TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0

COMPUTING THE HOMOLOGY OF KOSZUL COMPLEXES

Bernhard Köck

Abstract. Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each f. g. projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_{1} → P_{0} for all n ≥ 1, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^{2} Γ(P.) where Γ and N denote the functors occurring in the Dold-Kan correspondence.

Introduction

In the theory of exterior powers of f. g. projective modules over a commutative ring R, there are various formulas which, so far, can only be proved by using geometric means (e.g., the projective fibre bundle or blowing up). An example is the formula which computes the composition of exterior powers in the Grothendieck ring K_{0}(R) (e.g., see [FL]). The plethysm problem is to find (explicit) functorial short exact sequences which prove this formula within the framework of commutative algebra. It has been intensively studied in universal representation theory. In this paper, we study the analogous problem for the Adams-Riemann-Roch formula for regular closed immersions (between affine schemes).

We now recall the Adams-Riemann-Roch formula (see [FL]). Let i : Y ↪ X be a regular closed immersion of schemes with the locally free conormal sheaf C. Let ψ_{n} : K_{0}(X) → K_{0}(X) denote the n-th Adams operation on the Grothendieck ring K_{0}(X); it is defined as a certain integral polynomial in the exterior power operations on K_{0}(X). For any locally free O_{Y} -module V, we choose a locally free resolution P. of i_{∗}(V) on X, and we map the element [V] ∈ K_{0}(Y) to the Euler characteristic ∑_{k≥0}(−1)^{k}[P_{k}] ∈ K_{0}(X). This association induces a well-defined homomorphism i_{∗} : K_{0}(Y) → K_{0}(X). The Adams-Riemann-Roch theorem for i_{∗} now states that, for all y ∈ K_{0}(Y), we have:

ψ_{n}(i_{∗}(y)) = i_{∗}(θ^{n}(C) · ψ_{n}(y)) in K_{0}(X);

here, θ^{n}(C) denotes the so-called Bott element associated with C (see [FL]).

This formula (for only one n ≥ 2) implies the famous Grothendieck-Riemann-Roch theorem for i_{∗} in a rather formal manner (see [FL]). It has been formulated

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

2000 Mathematics Subject Classification. Primary 13D25, 19E20, 14C40, 13D15.

Key words and phrases. Koszul complex, Dold-Kan correspondence, cross effect functor, symmetric power operation, Adams-Riemann-Roch theorem, plethysm problem.

for the first time by Manin in [M], but it has been proved there only up to n^{2}- torsion in K_{0}(X). Moreover, as a formula in K_{0}(X) ⊗ ℚ, it follows already from Théorème 4.3 in Expose VII of [SGA 6]. The given version without denominators can be derived from Théorème 2.1 in Jouanolou’s paper [J] and can be found in Soulé’s paper [S] and in the book [FL] by Fulton and Lang. In all these sources, the deformation to the normal cone is the crucial tool in the proof. The object of this paper is to construct explicit short exact sequences of O_{X}-modules which prove this formula in the case codim(Y/X) = 1, n arbitrary and in the case codim(Y/X) = 2, n = 2. Since the proof of the Adams-Riemann-Roch formula in the general case can immediately be reduced to the case codim(Y/X) = 1 using blowing up and the excess intersection formula (see [SGA 6]), we in particular obtain a new, very natural and simple proof in the general case which, in contrast to [J], [S], or [FL], does not use the deformation any longer.

To illustrate our approach, we now rather completely describe it for codim(Y/X) = 1 and n = 2. As in the beginning of the introduction, we consider only the affine case; so, let X = Spec(R) and Y = Spec(R/I) where the ideal I of R is locally generated by a non-zero-divisor. Let σ_{2} and λ_{2} denote the second symmetric power and second exterior power operation. Let V be a f. g. projective R/I-module and

0 → P → Q → V → 0

an R-projective resolution of V. Since ψ_{2} = σ_{2} − λ_{2} and θ^{2}(I/I^{2}) = 1 + [I/I^{2}], it suffices to prove the formula

σ_{2}(i_{∗}([V])) = i_{∗}([Sym^{2}(V)] − [Λ^{2}(V) ⊗ I/I^{2}]) in K_{0}(X) (0.1)

and a similar formula for λ_{2}. Now, the main idea is to consider the Koszul complex

K : 0 → Λ^{2}(P) → P ⊗ Q → Sym^{2}(Q) → 0

which, for instance, can be defined as a subcomplex of the natural short exact sequence 0 → Λ^{2}(Q) → Q ⊗ Q → Sym^{2}(Q) → 0. It is immediately clear that the alternating sum of the objects of this complex equals the left hand side of the formula (0.1). Since the alternating sum of the objects equals the alternating sum of the homology modules, it suffices to show that H_{0}(K) ≅ Sym^{2}(V), H_{1}(K) ≅ Λ^{2}(V)⊗I/I^{2}, and H_{2}(K) ≅ 0. (Here, H_{0}(K), H_{1}(K), and H_{2}(K) denote the homology of K at the places Sym^{2}(Q), P ⊗Q, and Λ^{2}(P), respectively.) Whilst the isomorphisms H_{0}(K) ≅ Sym^{2}(V) and H_{2}(K) ≅ 0 can rather easily be proved (see Remark 3.3), the proof of the isomorphism H_{1}(K) ≅ Λ^{2}(V) ⊗ I/I^{2} is more complicated. The most important observation for this is the fact that H_{1}(K) does not depend on the chosen resolution and that the association V ↦ H_{1}(K) =: F (V) is even a functor. This can be proved as follows. Any homomorphism V → V′ between f. g. projective R/I-modules can be lifted to a homomorphism from a resolution of V to a resolution of V′, and this lifted homomorphism then induces a homomorphism between the corresponding Koszul complexes K and K′ in the obvious way. In order to prove that the induced homomorphism between the homology modules does not depend on the lifting, we consider the complex

L : 0 → P ⊗ P → Sym^{2}(P) ⊕ P ⊗ Q → Sym^{2}(Q) → 0

(with the obvious differentials) which is quasi-isomorphic to the complex K. One can show (see also Remark 2.6) that any homotopy between two homomorphisms between the resolutions induces a homotopy between the corresponding homomorphisms between the complexes L and L′. (The analogous statement for the complexes K and K′ is not true!) As usual in homological algebra, this implies that the association V ↦ H_{1}(L) ≅ F (V) is a well-defined functor. In particular, for any f. g. projective R/I-modules V, W, the direct sum F (V) ⊕ F (W) is a canonical direct summand of F (V ⊕ W). Let cr_{2}(F)(V, W) denote the complement. Then cr_{2}(F) is a bifunctor. It is called the second cross effect of F. (This terminology has been introduced by Eilenberg and MacLane in [EM].) One easily sees that we have cr_{2}(F) ≅ Tor_{1}^{R} in our situation. Hence, cr_{2}(F) is biadditive, and we have cr_{2}(F)(R/I, R/I) ≅ I/I^{2}. Furthermore, one immediately sees that F (R/I) ≅ 0. Eilenberg ([E]) and Watts ([Wa]) have proved that any additive functor F on the category of f. g. projective modules over a commutative ring A is isomorphic to the tensor functor F (A) ⊗ −. Similarly, one easily shows that any functor F with the three properties just proved is isomorphic to the functor Λ^{2}(−) ⊗ I/I^{2} (see Corollary 1.6). This finally proves the claimed isomorphism H_{1}(K) ≅ Λ^{2}(V) ⊗ I/I^{2} and hence the formula (1). The analogous formula for λ_{2} can be proved in a similar manner (see also Remark 3.6).

