Proc. Amer. Math. Soc. 129, 1563–572 (2001)

FREE SUMMANDS OF CONORMAL MODULES AND CENTRAL ELEMENTS IN HOMOTOPY LIE ALGEBRAS OF LOCAL RINGS

SRIKANTH IYENGAR

Abstract. If (Q, 𝔫) ↠ (R, 𝔪) is a surjective local homomorphism with kernel I, such that I ⊆ 𝔫^{2} and the conormal module I/I^{2} has a free summand of rank n, then the degree 2 central subspace of the homotopy Lie algebra of R has dimension greater than or equal to n. This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.

Let R be a noetherian local ring with maximal ideal 𝔪 and residue field k. Ring theoretic properties of R are reflected on the algebra structures carried by Tor^{R} (k, k) and Ext_{R} (k, k) . Recall that the former has the ⋔-product of Cartan and Eilenberg, and the latter the Yoneda multiplication. The k-algebra structure of Tor^{R} (k, k) is rather “simple”: It is free in the appropriate category, and so determined by the dimension of the k-vector space Tor_{i}^{R} (k, k) , for all integers i ≥ 0, that is to say, the Betti numbers of k over R. The multiplicative structure of Ext_{R} (k, k) is quite another matter.

The simplicity of the product on Tor^{R} (k, k) arises from the fact that it is endowed with additional structures: It is a commutative algebra, in the graded sense, with a family of divided powers, and has a diagonal map that is compatible with the divided powers algebra structure on it. In other words, the k-algebra Tor^{R} (k, k) is a commutative Hopf algebra with divided powers, and so is free as a divided powers algebra.

In contrast, the multiplication on Ext_{R} (k, k) is decidedly non-commutative, unless R happens to be a complete intersection of a special kind. The graded kdual of the product on Tor^{R} (k, k) turns Ext_{R} (k, k) = Tor^{R} (k, k)^{∗} into a Hopf algebra. Attention has focussed on a certain subspace of primitives of this Hopf algebra, denoted π(R), which is a Lie algebra with the bracket operation defined by the commutator in Ext_{R} (k, k) . This object is called the homotopy Lie algebra of R. The importance of this Lie algebra is attested to by its defining property: Its universal enveloping algebra is Ext_{R} (k, k) .

In this note, we are concerned with the centre of the homotopy Lie algebra of R, denoted ζ(R). This subspace, besides being a measure of the non-commutativity of π(R), and hence of Ext_{R} (k, k) , plays an important part in the change of rings properties of the homotopy Lie algebra, among other things; the interested reader is referred to [?], and the references therein, for additional information on this matter. The construction of π(R) outlined above is a compilation of work of André, Assmus, Avramov, Gulliksen, Levin, Sjödin, Schoeller, and others.

Our primary goal is to establish the following Main Theorem. Let (Q, 𝔫, k) be a local ring and I an ideal in Q. Set R = Q/I. If there is an ideal J in Q with I^{2} ⊆ J ⊆ I and the R-module I/J is free, then

Date: March 19, 2001.

1991 Mathematics Subject Classification. 13C, 13D, 18G.

Key words and phrases. homotopy Lie algebra, central elements, conormal module.

rank_{k} ζ^{2}(R) ≥ rank_{R}(\frac{I}{J}) − rank_{k} (\frac{I}{J + I ∩ 𝔫^{2}} ) .

Here is a special case of the theorem: If the ideal I is generated by a regular sequence, then I/I^{2} is a free R-module, so that

rank_{k} ζ^{2}(R) ≥ rank_{R}(\frac{I}{I^{2}} ) − rank_{k} (\frac{I}{I ∩ 𝔫^{2}} ) .

This well known fact has the following consequence: If the 𝔪-adic completion R̂ of a local ring (R, 𝔪, k) is such that R̂ ≅ Q/(f) with (Q, 𝔫, k) a local ring and f ∈ 𝔫^{2} a non-zero divisor, in other words, if R̂ has an embedded deformation, then π^{2}(R̂) has a central element, and since π(R) ≅ π(R̂), there is a central element in π^{2}(R). In [?], Avramov posed the following question: Does every element in ζ^{⩾ 2}(R) arise from an embedded deformation of R̂?

The graded version of this question has been settled in the negative: In [?], Dupont exhibits a graded ring R with a central element in π^{2}(R) and proves that there is no graded ring Q with a homogeneous non-zero divisor f such that R = Q/(f). The local case of this problem remains open, although Löfwall [?] and Avramov [?] have answered it in the affirmative in certain special situations.

