding section we obtain the following result; in particular, it implies that if (0 :_{M} 𝔪^{i}) ≠ (0 :_{M} 𝔪^{i+1}), then M/(0 :_{M} 𝔪^{i}) has extremal homological invariants.

Proposition 7. If M is a finite R–module, and L is a submodule such that L ⊆ (0 :_{M} 𝔪) ≠ (L :_{M} 𝔪), then the module N = M/L satisfies

proj cx_{R} N = inj cx_{R} N = gl cx R;
proj curv_{R} N = inj curv_{R} N = gl curv R .  □

A different approach to extremal quotients modules is given by

Theorem 8. Let M be a syzygy of k. If β: M ⟶ N is a homomorphism to a finite R–module, and the induced map β^{n}: Ext_{R}^{n}(M, k) ⟶ Ext_{R}^{n}(N, k) is non-zero for some n ≥ 0, then proj cx_{R} N = gl cx R and proj curv_{R} N = gl curv R.

Corollary 9. Each non-zero homomorphic image of a finite direct sum of syzygies of k has maximal projective complexity and curvature. □

Remark. The theorem implies that if R is not regular and β(M) ⊈ 𝔪N, then proj dim_{R} N = ∞: this is a recent result of Martsinkovsky [11; Proposition 8]. The corollary strengthens [11; Proposition 7] in a similar way.

Proof of Theorem 8. Let M = Syz_{m}^{R}(k), let β′ be a homomorphism of M′ = Syz_{m+n}(k) to N′ = Syz_{n}(N) induced by some extension of β to a morphism of

minimal free resolutions, and denote by γ the natural projection N′ ⟶^{γ} N′/𝔪N′. In cohomology we get a composition

ξ^{∗}: E^{∗} ⊗_{k} Hom (N′/𝔪N′, k) ⟶^{γ^{∗}} Ext_{R}^{∗}(N′, k) ⟶^{β′^{∗}} Ext_{R}^{∗}(M′, k) ⟶^{α^{∗}} E^{⩾m+n}

where α^{∗} is an iterated connected homomorphism, and thus bijective. Since γ^{0} is an isomorphism and β′^{0} = β^{n} is not trivial, we have ξ^{0} ≠ 0. After these adjustments, we revert to the notation introduced in the proof of Lemma 6.

If R is a complete intersection of codimension c, then π^{j} = 0 for j > 2, hence E^{∗} is a finite free module over the symmetric algebra U^{∗} = U(π^{2}) of the rank c vector space π^{2}. Thus, ξ^{∗} is a non-zero map of a free U^{∗}–module into a torsionfree one, and hence Im ξ^{∗} contains a non-zero free graded U^{∗}–module generated in degree 0. This yields (in)equalities of formal power series P_{N′}^{R}(t) ≽ ∑_{i} rank U^{i}t^{i} = 1/(1 − t^{2})^{c}. Together with Theorem 3 they imply numerical relations

proj cx_{R} N = proj cx_{R} N′ ≥ c = gl cx R

which shows that N has maximal complexity, and thus maximal curvature as well.

If R not not a complete intersection, then consider the subalgebra V^{∗} = U(π^{>m+n}) of E^{∗}. By the Poincaré-Birkhoff-Witt theorem, E^{∗} ≅ V^{∗} ⊗ W^{∗} as graded left V^{∗}–modules, where W^{∗} is the tensor product of the exterior algebra of the odd dimensional subspace of π^{⩽s} with with the symmetric algebra on its even dimensional subspace. This implies injectivity for the right hand vertical map in the commutative diagram of graded left V^{∗}–modules

E^{∗} ⊗_{k} Hom_{R} (N′/𝔪N′, k) ⟶^{ξ^{∗}} E^{⩾m+n}
↑  ↑
V^{∗} ⊗_{k} Hom_{R} (N′/𝔪N′, k) ⟶^{V^{∗}⊗_{k}ξ^{0}} V^{∗} ⊗_{k} E^{m+n} .

Thus, Im(ξ^{∗}) contains a copy of V^{∗}. Since ξ^{∗} factors over β′^{∗}, so does Ext_{R}^{∗}(N′, k).

We now have an inequality P_{N′}^{R}(t) ≽ ∑_{i} rank_{k} V^{i}t^{i}. On the other hand, the series W (t) = ∑_{i} rank W^{i}t^{i} is the expansion around t = 0 of the product

\frac{∏_{2i<m+n}(1 + t^{2i+1})^{e_{2i+1}}}{∏_{2i⩽m+n}(1 − t^{2i})^{e_{2i}}} where e_{j} = rank_{k} π^{j} .

Thus, the preceding inequality and the Poincaré-Birkhoff-Witt theorem yield

P_{N′}^{R}(t) · W (t) ≽ ( ∑_{i} rank_{k} V^{i}t^{i}) · W (t) = P_{k}^{R}(t) .

With ρ(A) denoting the radius of convergence of a power series A(t), we get

min{ρ(P_{N′}^{R}), 1} = min{ρ(P_{N′}^{R}), ρ(W )} ≤ ρ(P_{N′}^{R} ·W ) ≤ ρ(P_{k}^{R}) .

Taking inverses, and referring to Proposition 3, we conclude that

max{proj curv_{R} N, 1} = max{proj curv_{R} N′, 1} ≥ proj curv_{R} k = gl curv R .

As gl curv R > 1 by Theorem 3, we see that proj curv_{R} N and gl curv R are equal and greater than 1, and thus proj cx_{R} N and gl cx R are both infinite. □

4. Extremal algebras

The following result exhibits a new reason for the appearance of extremal resolutions: the passage from a ring with a bad singularity to a ring with a well behaved singularity creates complications in the module structure.