Now we briefly describe the idea of our approach in the case codim(Y/X) = 2, n = 2. Let V be a locally free O_{Y} -module and P. a locally free resolution of i_{∗}(V) on X. Here, we compute the homology of the complex L := NSym^{2}Γ(P.) where Γ and N denote the functors occurring in the Dold-Kan correspondence (see section 2). The idea to consider this complex is suggested by the observation that the complex L used in the case codim(Y/X) = 1 is isomorphic to the complex NSym^{2}Γ(P → Q) (see section 2). On the contrary, it seems that an analogue of the Koszul complex K does not exist in the case codim(Y/X) ≥ 2. We obtain the following result (see Theorem 6.4): H_{0}(L) ≅ i_{∗}(Sym^{2}(V)), H_{1}(L) ≅ i_{∗}(Λ^{2}(V)⊗C), H_{2}(L) ≅ i_{∗}(D^{2}(V)⊗ Λ^{2}(C)), and H_{k}(L) ≅ 0 for k ≥ 3. This computation of the homology then implies the Adams-Riemann-Roch formula in the case codim(Y/X) = 2, n = 2 as above (see section 6). The main tools we use in this computation are the Eilenberg-Zilber theorem and the universal form of the Cauchy decomposition developed by Akin, Buchsbaum and Weyman in [ABW]. We again verify certain properties of the corresponding cross effect functors which, together with easy abstract lemmas, imply the above isomorphisms.

In section 5, we present a further application of the theory of cross effect functors developed in section 1. We prove the Riemann-Roch formula for tensor power operations (see Theorem 5.3) which has been proved in the paper [K 2] using the deformation to the normal cone (and which in turn implies the Adams-Riemann-Roch formula in characteristic 0, see section 4 in [K 2]). Here, the codimension of Y in X may be arbitrary. For any locally free O_{Y} -module V and for any locally free resolution P. → i_{∗}(V) of i_{∗}(V) on X, we compute the homology of the total complex of the n-th tensor power P.^{⊗n} of P. together with the canonical action of the symmetric group (see Theorem 5.1).

Acknowledgments. I would like to thank D. Grayson and R. McCarthy for many very helpful discussions where I have learned, for example, the (advantages of) simplicial techniques. In particular, D. Grayson has contributed the basic idea for the important Corollary 2.5. Furthermore, R. McCarthy has explained the main features of cross effect functors to me. It is a pleasure to thank them for all this assistance.

1. On Cross Effect Functors

First, we recall the definition of the cross effect functors cr_{k}(F), k ≥ 0, associated with a functor F between additive categories (cf. [EM] and [JM]). Then, we introduce the diagonal and plus maps and describe these maps explicitly for the second and third symmetric power functor. The most important result of this section is Theorem 1.5 which states that any functor F (of finite degree) from the category of f. g. projective modules over a ring A to any abelian category is essentially determined by the objects cr_{k}(F)(A, …, A), k ≥ 0, and the diagonal and plus maps between these objects. Finally, we characterize the classical exterior power, symmetric power and divided power functors.

Let P be an additive category, M an abelian category, and F : P → M a functor with F (0) = 0. Definition 1.1. Let k ≥ 0. For any V_{1}, …, V_{k} ∈ P and 1 ≤ i ≤ k, let

p_{i} : V_{1} ⊕ … ⊕ V_{k} → V_{i} → V_{1} ⊕ … ⊕ V_{k}

denote the i-th projection. The k-functor

\begin{aligned}cr_{k}(F) : & P^{k} & → & M \\ & (V_{1}, …, V_{k}) & ↦ & Im ( ∑_{j=1}^{k}(−1)^{k−j} ∑_{1≤i_{1}<...<i_{j}≤k} F (p_{i_{1}} + … + p_{i_{j}})) \end{aligned}

is called the k-th cross effect of F; here, for any V_{1}, …, V_{k}, W_{1}, …, W_{k} ∈ P and f_{1} ∈ Hom_{P}(V_{1}, W_{1}), …, f_{k} ∈ Hom_{P}(V_{k}, W_{k}), the map

cr_{k}(F)(f_{1}, …, f_{k}) : cr_{k}(F)(V_{1}, …, V_{k}) → cr_{k}(F)(W_{1}, …, W_{k})

is induced by f_{1} ⊕ … ⊕ f_{k} ∈ Hom_{P}(V_{1} ⊕ … ⊕ V_{k}, W_{1} ⊕ … ⊕ W_{k}). F is said to be a functor of degree ≤ k if cr_{k+1}(F) is identically zero.

The cross effect functors cr_{k}(F), k ≥ 0, have the following properties. We obviously have cr_{0}(F) ≡ 0 and cr_{1}(F) = F. Furthermore, we have cr_{k}(F)(V_{1}, …, V_{k}) = 0, if V_{i} = 0 for one i ∈ {1, …, k} (see Theorem 9.2 on p. 79 in [EM]). The canonical action of the symmetric group Σ_{k} on V^{k} = V ⊕ … ⊕ V induces a natural action of Σ_{k} on cr_{k}(F)(V, …, V) (see Theorem 9.3 on p. 80 in [EM]). Any natural transformation F → G between functors F and G from P to M induces a natural transformation cr_{k}(F) → cr_{k}(G) (see Theorem 9.5 on p. 80 in [EM]). The most important property is the following proposition. Proposition 1.2. For any k, l ≥ 1 and V_{1}, …, V_{l} ∈ P, we have a canonical isomorphism

cr_{k}(F)(…, V_{1} ⊕ … ⊕ V_{l}, …)

≅ ⊕_{1≤j≤l} ⊕_{1≤i_{1}<...<i_{j}≤l} cr_{k+j}(F)(…, V_{i_{1}}, …, V_{i_{j}}, …)

which is functorial in V_{1}, …, V_{l}. In particular, F is of degree ≤ k, if and only if cr_{k}(F) is a k-additive functor.

Proof. See Theorem 9.1, Lemma 9.8 and Lemma 9.9 in [EM]. □

For instance, Proposition 1.2 states that

F (V_{1} ⊕ V_{2}) ≅ F (V_{1}) ⊕ F (V_{2}) ⊕ cr_{2}(F)(V_{1}, V_{2})

for all V_{1}, V_{2} ∈ P; i.e., cr_{2}(F) measures the deviation from additivity of the functor F. This isomorphism can also be used to define cr_{2}(F) (see section 3 in [JM]). Similarly, the higher cross effects can inductively be defined by the isomorphism

cr_{k}(F)(V_{1}, …, V_{k−1}, V_{k} ⊕ V_{k+1})
≅ cr_{k}(F)(V_{1}, …, V_{k−1}, V_{k}) ⊕ cr_{k}(F)(V_{1}, …, V_{k−1}, V_{k+1})
⊕cr_{k+1}(F)(V_{1}, …, V_{k}, V_{k+1})