One of the difficulties in addressing the question above is that, hitherto, other than embedded deformations, one had no good source of central elements in π(R). The theorem above is offered as procedure for generating rings with central elements in the homotopy Lie algebra which do not a priori arise from embedded deformations. It should be noted that this result builds on the work in [?].

The proof of the Main Theorem is given in Section 1. As an outgrowth of these investigations, we found another approach to certain well known results regarding the conormal module I/I^{2}. These are discussed in Section 2.

1. Central elements

The proofs in this paper rely on computations with Differential Graded algebras with divided powers, henceforth abbreviated to DG Γ-algebras. We refer to the monographs of Gulliksen and Levin [?], and Avramov [?], for definitions and basic results concerning these objects. We adhere to the following conventions:

1.1. Let Q be a commutative ring. A DG Γ -algebra over Q is a DG Γ-algebra A such that A_{0} is a Q-algebra. Any DG Γ-algebra A considered here will be such that the ring A_{0} is noetherian and each A_{i} is a finitely generated A_{0}-module; in particular, all rings, which we regard as DG Γ-algebras concentrated in degree 0, will be noetherian.

For a DG Γ-algebra A, the symbol A⟨X⟩ denotes a DG Γ-algebra obtained by successive adjunction of exterior and divided power variables, cf. [?, 6]. In this case, for any integer n, we write X_{n} for the variables of degree n, and A⟨X_{⩽ n}⟩ to denote the DG Γ-subalgebra of A⟨X⟩ obtained by adjunction of variables of degree ≤ n.

A derivation of a Γ-algebra A is a homogeneous Q-linear map θ: A → A such that

θ(xy) = θ(x)y + (−1)^{|θ||x|}xθ(y)  for x, y ∈ A;
θ(x^{(i)}) = θ(x)x^{(i−1)}  for x ∈ A_{even} and all i ≥ 1 .

When we refer to a derivation of a DG Γ-algebra, it is to be understood that it is a derivation of the underlying Γ-algebra, and, unless stated to the contrary, that it commutes with the differential: ∂ ∘ θ = (−1)^{|θ|}θ ∘ ∂.

For ease of reference, we record the following fact about derivations, cf. [?, 1.3.2].

1.2. A derivation θ on a DG Γ-algebra A extends to A⟨X | ∂x ∈ A, for x ∈ X⟩ if and only if θ(∂x) ∈ ∂(A) for each x ∈ X. If θ(∂x) = ∂(b_{x}) then one may extend θ to A⟨X⟩ with θ(x) = (−1)^{|θ|}b_{x}.

Next, we list various criteria that allow one to detect the existence of direct summands in the conormal module I/I^{2}; verifying these assertions is a routine exercise. Recall that the free rank of an R-module M is the maximal rank of a free direct summand of M; we denote it f-rank_{R}M.

1.3. Consider an ideal I in a local ring Q. Set R = Q/I.

(1) f-rank_{R}(I/I^{2}) ≥ n if and only if there is an ideal J in Q with I^{2} ⊆ J ⊆ I such that the R-module I/J is free of rank n;

(2) the classes of a_{1}, …, a_{n} ∈ I form a basis for a free summand of I/I^{2} if and only if they extend to a minimal set of generators a_{1}, …, a_{n}, b_{1}, …, b_{m} for I with the property that for any r_{1}, …, r_{n} in Q

∑_{i=1}^{n} r_{i}a_{i} ∈ (b_{1}, …, b_{m}) ⟹ r_{j} ∈ I for 1 ≤ j ≤ n, .

The following proposition adds to this list; it is a particular case of [?, 6.2.7]. We give a direct proof suited to the situation on hand. First, a piece of notation: For integers i, j, we set

δ_{ij} = \begin{cases}1 & if i = j \\ 0 & otherwise \end{cases} .

1.4. Proposition. Let Q be a ring, I ⊂ Q a proper ideal, and Q⟨X⟩ a DG Γ - algebra resolution of R = Q/I over Q. If f-rank_{R}(I/I^{2}) ≥ n, then there are elements x_{1}, ⋯, x_{n} ∈ Q⟨X⟩ of degree 1, and Q-linear derivations θ_{i} of Q⟨X⟩ such that θ_{i}(x_{j}) = δ_{ij} for 1 ≤ i, j ≤ n.