Proposition 10. Let R ⟶ S be a finite local homomorphism of local rings. (1) If S is regular, then proj cx_{R} S = gl cx R.

(2) If S is a complete intersection, then proj curv_{R} S = gl curv R.

The proof is immediate from Theorem 3 and the simple result below. □

Lemma 11. If R ⟶ S is a finite homomorphism of local rings, then

gl cx R ≤ gl cx S + proj cx_{R} S;
gl curv R ≤ max{gl curv S, proj curv_{R} S} .

Proof. The second page of the standard change of rings spectral sequence

2E_{p,q} = Ext_{S}^{p}(N, Ext_{R}^{q}(S, k)) ⟹ Ext_{R}^{p+q}(N, k)

for a finite S–module N can be rewritten as Ext_{S}^{∗}(N, l)⊗_{k} Ext_{R}^{∗}(S, k), where l is the residue field of S. The convergence of the spectral sequence implies an inequality of power series P_{N}^{S}(t) · P_{S}^{R}(t) ≽ P_{N}^{R}(t), from which one gets numerical inequalities

proj cx_{R} N ≤ proj cx_{S} N + proj cx_{R} S;
proj curv_{R} N ≤ max{proj curv_{S} N, proj curv_{R} S} .

Setting N = l, and remarking that as an R–module l is a finite direct sum of copies of k, one obtains the desired inequalities by applying Proposition 2. □

The homological invariants of the unique R–module of length 1 are listed in Proposition 2. Our last result provides an equally complete story in length 2.

Proposition 12. If S is a module of length 2 over a local ring R, then

proj cx_{R} S = inj cx_{R} S ≥ gl cx R − 1 ;
proj curv_{R} S = inj curv_{R} S = gl curv R .

If proj dim_{R} S < ∞ or inj dim_{R} S < ∞, then R is a hypersurface of multiplicity at most 2. Conversely, each hypersurface of multiplicity at most 2 has modules of length 2 and of finite projective and injective dimensions.

Proof. For S ≅ k ⊕ k all the assertions are clear. Else, S ≅ R/(x^{2}, x_{1}, …, x_{d}) for some minimal set of generators x, x_{1}, …, x_{d} of 𝔪. This R–module is isomorphic to its Matlis dual, so μ_{n}^{R}(S) = β_{n}^{R}(S) each n, and the injective invariants of S are equal to its projective one. The map R ⟶ S is a surjective ring homomorphism. The periodic resolution … ⟶ S ⟶^{x} S ⟶^{x} … of k over S yields gl cx S = gl curv S = 1, so the desired relations for proj cx_{R} S and proj curv_{R} S follow from Lemma 11.

When proj dim_{R} S < ∞, the first inequality becomes gl cx R ≤ 1, so R is a codimension 1 complete intersection by Theorem 3. Completing R if necessary, we may further assume that R = Q/(f), where (Q, 𝔫) is a (d + 1)-dimensional regular local ring and f ∈ 𝔫^{2}. Then S = Q/𝔞, with 𝔞 generated by a maximal Q–regular sequence. Tate [17; Theorem 4] provides an infinite free resolution of the R–module S, which is minimal if f ∈ 𝔫𝔞. Thus, f ∉ 𝔫𝔞 by the finiteness of proj dim_{R} S. But 𝔫^{2}S = 0 implies 𝔞 ⊇ 𝔫^{2}, hence f ∉ 𝔫^{3}, so the multiplicity of R is at most 2.

Conversely, let R be a hypersurface of multiplicity at most 2 and dimension d. There is nothing to prove when R is regular. Otherwise, the associated graded ring of R for the 𝔪-adic filtration is isomorphic to G = k[y_{1}, …, y_{d+1}]/(q), where q is a quadratic form. The set D ⊆ G_{1} of linear forms dividing q consists either of the origin, or of a single line, or of the union of two lines in the k–vector space G_{1}, hence D ≠ G_{1} as long as d > 1. This means that, by induction on d, we can find a G–regular sequence of linear forms g_{1}, …, g_{d}, such that rank_{k} G/(g_{1}, …, g_{d}) = 2. For i = 1, …, d choose x_{i} ∈ 𝔪 with initial form g_{i}. The sequence x_{1}, …, x_{d} is then R–regular, and R/(x_{1}, …, x_{d}) has length 2. □

To finish, we note that the last result cannot be extended to length 3 or higher.

Examples 13. (1) In the ring R = k[[x, y, z]]/(x^{2}, xy, y^{2} −xz) the element z is regular and S = R/(z) ≅ k[[x, y]]/(x^{2}, xy, y^{2}), so proj dim_{R} S = 1 and length_{R} S = 3. A well known change of rings theorem [14; (27.5)] gives the first equality below

P_{k}^{R}(t) = (1 + t) P_{k}^{S}(t) = \frac{1 + t}{1 − 2t} .

The second one follows from the recurrent relation P_{k}^{S}(t) − 1 = 2t · P_{k}^{S}(t), obtained from the cohomology exact sequence associated with 0 ⟶ 𝔪S ⟶ S ⟶ k ⟶ 0, and the isomorphism 𝔪S ≅ k^{2}. Thus, gl cx R = ∞ and gl curv R = 2.

(2) The hypersurface R = 𝔽_{2}[[x, y)]]/(xy(x + y)) has multiplicity 3 and the R– module S = R/(x(1 + x + y) + y^{2})) has projective dimension 1 and length 4, but R has no length 3 module of finite projective dimension.

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EXTREMAL RESOLUTIONS 9

17. J. Tate, Homology of noetherian rings and of local rings, Ill. J. Math. 1 (1957), 14–27. Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U.S.A. E-mail: avramov@math.purdue.edu