(see section 7 in [JM]). In fact, we will use this definition whenever we have to compute cross effect functors. Definition 1.3. Let l ≥ k ≥ 1, ε = (ε_{1}, …, ε_{k}) ∈ {1, …, l}^{k} with |ε| = ε_{1} + … + ε_{k} = l and V_{1}, …, V_{k} ∈ P. The composition

\xymatrix{Δ_{ε} : cr_{k}(F)(V_{1}, …, V_{k}) \ar[r]^{cr_{k}(F)(Δ,... ,Δ)} & cr_{k}(F)(V_{1}^{ε_{1}}, …, V_{k}^{ε_{k}})}
[figure: π; cr_{l}(F)(V_{1}, …, V_{1}, …, V_{k}, …, V_{k})]

of the map cr_{k}(F)(Δ, …, Δ) (induced by the diagonal maps Δ : V_{i} → V_{i}^{ε_{i}}, i = 1, …, k) with the canonical projection π (according to Proposition 1.2) is called diagonal map associated with ε. The analogous composition

\xymatrix{+_{ε} : cr_{l}(F)(V_{1}, …, V_{1}, …, V_{k}, …, V_{k}) \ar@{^{(}->}[r] & cr_{k}(F)(V_{1}^{ε_{1}}, …, V_{k}^{ε_{k}})}
[figure: cr_{k}(F)(+,... ,+); cr_{k}(F)(V_{1}, …, V_{k})]

is called plus map associated with ε.

Obviously, the maps Δ_{ε} and +_{ε} form natural transformations between the functors cr_{k}(F) and cr_{l}(F) ∘ (Δ_{ε_{1}}, …, Δ_{ε_{k}}) from P^{k} to M. One easily sees that the map Δ_{ε} can be decomposed into a composition of maps Δ_{δ} with δ ∈ {1, 2}^{j} such that |δ| = j + 1 and j ∈ {k, …, l − 1}. The same holds for +_{ε}. Example 1.4. Let P = M be the category of modules over a commutative ring A. For any n ≥ 1, let Sym^{n} : P → M denote the n-th symmetric power functor.

(a) For all k ≥ 1 and V_{1}, …, V_{k} ∈ P, we have:

cr_{k}(Sym^{n})(V_{1}, …, V_{k}) ≅ ⊕_{{(n_{1},... ,n_{k})∈\{1,... ,n\}^{k} \atop n_{1}+...+n_{k}=n}} Sym^{n_{1}}(V_{1}) ⊗ … ⊗ Sym^{n_{k}}(V_{k}).

(b) Let n = 2. Then the following diagrams commute for all V ∈ P:

[figure: cr_{2}(Sym^{2})(V, V); Sym^{2}(V); Δ_{2}; V ⊗ V; v_{1}v_{2}; v_{1} ⊗ v_{2} + v_{2} ⊗ v_{1}; and; cr_{2}(Sym^{2})(V, V); +_{2}; V ⊗ V; Sym^{2}(V); v_{1} ⊗ v_{2}; v_{1}v_{2}.]

(c) Let n = 3. By virtue of the isomorphisms given in (a), the diagonal and plus maps can be described as follows (for V, W ∈ P):

\begin{aligned}Δ_{3} : & Sym^{3}(V) & → & V ⊗ V ⊗ V \\ & v_{1}v_{2}v_{3} & ↦ & ∑_{σ∈Σ_{3}} v_{σ(1)} ⊗ v_{σ(2)} ⊗ v_{σ(3)}. \end{aligned}

\begin{aligned}Δ_{(2,1)} : & Sym^{2}(V) ⊗ W ⊕ V ⊗ Sym^{2}(W) & → & V ⊗ V ⊗ W \\ & (v_{1}v_{2} ⊗ w, v ⊗ w_{1}w_{2}) & ↦ & v_{1} ⊗ v_{2} ⊗ w + v_{2} ⊗ v_{1} ⊗ w. \end{aligned}

\begin{aligned}Δ_{(1,2)} : & Sym^{2}(V) ⊗ W ⊕ V ⊗ Sym^{2}(W) & → & V ⊗ W ⊗ W \\ & (v_{1}v_{2} ⊗ w, v ⊗ w_{1}w_{2}) & ↦ & v ⊗ w_{1} ⊗ w_{2} + v ⊗ w_{2} ⊗ w_{1}. \end{aligned}

Δ_{2} : Sym^{3}(V) → Sym^{2}(V) ⊗ V ⊕ V ⊗ Sym^{2}(V)

v_{1}v_{2}v_{3} ↦ (v_{2}v_{3} ⊗ v_{1} + v_{1}v_{3} ⊗ v_{2} + v_{1}v_{2} ⊗ v_{3}, v_{1} ⊗ v_{2}v_{3} + v_{2} ⊗ v_{1}v_{3} + v_{3} ⊗ v_{1}v_{2}).

\begin{aligned}+_{3} : & V ⊗ V ⊗ V & → & Sym^{3}(V) \\ & v_{1} ⊗ v_{2} ⊗ v_{3} & ↦ & v_{1}v_{2}v_{3}. \end{aligned}

\begin{aligned}+_{(2,1)} : & V ⊗ V ⊗ W & → & Sym^{2}(V) ⊗ W ⊕ V ⊗ Sym^{2}(W), \\ & v_{1} ⊗ v_{2} ⊗ w & ↦ & (v_{1}v_{2} ⊗ w, 0). \end{aligned}

\begin{aligned}+_{(1,2)} : & V ⊗ W ⊗ W & → & Sym^{2}(V) ⊗ W ⊕ V ⊗ Sym^{2}(W) \\ & v ⊗ w_{1} ⊗ w_{2} & ↦ & (0, v ⊗ w_{1}w_{2}). \end{aligned}

+_{2} : Sym^{2}(V) ⊗ V ⊕ V ⊗ Sym^{2}(V) →  Sym^{3}(V)
(v_{1}v_{2} ⊗ v_{3}, v′_{1} ⊗ v′_{2}v′_{3})  ↦ v_{1}v_{2}v_{3} + v′_{1}v′_{2}v′_{3}.

Proof. Straightforward. □

The following theorem states that any functor F of finite degree from the category of all f. g. projective modules over a ring A to any abelian category M is determined by the objects cr_{i}(F)(A, …, A), i ≥ 0, the diagonal and plus maps between these objects, and the actions

A → End_{M}(cr_{i}(F)(A, …, A)), a ↦ cr_{i}(F)(1, …, 1, a, 1, …, 1),

of the multiplicative monoid A on these objects. (Here, 1 and a denote the multiplication with 1 and a on A, respectively.) Theorem 1.5. Let A be a ring, M an abelian category, d ∈ ℕ, and

\xymatrix{F, G : (f. g. projective A-modules) \ar[r] \ar[r] & M}

two functors of degree ≤ d with F (0) = 0 = G(0). Suppose that there exist isomorphisms

\xymatrix{α_{i}(A, …, A) : cr_{i}(F)(A, …, A) \ar[r]^{∼} & cr_{i}(G)(A, …, A),} i = 1, …, d,

which are compatible with the action of A in each component and which make the following diagrams commute for all i ∈ {1, …, d−1} and ε ∈ {1, 2}^{i} with |ε| = i+1:

\xymatrix{cr_{i}(F)(A, …, A) \ar[r]^{∼} \ar[d]^{Δ_{ε}} & cr_{i}(G)(A, …, A) \ar[d]^{Δ_{ε}} \\ cr_{i+1}(F)(A, …, A) \ar[r]^{∼} & cr_{i+1}(G)(A, …, A)}

and

\xymatrix{cr_{i+1}(F)(A, …, A) \ar[r]^{∼} \ar[d]^{+_{ε}} & cr_{i+1}(G)(A, …, A) \ar[d]^{+_{ε}} \\ cr_{i}(F)(A, …, A) \ar[r]^{∼} & cr_{i}(G)(A, …, A).}

Then the functors F and G are isomorphic.