If the classes of a_{1}, …, a_{n} form a basis for a free direct summand of I/I^{2}, then the x_{i} may be chosen with ∂(x_{i}) = a_{i} for 1 ≤ i ≤ n.

Proof. Let a_{1}, …, a_{n}, b_{1}, …, b_{m} be the minimal generating set for the ideal I provided by ??.2. Since ∂(X_{1}) generates the ideal I, by a linear change of variables we may assume that X_{1} = {x_{1}, …, x_{n}, y_{1}, …, y_{m+p}} with ∂(x_{i}) = a_{i}, for 1 ≤ i ≤ n, and ∂(y_{j}) = b_{j}, for 1 ≤ j ≤ m, and ∂(y_{m+k}) = 0, for 1 ≤ k ≤ p.

For 1 ≤ i ≤ n, set θ_{i}: Q → Q to be the trivial Q-linear derivation with θ_{i}(r) = 0 for all r ∈ Q. By ?? these extend to Q-linear Γ-derivations on Q⟨X_{1}⟩ with θ_{i}(x_{j}) = δ_{ij} and θ_{i}(y_{k}) = 0. We prove that these extend to Q⟨X⟩.

To begin with, for any element x ∈ X_{2}, one has ∂x = (∑_{i=1}^{n} r_{i}x_{i}) + y, for some elements r_{1}, …, r_{n} in Q and y ∈ ∑_{j=1}^{m+p} Qy_{j}. Since ∂^{2}(x) = 0, one has

∑_{i=1}^{n} r_{i}a_{i} ∈ ∑_{j=1}^{m+p} Q∂(y_{j}) = (b_{1}, …, b_{m}) .

From ??.2 we derive that the elements r_{1}, …, r_{n} are in I, and so θ_{i}(∂(x)) = r_{i} ∈ ∂(Q⟨X_{1}⟩). Thus, by ?? the θ_{i} extend to Γ-derivations on Q⟨X_{⩽ 2}⟩.

Now assume that the θ_{i} have been extended to Q⟨X_{⩽ k}⟩, for some k ≥ 2. For any x ∈ X_{k+1} the element θ_{i}(∂x) in Q⟨X_{⩽ k}⟩ is a cycle:

∂θ_{i}(∂x) = −θ_{i}∂(∂x) = 0 .

Since the degree of θ_{i}(∂x) is k − 1 ≥ 1, and Q⟨X_{⩽ k}⟩ no homology in degrees 1, …, k − 1, we find that θ_{i}(∂x) ∈ ∂(Q⟨X_{⩽ k}⟩). Thus, by ??, the θ_{i} extend to derivations on Q⟨X_{⩽ k+1}⟩. A direct limit argument completes the proof. □

The proof of the Main Theorem hinges on a well known link between the Lie algebra π(R) and the action of certain derivations of the acyclic closure of R; confer [?, 6.3.1], or [?, 1.9], for the details on acyclic closures. We give a precis of this relationship in a form suited to our use, modeled along the lines of [?, 10.2.1].

1.5. Let A be a DG Γ-algebra such that A_{0} is a local ring, with maximal ideal 𝔪 and residue field k. As in the case of local rings, the k-algebra Ext_{A} (k, k) is the universal enveloping algebra of a graded Lie algebra, which is unique up to isomorphism, and it is denoted π(A).

Let A⟨X⟩ be an acyclic closure of k over A, where X = {x_{i}}_{i ⩾ 1} is ordered such that |x_{i}| ≤ |x_{j}| for i < j. The A-module D consisting of all the A-linear Γderivations of A⟨X⟩ is a DG Lie algebra - here we do not assume that the derivations commute with the differential on A⟨X⟩ - with Lie bracket defined by

[θ, ϑ] = θ ∘ ϑ − (−1)^{|θ||ϑ|}ϑ ∘ θ .

The following statements hold:

(1) π(A) ≅ H(D) as Lie algebras;

(2) for i ≥ 1, there are derivations, ϑ_{i} ∈ D, that commute with the differential on A⟨X⟩, with ϑ_{i}(x_{j}) = δ_{ij} for j ≤ i.

(3) each set of derivations {ϑ_{i}}_{i ⩾ 1} that satisfy condition (2) maps to a k-basis

for H(D). In the light of these results, henceforth we identify π(A) with H(D).