Proof. For any i ≥ 1 and n_{1}, …, n_{i} ≥ 1, we write cr_{i}(F)(A^{n_{1}}, …, A^{n_{i}}) as a direct sum of copies of cr_{i}(F)(A, …, A), …, cr_{d}(F)(A, …, A) according to Proposition 1.2; in the same way, we write cr_{i}(G)(A^{n_{1}}, …, A^{n_{i}}) as a direct sum of copies of cr_{i}(G)(A, …, A), …, cr_{d}(G)(A, …, A). The corresponding sum of copies of the isomorphisms α_{i}(A, …, A), …, α_{d}(A, …, A) then yields an isomorphism

\xymatrix{α_{i}(A^{n_{1}}, …, A^{n_{i}}) : cr_{i}(F)(A^{n_{1}}, …, A^{n_{i}}) \ar[r]^{∼} & cr_{i}(G)(A^{n_{1}}, …, A^{n_{i}}).}

Now, we show by descending induction on i that, for all m_{1}, …, m_{i}, n_{1}, …, n_{i} ≥ 1 and for all matrices A_{1} ∈ Hom(A^{n_{1}}, A^{m_{1}}), …, A_{i} ∈ Hom(A^{n_{i}}, A^{m_{i}}), the following diagram commutes:

\xymatrix{cr_{i}(F)(A^{n_{1}}, …, A^{n_{i}}) \ar[r]^{α_{i}(A^{n_{1}} ,... ,A^{n_{i}})} \ar[d]^{cr_{i}(F)(A_{1},... ,A_{i})} & cr_{i}(G)(A^{n_{1}}, …, A^{n_{i}}) \ar[d]^{cr_{i}(G)(A_{1},... ,A_{i})} \\ cr_{i}(F)(A^{m_{1}}, …, A^{m_{i}}) \ar[r]^{α_{i}(A^{m_{1}} ,... ,A^{m_{i}})} & cr_{i}(G)(A^{m_{1}}, …, A^{m_{i}}).}

For i = d+1, this assertion is clear since cr_{d+1}(F) and cr_{d+1}(G) are identically zero. So, we may assume that i ≤ d and that the assertion is already proved for i + 1. Writing (A_{1}, …, A_{i}) as the composition of (A_{1}, id, …, id), …, (id, …, id, A_{i}) and using symmetry, we may furthermore assume that A_{2}, …, A_{i} are the identity matrices. By construction of α_{i}(A^{n_{1}}, …, A^{n_{i}}), we may finally assume that n_{2} = … = n_{i} = 1. Now, we write A for A_{1}, cr_{i}(F)(A^{n}) for cr_{i}(F)(A^{n_{1}}, A, …, A), and similarly for cr_{i}(G)(A^{n_{1}}, A, …, A), etc. By construction, the above diagram commutes, if A is a standard projection A^{n} → A or a standard embedding A → A^{m}. Furthermore, the diagram commutes if n = m = 1 by assumption. Hence, the diagram commutes for all matrices A which have at most one entry which is different from zero. Thus it suffices to show that the diagram commutes for the matrix A+B if it commutes for the matrices A and B. For this, we decompose A + B into the composition

\xymatrix{A^{n} \ar[r]^{Δ} & A^{n} ⊕ A^{n} \ar[r]^{A⊕B} & A^{m} ⊕ A^{m} \ar[r]^{+} & A^{m}}

and identify cr_{i}(−)(A^{n} ⊕ A^{n}) with cr_{i}(−)(A^{n}) ⊕ cr_{i}(A^{n}) ⊕ cr_{i+1}(−)(A^{n}, A^{n}) according to Proposition 1.2. By construction and assumption, the following diagram commutes:

\xymatrix{cr_{i}(F)(A^{n}) \ar[r]^{α_{i}(A^{n})} \ar[d]^{cr_{i}(F)(Δ)} & cr_{i}(G)(A^{n}) \ar[d]^{cr_{i}(G)(Δ)} \\ cr_{i}(F)(A^{n} ⊕ A^{n}) \ar[r]^{α_{i}(A^{n})⊕α_{i}(A^{n})⊕α_{i+1}(A^{n},A^{n})} & cr_{i}(G)(A^{n} ⊕ A^{n}).}

(To see this, one decomposes the map cr_{i}(F)(Δ) into a direct sum of copies of the maps

cr_{i+j}(F)(Δ, …, Δ) : cr_{i+j}(F)(A, …, A) → cr_{i+j}(F)(A ⊕ A, …, A ⊕ A), j ≥ 0,

according to Proposition 1.2. The composition of such a map with a natural projection cr_{i+j}(F)(A ⊕ A, …, A ⊕ A) → cr_{i+j+k}(F)(A, …, A) (again according to Proposition 1.2) is then a diagonal map Δ_{ε} for some ε ∈ {1, 2}^{i+j} with |ε| = i+j +k.) Likewise, the corresponding diagram for + in place of Δ commutes.

By the assumption on A and B and by the induction hypothesis, also the following diagram commutes:

\xymatrix{cr_{i}(F)(A^{n} ⊕ A^{n}) \ar[r]^{α_{i}(A^{n})⊕α_{i}(A^{n})⊕α_{i+1}(A^{n},A^{n})} \ar[d]^{cr_{i}(F)(A⊕B)} & cr_{i}(G)(A^{n} ⊕ A^{n}) \ar[d]^{cr_{i}(G)(A⊕B)} \\ cr_{i}(F)(A^{m} ⊕ A^{m}) \ar[r]^{α_{i}(A^{m})⊕α_{i}(A^{m})⊕α_{i+1}(A^{m},A^{m})} & cr_{i}(G)(A^{m} ⊕ A^{m}).}

Now, the proof of the above assertion is complete. This assertion implies in particular that, for all n ≥ 1 and for all projectors A ∈ End(A^{n}), the isomorphism

\xymatrix{F (A^{n}) = cr_{1}(F)(A^{n}) \ar[r]^{α_{1}(A^{n})} & cr_{1}(G)(A^{n}) = G(A^{n})}

induces an isomorphism

\xymatrix{α_{1}(Im(A)) : F (Im(A)) ≅ Im(F (A)) \ar[r]^{∼} & Im(G(A)) ≅ G(Im(A)).}

If B ∈ End(A^{m}) is a further projector and f : Im(A) → Im(B) is an A-homo-morphism, then the diagram

\xymatrix{F (Im(A)) \ar[r]^{α_{1}(Im(A))} \ar[d]^{F (f)} & G(Im(A)) \ar[d]^{G(f)} \\ F (Im(B)) \ar[r]^{α_{1}(Im(B))} & G(Im(B))}

commutes (again by the above assertion). Finally, if V is an arbitrary f. g. projective A-module, we choose a projector A as above and an isomorphism V ≅ Im(A) and define α_{1}(V) to be the composition

\xymatrix{F (V) ≅ F (Im(A)) \ar[r]^{α_{1}(Im(A))} & G(Im(A)) ≅ G(V).}

It is clear that α_{1}(V) does not depend on the chosen projector and on the chosen isomorphism and that α_{1} is an isomorphism of functors. This proves Theorem 1.5. □

Corollary 1.6 (Characterization of the exterior power functors). Let A be a commutative ring, B a further ring, d ∈ ℕ, and

F : P_{A} := (f. g. projective A-modules) → (B-modules) =: M_{B}

a functor with F (0) = 0 and with the following properties:

(i) cr_{d}(F) is d-additive, i.e., cr_{d+1}(F) ≡ 0. (ii) The d A-module structures on cr_{d}(F)(A, …, A) coincide. (iii) F (A^{d−1}) = 0. Then we have for all V ∈ P_{A}:

F (V) ≅ Λ^{d}(V) ⊗_{A} cr_{d}(F)(A, …, A).