1.6. Following André, we say that a cycle z in a DG Γ-algebra B is a special cycle if there is a derivation θ: B → B such that θ(z) = 1. Our interest in special cycles arises from a result established by André [?] which, tailored to our needs, reads:

If z_{1}, …, z_{n} are special cycles in a DG Γ-algebra B, with associated derivations θ_{1}, …, θ_{n}, then there is a DG Γ-subalgebra A ⊆ B such that

(a) ⋂_{i=1}^{n} Ker(θ_{i}) ⊆ A;

(b) the morphism of DG Γ-algebra, extending the inclusion A ⊆ B

A⟨x_{1}, …, x_{n} | ∂(x_{i}) = 0⟩ → B with x_{i} ↦ z_{i} ,

is an isomorphism. When we encounter such a situation, we write B = A⟨z_{1}, …, z_{n}⟩, although this constitutes an abuse of notation.

For our proof of the Main Theorem, it is expedient to isolate the following

1.7. Observation. Consider an ideal I in a local ring (Q, 𝔫, k) such that I^{2} = 0 and the Q/I-module I is free. Set 𝔪 = 𝔫/I, and

n = μ(I) − rank_{k} (\frac{I}{I ∩ 𝔫^{2}} ) .

There are elements a_{1}, …, a_{n}, b_{1}, …, b_{m}, and r_{1}, …, r_{e} in 𝔫 such that (a) a_{1}, …, a_{n} minimally generate I ∩ 𝔫^{2};

(b) a_{1}, …, a_{n}, b_{1}, …, b_{m} is a minimal set of generators for I;

(c) r_{1}, …, r_{e} map to a minimal set of generators for 𝔪;

(d) 𝔫 = (r_{1}, …, r_{e}, b_{1}, …, b_{m}).

Indeed, from the exact sequence of k-vector spaces

0 → \frac{I ∩ 𝔫^{2}}{𝔫I} → \frac{I}{𝔫I} → \frac{I}{I ∩ 𝔫^{2}} → 0 .

we deduce that the rank of the k-vector space (I ∩ 𝔫^{2})/𝔫I is n. Select elements a_{1}, …, a_{n} in I ∩ 𝔫^{2} which map to a basis for (I ∩ 𝔫^{2})/𝔫I, and elements b_{1}, …, b_{m} in I which map to a basis for I/(I ∩ 𝔫^{2}). The exact sequence above implies that a_{1}, …, a_{n}, b_{1} …, b_{m} is a minimal system of generators for I. Conditions (a) and (b) are then fulfilled. Now, pick elements r_{1}, …, r_{e} which map to a minimal set of of generators for 𝔪. Note that

𝔫 = (r_{1}, …, r_{e}) + I
= (r_{1}, …, r_{e}) + (b_{1}, …, b_{m}) + (a_{1}, …, a_{n})
= (r_{1}, …, r_{e}) + (b_{1}, …, b_{m}) ,

where the second equality is by (b), and the third follows from Nakayama’s lemma, since (a_{1}, …, a_{n}) ⊆ 𝔫^{2}. This justifies (d).

1.8. Proof of the Main Theorem. Consider the ring \overline{Q} = Q/J, and the ideal \overline{I} = I/J in \overline{Q}. Observe that R = \overline{Q}/\overline{I} with (\overline{I})^{2} = 0, and that the R-module \overline{I} is free. Moreover, with \overline{𝔫} = 𝔫/J, one has an isomorphism

\frac{\overline{I}}{\overline{I} ∩ (\overline{𝔫})^{2}} ≅ \frac{I}{J + I ∩ 𝔫^{2}} .

Therefore, by descending to the ring \overline{Q} we assume that J = 0, so that I^{2} = 0 and I is a free R-module. This places us in the situation of Observation ??. For the rest of this proof we utilize the notation and conclusion of the same.

Let Q⟨X⟩ → R be a DG Γ-resolution of R over Q. By definition, the canonical map

π(R) → π(Q⟨X⟩)

is an isomorphism of Lie algebras. Thus, it suffices to establish that the rank of ζ^{2}(Q⟨X⟩) is at least n.

The elements r_{1}, …, r_{e} map to a minimal set of generators for the maximal ideal of R, so that the first stage in the acyclic closure of k over Q⟨X⟩ is given by

Q⟨X, Y_{1}⟩ = Q⟨X⟩ ⊗_{Q}Q⟨y_{1}, …, y_{e} | ∂(y_{i}) = r_{i}⟩ .

Since a_{1}, …, a_{n}, b_{1}, …, b_{m} is a minimal system of generators for I, there are elements x_{1}, …, x_{n+m} ∈ X_{1} with ∂(x_{i}) = a_{i}, for 1 ≤ i ≤ n, and ∂(x_{n+j}) = b_{j}, for 1 ≤ j ≤ m. From part (d) of the observation above we obtain a_{i} ∈ 𝔫 = (r_{1}, …, r_{e}, b_{1}, …, b_{m}), for 1 ≤ i ≤ n, and hence there are elements

c_{i} ∈ ∑_{j=1}^{e} Qy_{j} + ∑_{k=1}^{m} Qx_{n+k} with ∂(c_{i}) = a_{i} .

Therefore, the elements z_{i} = x_{i} − c_{i}, for 1 ≤ i ≤ n, are cycles in Q⟨X, Y_{1}⟩.

Let θ_{1}, …, θ_{n} be the Q-linear derivations of Q⟨X⟩ supplied by Proposition ??, corresponding to the elements a_{1}, …, a_{n}. Each θ_{i} extends to a Q⟨Y_{1}⟩-linear derivation θ̃_{i}: Q⟨X, Y_{1}⟩ → Q⟨X, Y_{1}⟩. Note that for 1 ≤ i, j ≤ n and 1 ≤ k ≤ e one has

θ̃_{i}(z_{j}) = δ_{ij} and θ̃_{i}(y_{k}) = 0 .

In particular, the elements z_{1}, …, z_{n} are special cycles of degree 1 in Q⟨X, Y_{1}⟩.

From this point onwards the proof develops along the lines of that of [?, 1]. Given the θ̃_{i}, one invokes ?? to write

Q⟨X, Y_{1}⟩ = A⟨Z⟩ = A⟨z_{1}, ⋯, z_{n} | ∂(z_{i}) = 0⟩ ,

where A is a DG Γ-subalgebra of Q⟨X, Y_{1}⟩ containing y_{1}, …, y_{e}. Note that H_{0}(A) = k. Let A⟨Y_{⩾ 2}⟩ = A⟨y_{e+1}, y_{e+2}, …⟩ be an acyclic closure of k over A, with |y_{i}| ≤ |y_{j}| if i < j. Set

Q⟨X, Y ⟩ = Q⟨X, Y_{1}, Y_{⩾ 2}⟩ = Q⟨X, Y_{1}⟩ ⊗_{A}A⟨Y_{⩾ 2}⟩ .

Observe that the DG Γ-algebra Q⟨X, Y ⟩ = A⟨Y_{⩾ 2}⟩⟨Z⟩ is quasi-isomorphic to k⟨Z⟩, so that

Q⟨X, Y, U⟩ = Q⟨X, Y ⟩⟨u_{1}, …, u_{n} | ∂(u_{i}) = z_{i}⟩ ,

is an acyclic closure of k over Q⟨X⟩. Now, by ??.3, derivations corresponding to the variables in Y ∪ U, that satisfy condition ??.2, form a basis for π(Q⟨X⟩).

Claim. The variables u_{1}, …, u_{n} correspond to central elements in π^{2}(Q⟨X⟩).

It suffices to construct Q⟨X⟩-linear derivations {ω_{i}}_{i=1}^{n} and {ϑ_{j}}_{j ⩾ 1}, of the DG Γ-algebra Q⟨X, Y, U⟩ such that

(a) ω_{i}(u_{j}) = δ_{ij} for u_{j} ∈ U and ω_{i}(A⟨Y_{⩾ 2}⟩) = 0;

(b) ϑ_{j}(y_{j}) = 1 and ϑ_{j}(v) ∈ A⟨Y_{⩾ 2}⟩ for v ∈ Y ∪ U.

Indeed, if such ω_{i} and ϑ_{j} exist, then, according to ??, the classes of these elements form a basis for π(Q⟨X⟩). For any 1 ≤ i ≤ n and v ∈ Y ∪ U, one has

[ω_{i}, ϑ_{j}](v) = ω_{i}(ϑ_{j}(v)) − ϑ_{j}(ω_{i}(v))
= ω_{i}(ϑ_{j}(v))
= 0 ,

where the second equality holds since, by (a), ω_{i}(v) ∈ {0, 1}, and the third equality follows from (a) and (b). Since [ω_{i}, ϑ_{j}] is a Q⟨X⟩-linear derivation, this entails [ω_{i}, ϑ_{j}] = 0 on Q⟨X, Y, U⟩. A similar computation proves that [ω_{i}, ω_{j}] = 0 for 1 ≤ j ≤ n, so that the class of ω_{i} is central in π(Q⟨X⟩) (n.b. |ω_{i}| = 2 for each i.)