Proof. The assumption F (A^{d−1}) = 0 implies

cr_{1}(F)(A) = 0, …, cr_{d−1}(F)(A, …, A) = 0

by Proposition 1.2. Similarly, for the functor G defined by G(V) := Λ^{d}(V) ⊗_{A} cr_{d}(F)(A, …, A), we have cr_{1}(G)(A) = 0, …, cr_{d−1}(G)(A, …, A) = 0. Furthermore, we have cr_{d}(G)(A, …, A) ≅ cr_{d}(F)(A, …, A). Thus, Corollary 1.6 follows from Theorem 1.5. □

Example 1.7. In the case d = 1, Corollary 1.6 yields the following statement which has already been proved in the papers [E] and [Wa]: Each additive functor F : P_{A} → M_{B} is isomorphic to the tensor functor − ⊗ F (A). By induction, we obtain the following generalization: Let F : P_{A}^{d} → M be a d-additive functor such that the d A-module structures on F (A, …, A) coincide. Then we have for all V_{1}, …, V_{d} ∈ P_{A}:

F (V_{1}, …, V_{d}) ≅ V_{1} ⊗_{A} … ⊗_{A} V_{d} ⊗_{A} F (A, …, A).

Proposition 1.8 (Characterization of the symmetric power functors). Let A, B, d, and F as in Corollary 1.6. In contrast to Corollary 1.6, we assume however that F has the following property in place of property (iii): For all i = 1, …, d − 1, the B-module homomorphism

\xymatrix{plus_{i} : ⊕_{ε∈\{1,... ,d\}^{i}, |ε|=d} cr_{d}(F)(A, …, A) \ar[r]^{(+_{ε})_{ε∈\{1,... ,d\}^{i},|ε|=d}} & cr_{i}(F)(A, …, A)}

is bijective. Then we have for all V ∈ P_{A}:

F (V) ≅ Sym^{d}(V) ⊗_{A} cr_{d}(F)(A, …, A).

Proof. We view cr_{d}(F)(V, …, V) as a Σ_{d}-module with the action introduced above and F (V) as a Σ_{d}-module with trivial action. Then, the plus map

+_{d} : cr_{d}(F)(V, …, V) → F (V)

is obviously compatible with these Σ_{d}-actions. Since +_{d} : cr_{d}(F)(A, …, A) → F (A) is bijective by assumption, the symmetric group Σ_{d} acts trivially on the object cr_{d}(F)(A, …, A). Hence, the composition

V^{⊗d} ⊗_{A} cr_{d}(F)(A, …, A) ≅ cr_{d}(F)(V, …, V) ⟶^{+_{d}} F (V)

of the isomorphism given in Example 1.7 with the plus map +_{d} induces a B-module homomorphism

α(V) : Sym^{d}(V) ⊗_{A} cr_{d}(F)(A, …, A) → F (V)

which is functorial in V. One easily proves that, for all i = 1, …, d, the homomorphism cr_{i}(α)(A, …, A) coincides with the isomorphism plus_{i} by virtue of the isomorphism

cr_{i}(Sym^{d})(A, …, A) ≅ ⊕_{ε∈{1,... ,d}^{i}, |ε|=d} A

(see Example 1.4(a)). Hence, cr_{i}(α)(A, …, A) is bijective. Then, by Proposition 1.2, α(V) is bijective for all f. g. free A-modules V. Again by Proposition 1.2, the same then holds for all V ∈ P_{A}. This proves Proposition 1.8. □

Proposition 1.9 (Characterization of the divided power functors). Let A, B, d, and F be as in Corollary 1.6. In contrast to Corollary 1.6, we assume however that F has the following property in place of property (iii): For all i = 1, …, d − 1, the B-module homomorphism

\xymatrix{diag_{i} : cr_{i}(F)(A, …, A) \ar[r]^{(Δ_{ε})_{ε∈\{1,... ,d\}^{i},|ε|=d}} & ⊕_{ε∈\{1,... ,d\}^{i},|ε|=d} cr_{d}(F)(A, …, A)}

is bijective. Then we have for all V ∈ P_{A}:

F (V) ≅ D^{d}(V) ⊗_{A} cr_{d}(F)(A, …, A).

Proof. The d-th divided power D^{d}(V) of a f. g. projective A-module V is the module of fixed elements under the natural action of the symmetric group Σ_{d} on V^{⊗d}. Thus, Proposition 1.9 can be proved analogously to Proposition 1.8. □

Remark 1.10. For any functor F : P_{A} → M_{B} with F (0) = 0, the composition

\xymatrix{F (A) \ar[r]^{Δ_{2}} & cr_{2}(F)(A, A) \ar[r]^{+_{2}} & F (A)} equals F (2 · id_{A}) − 2 · id_{F (A)}. Now assume that d = 2. Under the assumptions of Proposition 1.8, the endomorphism F (2 · id_{A}) of F (A) then corresponds to the endomorphism cr_{2}(F)(2 · id_{A}, 2 · id_{A}) = 4 · id of cr_{2}(F)(A, A); hence, F (2 · id_{A}) = 4 · id_{F (A)}. Thus, +_{2} ∘ Δ_{2} = 2 · id_{F (A)}. This proves that the assumptions of Proposition 1.8 also determine the map Δ_{2}. Thus, Proposition 1.8 follows from Theorem 1.5 in the case d = 2. The same holds for Proposition 1.9. It seems that a simple generalization of this argument to the case d ≥ 2 does not exist.

2. Simplicial Modules and Koszul Complexes

First, we recall the Dold-Kan correspondence; in particular, we introduce the functors Γ and N between the category of simpicial modules and the category of complexes. Then, we recall the definition of the n-th Koszul complex Kos^{n}(f) associated with a homomorphism f : P → Q between the projective modules P, Q over a commutative ring R. Finally, we construct an explicit complex homomorphism u^{n}(f) from Kos^{n}(f) to NSym^{n}Γ(… → P →^{f} Q) and prove that u^{n}(f) is a quasi-isomorphism.