It remains to construct derivations that satisfy the conditions listed above.

For each element u_{i} ∈ U, one has ∂(u_{i}) ∈ Q⟨X, Y ⟩, so that we define ω_{i} to be the Q⟨X, Y ⟩-linear derivation of Q⟨X, Y, U⟩ with ω_{i}(u_{j}) = δ_{ij}. This ω_{i} has the required properties.

Fix an integer j ≥ 1. Let ϑ_{j} be the Q⟨X⟩-linear derivation on Q⟨X⟩⟨y_{1}, …, y_{j}⟩ defined by ϑ_{j}(y_{k}) = δ_{jk}, for k ≤ j. We resort to an induction argument to extend this derivation to Q⟨X, Y ⟩ with the property that ϑ_{j}(A⟨Y_{⩾ 2}⟩) ⊆ A⟨Y_{⩾ 2}⟩.

Since the variables y_{1}, …, y_{e} are contained in A ⊂ Q⟨X, Y_{1}⟩, we may write A = A′⟨y_{1}, …, y_{e}⟩ with A′ a DG Γ-subalgebra of Q⟨X⟩. In particular, ϑ_{j}(A′) = 0, so that ϑ_{j}(A′⟨y_{1}, …, y_{j}⟩) ⊆ A′⟨y_{1}, …, y_{j}⟩. This is the basis of our induction. Now assume that ϑ_{j} has been defined on y_{i} for all i ≤ k in such a manner that ϑ_{j}(A′⟨y_{1}, …, y_{k}⟩) ⊆ A′⟨y_{1}, …, y_{k}⟩. By construction, ∂(y_{k+1}) ∈ A′⟨y_{1}, …, y_{k}⟩, and the induction hypothesis yields that the element ϑ_{j}(∂(y_{k+1})) is a cycle in A′⟨y_{1}, …, y_{k}⟩. Since A′⟨Y ⟩ = A⟨Y_{⩾ 2}⟩ is acyclic, there is an element v ∈ A′⟨Y ⟩ such that ∂(v) = ϑ_{j}(∂(y_{k+1})); the degree of ∂(v) entails v ∈ A′⟨y_{1}, …, y_{k}⟩. Thus, by ??, one can extend ϑ_{j} with ϑ_{j}(y_{k+1}) = (−1)^{|ϑ_{j}|}v, and so ϑ_{j}(A′⟨y_{1}, …, y_{k+1}⟩) ⊆ A′⟨y_{1}, …, y_{k+1}⟩. This completes the induction. Now by a direct limit argument we construct a derivation ϑ_{j} on Q⟨X, Y ⟩ such that ϑ_{j}(A⟨Y_{⩾ 2}⟩) ⊆ A⟨Y_{⩾ 2}⟩.

Finally, since Q⟨X, Y ⟩ = A⟨Y_{⩾ 2}⟩⟨Z⟩, for each u_{k} ∈ U, the degree of the element ϑ_{j}(∂(u_{k})) = ϑ_{j}(z_{k}) dictates that it lie in A⟨Y_{⩾ 2}⟩ - which is acyclic. Thus, ∂(v_{k}) = ϑ_{j}(∂(u_{k})) for some element v_{k} ∈ A⟨Y_{⩾ 2}⟩, and, by ??, we extend ϑ_{j} to Q⟨X, Y, U⟩ with ϑ_{j}(u_{k}) = (−1)^{|ϑ_{j}|}v_{k}. The ϑ_{j} thus defined has all the required properties. □

Example. Consider the ideal I = (ab, c^{2}, cd, a^{2}c + d^{3}, b^{2}d + ad^{2}) in the polynomial ring Q = k[a, b, c, d], over a field k. The element ab ∈ I generates a free summand of I/I^{2}. Indeed, according to MACAULAY, the beginning of the minimal resolution

of I is Q^{8} ⟶^{ϕ} Q^{5} ⟶^{ψ} I → 0 where

ψ = (ab c^{2} cd a^{2}c + d^{3} b^{2}d + ad^{2})

ϕ = (\begin{matrix}0 & 0 & 0 & −cd & −c^{2} & a^{2}c + d^{3} & abc & b^{2}d + ad^{2} \\ −d & 0 & a^{2} & 0 & ab & 0 & 0 & 0 \\ c & b^{2} + ad & d^{2} & ab & 0 & 0 & a^{3} & 0 \\ 0 & 0 & −c & 0 & 0 & −ab & −b^{2} − ad & 0 \\ 0 & −c & 0 & 0 & 0 & 0 & d^{2} & −ab \end{matrix})

The entries in the first row in the relations matrix ϕ of the ideal I are contained in I, so that, by ??.2, the minimal generator ab generates a free summand in the Q/Imodule I/I^{2}, and thus a free summand in the R = (Q/I)_{𝔫}-module (I/I^{2})_{𝔫}, where 𝔫 = (a, b, c, d). Now, since ab ∈ 𝔫^{2}, the Main Theorem yields that rank_{k} ζ^{2}(R) ≥ 1. A propos the question of Avramov, it is of interest to know if the 𝔫-adic completion R̂ has an embedded deformation.

The fibres of flat homomorphisms are a nice source of rings with central elements:

Example. Suppose that ϕ: (R, 𝔪, k) → (S, 𝔫, l) is a flat local homomorphism. Flatness implies that the S/𝔪S-module 𝔪S/(𝔪S)^{2} is free of rank equal to edim R, the embedding dimension of R. Therefore, the Main Theorem yields that

rank_{l} ζ^{2}(S/𝔪S) ≥ edim R − rank_{l} (\frac{𝔪S}{𝔪S ∩ 𝔫^{2}} )
= edim R − (edim S − edim(S/𝔪S))
= edim R + edim(S/𝔪S) − edim S .

The same lower bound may be obtained as follows: According to [?, 1.1], (rather, its dual) there is an exact sequence of l-vector spaces

0 → π^{1}(S/𝔪S) → π^{1}(S) → π^{1}(R) ⊗_{k} l ⟶^{δ} π^{2}(S/𝔪S) → π^{2}(S) → π^{2}(R) ⊗_{k} l → 0 ,

where for any local ring (Q, 𝔮), one has rank_{Q/𝔮Q} π^{1}(Q) = edim(Q). The desired inequality follows from the fact that the image of δ lies in ζ^{2}(S/𝔪S), cf. [?, 1]. This is no coincidence: The central elements detected by the Main Theorem as well as those in Im(δ) arise from special cycles.

2. On free summands in the conormal module

The technique utilized in the proof of the Main Theorem leads to another point of view on certain results regarding the conormal module. This is based on the following observation. For a complex B we denote sup(B) the supremum of the set {i | H_{i}(B) ≠ 0}; in particular, H(B) = 0 if and only if sup(B) = −∞.

2.1. Proposition. Let I be a proper ideal in a ring Q, and let Q⟨X⟩ be a DG Γ -algebra resolution of R = Q/I. Set n = f-rank_{R}(I/I^{2}). If A is a DG Γ -algebra over Q such that IA_{0} ⊆ ∂(A_{1}), then

A⟨X⟩ = A ⊗_{Q} Q⟨X⟩ ≅ W ⟨z_{1}, …, z_{n} | ∂(z_{i}) = 0⟩ ,

with W a DG Γ -subalgebra of A⟨X⟩. Therefore, if H(A) ≠ 0, then sup(A⟨X⟩) ≥ n.

Proof. Let x_{1}, ⋯, x_{n}, respectively θ_{1}, ⋯, θ_{n}, be the elements in X_{1}, respectively the Q-linear derivations of Q⟨X⟩, supplied by Proposition ??. The derivations θ_{i} extend to A-linear derivations θ̃_{i}: A⟨X⟩ → A⟨X⟩, for each i.