Let M be an additive category. Let Δ denote the category whose objects are the sets [n] = {0 < 1 < … < n}, n ≥ 0, and whose morphisms are the order preserving set maps. For 0 ≤ i ≤ n + 1, let δ_{i}^{n} : [n] → [n + 1] denote the injective order preserving map given by Im(δ_{i}^{n}) = [n + 1]\{i}. For 0 ≤ i ≤ n, let σ_{i}^{n} : [n + 1] → [n] denote the surjective order preserving map given by (σ_{i}^{n})^{−1}(i) = {i, i + 1}. A simplicial object X. in M is a contravariant functor from Δ to M. We write X_{n} for X.([n]), d_{i} : X_{n+1} → X_{n} for X.(δ_{i}^{n}) and s_{i} : X_{n} → X_{n+1} for X.(σ_{i}^{n}). A simplicial homomorphism f. : X. → Y. between two simplicial objects X., Y. in M is a natural transformation. A simplicial homotopy between simplicial homomorphisms f, g : \xymatrix{X. \ar[r] \ar[r] & Y} . consists of homomorphisms h_{i}(n) : X_{n} → Y_{n+1}, n ≥ 0, 0 ≤ i ≤ n, such that d_{0}h_{0}(n) = f_{n} and d_{n+1}h_{n}(n) = g_{n} for all n ≥ 0 and such that certain further relations d_{i}h_{j}(n) = … and s_{i}h_{j}(n) = … hold (see 8.3.11 in [W] or (2.3) in [JM]). A complex K. in M is a contravariant functor from the ordered set ℕ_{0} to the category M such that ∂^{2} = 0 where ∂ : K_{i+1} → K_{i} denotes the differential. The normalized complex N(X.) of a simplicial object X. in an abelian category M is given by

N(X.)_{n} := X_{n} /∑_{i=0}^{n−1} Im(s_{i} : X_{n−1} → X_{n}) ;

the differential in N(X.) is

∂ = ∑_{i=0}^{n}(−1)^{i}d_{i} : N(X.)_{n} → N(X.)_{n−1}

(for all n ≥ 0). Thus, we have a functor

N : Simp(M) → Compl(M)

from the category Simp(M) of simplicial objects in M to the category Compl(M) of complexes in M. We define a functor

Γ : Compl(M) → Simp(M)

in the reverse direction as follows. For n ≥ k, let Sur([n], [k]) denote the set of surjective order preserving maps from [n] to [k]. For (M., ∂) ∈ Compl(M) and n ≥ 0, let Γ(M.)_{n} be defined by

Γ(M.)_{n} := ⊕_{k=0}^{n} ⊕_{σ∈Sur([n],[k])} M_{k};

furthermore, d_{i} : Γ(M.)_{n} → Γ(M.)_{n−1}, i ∈ {0, …, n}, and s_{i} : Γ(M.)_{n} → Γ(M.)_{n+1}, i ∈ {0, …, n}, are defined by

d_{i}(m, σ) := \{ \begin{matrix}(m, σδ_{i}) & if σδ_{i} ∈ Sur([n − 1], [k]) \\ (∂(m), σ̂) & if σδ_{i} = δ_{0}σ̂ (where σ̂ ∈ Sur([n − 1], [k − 1])) \\ 0 & if σδ_{i} = δ_{j}σ̂ with j ≥ 1 \end{matrix}
s_{i}(m, σ) := (m, σσ_{i})

(for k ∈ {0, …, n} and (m, σ) ∈ M_{k} × Sur([n], [k])). One easily verifies that these definitions determine a simplicial object Γ(M.) in M (cf. section 2 in [JM] or section 8.4 in [W]). Proposition 2.1 (Dold-Kan correspondence). For any abelian category M, the functors N and Γ are inverse to each other (up to canonical isomorphisms). A simplicial homotopy between two simplicial homomorphisms induces a homotopy between the corresponding complex homomorphisms and vice versa.

Proof. See section 8.4 in [W]. □

Now, let F : P → M be a functor from an additive category P to an abelian category M with F (0) = 0. If F is additive, then the induced functor Compl(P) → Compl(M), K. ↦ F ∘ K., maps homotopic complex homomorphisms to homotopic complex homomorphisms. In the simplicial world, the analogous statement holds even for an arbitrary functor F: The induced functor Simp(P) → Simp(M), X. ↦ F X. := F ∘ X., maps simplicial homotopies to simplicial homotopies since only compositions (and no sums) are involved in the defining relations of a simplicial homotopy. Hence, using the Dold-Kan correspondence, we obtain a functor

Compl(P) → Compl(M), K. ↦ NF Γ(K.),

which maps homotopic complex homomorphisms to homotopic complex homomorphisms and which is obviously isomorphic to the functor K. ↦ F ∘ K., if F is additive. (This basic fact has already been observed by Dold and Puppe in [DP].) If K. is a complex in P with K_{n} = 0 for all n ≥ 2, then the complex NF Γ(K.) can be described as follows by using the cross effect functors introduced in section 1.

Lemma 2.2. Let f : P → Q be a homomorphism in P. We consider f as a complex with Q at the place 0 and P at the place 1. Then we have for all n ≥ 0:

NF Γ(P → Q)_{n} ≅ cr_{n}(F)(P, …, P) ⊕ cr_{n+1}(F)(Q, P, …, P);

the differential ∂ : NF Γ(P → Q)_{n} → NF Γ(P → Q)_{n−1} is given by the following diagram:

[figure: cr_{n}(F)(P, …, P); ⊕; cr_{n−1}(F)(P, …, P); ⊕; cr_{n+1}(F)(Q, P, …, P); cr_{n}(F)(Q, P, …, P);]

here, the upper horizontal map is the alternating sum ∑_{i=1}^{n−1}(−1)^{i}(+_{ε^{i}}) of the plus maps associated with ε^{1} = (2, 1, …, 1), …, ε^{n−1} = (1, …, 1, 2); the lower horizontal map is ((+_{(2,1,... ,1)}) ∘ cr_{n+1}(F)(id_{Q}, f, id_{P} , …, id_{P})) + ∑_{i=1}^{n−1}(−1)^{i}(+_{(1,ε^{i})}); the diagonal map is cr_{n}(F)(f, id_{P} , …, id_{p}).

Proof. By construction of Γ, we have Γ(P → Q)_{n} ≅ Q ⊕ P^{n} for all n ≥ 0. By virtue of these isomorphisms, the maps d_{i} : Γ(P → Q)_{n} → Γ(P → Q)_{n−1} and s_{i} : Γ(P → Q)_{n} → Γ(P → Q)_{n+1} can be described as follows:

d_{i}(q, p_{1}, …, p_{n}) = \{ \begin{matrix}(q + f(p_{1}), p_{2}, …, p_{n}) & if i = 0 \\ (q, p_{1}, …, p_{i−1}, p_{i} + p_{i+1}, p_{i+2}, …, p_{n}) & if 1≤i≤n − 1 \\ (q, p_{1}, …, p_{n−1}) & if i = n \end{matrix}
s_{i}(q, p_{1}, …, p_{n}) = (q, p_{1}, …, p_{i}, 0, p_{i+1}, …, p_{n}) if 0 ≤ i ≤ n

(see also Lemma A.2 in [JM]). Hence, by Proposition 1.2, we have:

F Γ(P → Q)_{n} ≅ F (Q) ⊕ F (P^{n}) ⊕ cr_{2}(F)(Q, P^{n})

≅ ⊕_{k=0}^{n} \begin{matrix}( & {n \atop k} & ) \\ ⊕ \end{matrix} (cr_{k}(F)(P, …, P) ⊕ cr_{k+1}(F) + (Q, P, …, P)) .