Since ∂(x_{i}) ∈ IA_{0} and IA_{0} ⊆ ∂(A_{1}), there are elements a_{1}, ⋯, a_{n} in A_{1} such that ∂(a_{i}) = ∂(x_{i}) for 1 ≤ i ≤ n. Therefore, for each i, the elements z_{i} = x_{i} − a_{i} in A⟨X⟩ are cycles of degree 1. Observe that θ̃_{i}(z_{j}) = δ_{ij}, that is to say, z_{1}, …, z_{n} are special cycles in A⟨X⟩. Now we invoke the result of André, recalled in ??, to obtain the desired isomorphism. It remains to note that if H(A) ≠ 0, then H_{0}(A) ≠ 0, so that H_{0}(A⟨X⟩) = H_{0}(A) ⊗_{Q} R = H_{0}(A) ≠ 0, since IA_{0} ⊆ ∂(A_{1}). In particular, H(W) ≠ 0, and so sup(A⟨X⟩) = sup(W) + n ≥ n. □

Remark. In this note, we shall have occasion to use only the bound sup(A⟨X⟩) ≥ n, and this can be established without resorting to André’s result. Indeed, the cycle

z = z_{1} · z_{2} ⋯ z_{n} ∈ A⟨X⟩_{n}

is not a boundary. This assertion may be justified as follows: Set θ = θ̃_{n} ∘ ⋯ ∘ θ̃_{1}, and note that θ(z) = 1. Now, if z = ∂(b) for some element b ∈ A⟨X⟩_{n+1}, then

1 = θ(z) = θ(∂b) = (−1)^{n}∂(θ(b)) ,

so that H_{0}(A⟨X⟩) = 0, and this implies that H(A⟨X⟩) = 0. This is a contradiction.

The following result has been established, using different techniques, by Vasconcelos [?] and by Ferrand [?]. For a finitely generated Q-module M, we write μ_{Q}(M) for the minimal number of generators of M, grade_{Q}(I, M) for the maximal length of a M-regular sequence in an ideal I ⊂ Q, and pd_{Q}(M) for the projective dimension of M over Q.

2.2. Corollary. Let I be an ideal in a local ring (Q, 𝔫, k), and set R = Q/I. If pd_{Q}(R) is finite, then

grade_{Q}(I, Q) ≥ f-rank_{R}(I/I^{2}) .

In particular, if I/I^{2} is a free R-module, then I is generated by a regular sequence.

Proof. Set n = f-rank_{R}(I/I^{2}), and let Q⟨X⟩ be a DG Γ-algebra resolution of R. Proposition ?? gives

pd_{Q} R = sup Tor^{Q} (k, R) = sup(k ⊗_{Q} Q⟨X⟩) ≥ n .

In particular, the Auslander-Buchsbaum formula: depth Q = pd_{Q} R + depth R, allows us to conclude that depth Q ≥ n.

Now, for each prime ideal 𝔭 ⊇ I, the R_{𝔭}-module I_{𝔭}/(I_{𝔭})^{2} contains a free summand of rank n. Since pd_{Q_{𝔭}} R_{𝔭} is finite as well, we deduce that depth Q_{𝔭} ≥ n, and hence

grade_{Q}(I, Q) = inf{depth Q_{𝔭} | 𝔭 ⊇ I} ≥ n .

Finally, if I/I^{2} is free, then f-rank_{R}(I/I^{2}) = μ(I), so that grade_{Q}(I, Q) = μ_{Q}(I), and hence I is generated by a regular sequence. □

The next result also has predecessors: Vasconcelos [?] and Lech [?] have proved, using different techniques, that if I/I^{2} is a free R-module, then μ_{Q}(𝔫) ≥ μ_{Q}(I).

2.3. Corollary. Let I be an ideal in a local ring (Q, 𝔫, k), and set R = Q/I. For each proper ideal J ⊇ I, one has

μ(J) ≥ f-rank_{R}(I/I^{2}) + grade_{Q}(J, R) .

Proof. Let Q⟨X⟩ be a DG Γ-algebra resolution of R, and let K be the Koszul complex on a minimal set of generators for J. Since K is a bounded complex of free Q-modules, the canonical map K ⊗_{Q} Q⟨X⟩ → K ⊗_{Q} R is a quasi-isomorphism. As I ⊆ J = ∂(K_{1}), Proposition ?? yields

sup(K ⊗_{Q} R) = sup(K ⊗_{Q} Q⟨X⟩) ≥ f-rank_{R}(I/I^{2}) .

It remains to note that since the Koszul complex is depth sensitive, μ_{Q}(J) = grade_{Q}(J, R) + sup(K ⊗_{Q} R). □

Acknowledgement. It is a pleasure to thank Luchezar Avramov for kind encouragement, and for useful conversations regarding the material presented here.

References

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Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.

E-mail address: iyengar@math.missouri.edu