Thus we have

NF Γ(P → Q)_{n} ≅ F Γ(P → Q)_{n} /(∑_{i=0}^{n−1} Im(s_{i}))

≅ cr_{n}(F)(P, …, P) ⊕ cr_{n+1}(F)(Q, P, …, P).

Finally, one can easily check now that the differential has the claimed form. □

For any object P in P and k ≥ 0, we write P [−k] for the complex which has P at the place k and 0 else. Lemma 2.2 states in particular that NF Γ(P [0]) ≅ F (P)[0] and that NF Γ(P [−1]) is isomorphic to the complex

… ⟶ cr_{n}(F)(P, …, P) ⟶^{∂_{n}} cr_{n−1}(F)(P, …, P) ⟶^{∂_{n−1}} … ⟶^{∂_{2}} F (P) ⟶ 0

where ∂_{n} = ∑_{i=1}^{n−1}(−1)^{i}(+_{ε^{i}}).

Now, we recall the definition of the Koszul complex (see I 4.3.1.3 in [I 1], V.1.3 in [ABW], or section 2 in [G]). Let n ∈ ℕ, R a commutative ring, and f : P → Q a homomorphism between projective R-modules P, Q.

Definition 2.3. For any k ∈ {0, …, n + 1}, let

d_{k+1} : Λ^{k+1}(P) ⊗ Sym^{n−k−1}(Q) → Λ^{k}(P) ⊗ Sym^{n−k}(Q)

denote the Koszul differential given by

p_{1} ∧ … ∧ p_{k+1} ⊗ q_{k+2} ⋯ q_{n}

↦ ∑_{i=1}^{k+1}(−1)^{k+1−i}p_{1} ∧ … ∧ p̂_{i} ∧ … ∧ p_{k+1} ⊗ f(p_{i})q_{k+2} ⋯ q_{n}.

The complex

0 → Λ^{n}(P) →^{d_{n}} Λ^{n−1}(P) ⊗ Q →^{d_{n−1}} … →^{d_{2}} P ⊗ Sym^{n−1}(Q) →^{d_{1}} Sym^{n}(Q) → 0

is called n-th Koszul complex associated with f and is denoted by Kos^{n}(f).

One can prove analogously to Lemme 2.1.2.1 on page 277 in [I 2] that the complexes Kos^{n}(f) and NSym^{n}Γ(P → Q) are isomorphic in the derived category of R. Whereas Illusie uses three complex homomorphisms (which even can not be composed since they have different directions) to realize this isomorphism, we now explicitly construct a single complex homomorphism u^{n}(f) from Kos^{n}(f) to NSym^{n}Γ(P → Q) and show that u^{n}(f) is a quasi-isomorphism. For k = 0, …, n, let u_{k}^{n}(f) denote the composition

Λ^{k}(P) ⊗ Sym^{n−k}(Q) ⟶^{a_{k}⊗id} P^{⊗k} ⊗ Sym^{n−k}(Q)

˜ ⟶^{(1.4)(a)} cr_{k}(Sym^{k})(P, …, P) ⊗ Sym^{n−k}(Q)

↪^{can} ⊕_{i=0}^{n−k} cr_{k}(Sym^{k+i})(P, …, P) ⊗ Sym^{n−k−i}(Q)

˜ ⟶^{(1.4)(a)} cr_{k}(Sym^{n})(P, …, P) ⊕ cr_{k+1}(Sym^{n})(Q, P, …, P)

˜ ⟶^{(2.2)} NSym^{n}Γ(P → Q)_{k};

here, a_{k} : Λ^{k}(P) → P^{⊗k}, p_{1} ∧…∧p_{k} ↦ ∑_{σ∈Σ_{k}}(−1)^{sgn(σ)}p_{σ(1)} ⊗…⊗p_{σ(k)}, denotes the antisymmetrization map; the other homomorphisms are explained under the indicated example or lemma.

Proposition 2.4. The homomorphisms u_{k}^{n}(f), k = 0, …, n, define a quasi-iso-morphism

u^{n}(f) : Kos^{n}(f) ˜→ NSym^{n}Γ(P → Q).

Proof. We write S for Sym, and we set L. := NS^{n}Γ(P → Q). The following picture illustrates u^{n}(f):

[figure: Λ^{n}(P); a_{n}; Λ^{n−1}(P) ⊗ Q; a_{n−1}⊗id; …; P ⊗ S^{n−1}(Q); a_{1}⊗id; S^{n}(Q); id; S^{n}(Q); ⊕; P ⊗ S^{n−1}(Q); ⊕; 0; ⊕; _{...}; ⋮; ⊕; ⋮; ⊕; P^{⊗(n−1)} ⊗ Q; ⊕; …; S^{n−1}(P) ⊗ Q; ⊕; 0; ⊕; P^{⊗n}; cr_{n−1}(S^{n})(P, …, P); …; S^{n}(P); 0; L_{n}; ∂; L_{n−1}; ∂; …; ∂; L_{1}; ∂; L_{0};]

here, the differentials in the middle complex are given by Lemma 2.2. For all k = 0, …, n, the composition

\xymatrix{Λ^{k}(P) \ar[r]^{a_{k}} & P^{⊗k} ≅ cr_{k}(S^{k})(P, …, P) \ar[r]^{∑_{i=1}^{k−1}(−1)^{i}(+_{ε^{i}})} & cr_{k−1}(S^{k})(P, …, P)}

≅ (S^{2}(P) ⊗ P ⊗ … ⊗ P ) ⊕ … ⊕ (P ⊗ … ⊗ P ⊗ S^{2}(P))

is obviously the zero map; furthermore, one easily sees that the diagram

\xymatrix{Λ^{k+1}(P) ⊗ S^{n−k−1}(Q) \ar[rr]^{d_{k+1}} \ar[d]^{a_{k+1}⊗id} &  & Λ^{k}(P) ⊗ S^{n−k}(Q) \ar[d]^{a_{k}⊗id} \\ P^{⊗(k+1)} ⊗ S^{n−k−1}(Q) \ar[r]^{id⊗f⊗id} & P^{⊗k} ⊗ Q ⊗ S^{n−k−1}(Q) \ar[r]^{can} & P^{⊗k} ⊗ S^{n−k}(Q)}

commutes. This proves that u is a complex homomorphism. Obviously, the complexes Kos^{n}(f) and NS^{n}Γ(P → Q) can be filtered by subcomplexes such that u^{n}(f) is compatible with these filtrations and such that, for k = 0, …, n, the map between the k-th successive quotients induced by u^{n}(f) is the complex homomorphism

u^{k}(0) ⊗ id : Λ^{k}(P)[−k] ⊗ S^{n−k}(Q) → NS^{k}Γ(P → 0) ⊗ S^{n−k}(Q).

Thus it suffices to show that, for all n ≥ 0, the complex homomorphism

u^{n}(0) : Λ^{n}(P)[−n] → NS^{n}Γ(P → 0)

is a quasi-isomorphism, i.e., that Proposition 2.4 is true in the case Q = 0. We prove this by induction on n. For n = 0 this is clear. So let n ≥ 1. The homomorphism u^{n}(id_{P}) is a quasi-isomorphism since Kos^{n}(id_{P}) is exact and the complex NS^{n}Γ(P →^{id} P) is homotopic to the zero complex. Furthermore, the complex homomorphisms

u^{k}(0) ⊗ id : Λ^{k}(P)[−k] ⊗ S^{n−k}(P) → NS^{k}Γ(P → 0) ⊗ S^{n−k}(P),

k = 0, …, n − 1, are quasi-isomorphisms by the induction hypothesis. Using the filtrations already considered above, we obtain that u^{n}(0) is a quasi-isomorphism. (Alternatively, this also follows from Proposition I 4.3.2.1(i) on p. 111 in [I 1] and the following fact which is a little bit tedious to prove: The complex homomorphism NS^{n}Γ(P → 0) → Λ^{n}(P)[−n] constructed in [I 1] is left inverse to u^{n}(0).) This ends the proof of Proposition 2.4. □

Any commutative square

\xymatrix{P \ar[r]^{f} \ar[d]^{α_{P}} & Q \ar[d]^{α_{Q}} \\ P′ \ar[r]^{f′} & Q′}

of homomorphisms between projective R-modules induces a homomorphism

Kos^{n}(α) : Kos^{n}(f) → Kos^{n}(f′)

between the corresponding Koszul complexes in the obvious way. Corollary 2.5. If (α_{P}^{1} , α_{Q}^{1}) and (α_{P}^{2} , α_{Q}^{2}) are homotopic complex homomorphisms

between the complexes P →^{f} Q and P′ →^{f′} Q′, then the induced homomorphisms between the k-th homology modules of the corresponding n-th Koszul complexes are equal:

H_{k}(Kos^{n}(α^{1})) = H_{k}(Kos^{n}(α^{2})) : H_{k}(Kos^{n}(f)) → H_{k}(Kos^{n}(f′))

(for all k ≥ 0).

Proof. Obviously, the diagram

\xymatrix{Kos^{n}(f) \ar[r]^{u^{n}(f)} \ar[d]^{Kos^{n}(α^{i})} & NSym^{n}Γ(P →^{f} Q) \ar[d]^{NSym^{n}Γ(α^{i})} \\ Kos^{n}(f′) \ar[r]^{u^{n}(f′)} & NSym^{n}Γ(P′ →^{f′} Q′)}

commutes for i = 1, 2. Furthermore, the complex homomorphisms NSym^{n}Γ(α^{1}) and NSym^{n}Γ(α^{2}) are homotopic to each other. Thus, Corollary 2.5 follows from Proposition 2.4. □

Remark 2.6. In the next section, we will use only Corollary 2.5 (and not Proposition 2.4). The simplicial methods used here to prove Corollary 2.5 are actually not necessary: Alternatively, one can use use Lemma 2.2 to define the complex NF Γ(P → Q) and one can directly show that homotopic complex homomorphisms between P → Q and P′ → Q′ induce homotopic complex homomorphisms between NF Γ(P → Q) and NF Γ(P′ → Q′). However, the simplicial definition of NF Γ(K.) has the advantage that it yields a more natural argument for the proof of Corollary 2.5 and that it can be applied also to complexes K. of length > 1 (see also section Remark 2.7. One can prove analogously to Proposition 2.4 that the Koszul complex

K̃os^{n}(f) : 0 → D^{n}(P) → D^{n−1}(P) ⊗ Q → … → P ⊗ Λ^{n−1}(Q) → Λ^{n}(Q) → 0

(defined analogously to 2.3) is quasi-isomorphic to the complex NΛ^{n}Γ(P → Q).

3. Computation of the Homology of Koszul Complexes

Let R be a commutative ring and n ∈ ℕ. In this section, we compute the homology modules of the n-th Koszul complex Kos^{n}(f) for each homomorphism f : P → Q between f. g. projective R-modules which has the following properties: f is injective and Q/f(P) is a f. g. projective R/I-module for some ideal I in R which is locally generated by a non-zero-divisor.

First, we recall the definition of the Schur modules corresponding to Young diagrams of hook type. Definition 3.1. Let k ∈ {0, …, n} and V a f. g. projective R-module. Let

d_{k+1} : Λ^{k+1}(V) ⊗ Sym^{n−k−1}(V) → Λ^{k}(V) ⊗ Sym^{n−k}(V)

denote the differential in the Koszul complex Kos^{n}(id_{V}) (see Definition 2.3). The module

L_{k}^{n}(V) := Im(d_{k+1}) ⊆ Λ^{k}(V) ⊗ Sym^{n−k}(V)

is called Schur module associated with V and the Young diagram (k + 1, 1, …, 1) ((n − k − 1) 1s).

Since Kos^{n}(id_{V}) is exact, L_{k}^{n}(V) is a f. g. projective R-module for all k = 0, …, n. For instance, we have L_{0}^{n}(V) ≅ Sym^{n}(V), L_{n−1}^{n}(V) ≅ Λ^{n}(V), and L_{n}^{n}(V) ≅ 0. Obviously, the definition given here agrees with the definition on p. 220 in [ABW]. Theorem 3.2. Let I be an ideal in R which is locally generated by a non-zero-divisor. Let 0 → P →^{f} Q → V → 0 be an R-projective resolution of a f. g. projective R/I-module V. Then, for all k = 0, …, n, we have a canonical isomorphism

H_{k}(Kos^{n}(f)) ≅ L_{k}^{n}(V) ⊗ (I/I^{2})^{⊗k}

between the k-th homology module of the n-th Koszul complex and the tensor product of the Schur module L_{k}^{n}(V) with the k-th tensor power of the conormal module I/I^{2}. In particular, H_{k}(Kos^{n}(f)) is a projective R/I-module. Remark 3.3. We consider the following extreme cases:

(a) If k = n, Theorem 3.2 states that the differential

d_{n} : Λ^{n}(P) → Λ^{n−1}(P) ⊗ Q

is injective. This immediately follows from the exactness of the Koszul complex Kos^{n}(id_{P}) and the injectivity of f.

(b) If k = 0, Theorem 3.2 states that the sequence

P ⊗ Sym^{n−1}(Q) ⟶^{d_{1}} Sym^{n}(Q) ⟶^{can} Sym^{n}(V) ⟶ 0

is exact. This already follows from the construction of symmetric powers. In both cases, the assumption on the ideal I is not used.

Theorem 3.2. We fix k ∈ {0, …, n}. For any f. g. projective R/I-module V, we choose an R-projective resolution 0 → P_{V} →^{f_{V}} Q_{V} → V → 0 of V. For any homomorphism α : V → W between f. g. projective R/I-modules V, W, there is a complex homomorphism

\xymatrix{P_{V} \ar[r] \ar[d]^{α_{P}} & Q_{V} \ar[d]^{α_{Q}} \\ P_{W} \ar[r] & Q_{W}}

which is compatible with α; this complex homomorphism is unique up to homotopy by a standard argument of homological algebra. Thus, by Corollary 2.5, we obtain a well-defined functor

F : P_{R/I} → (R-modules), V ↦ H_{k}(Kos^{n}(f_{V})).

Let G denote the functor

G : P_{R/I} → (R-modules), V ↦ L_{k}^{n}(V) ⊗ (I/I^{2})^{⊗k}.

We have to show that F (V) ≅ G(V) for all V ∈ P_{R/I}. For this, we verify the assumptions of Theorem 1.5. One easily checks (see also 4.3.1.5 on p. 109 in [I 1]) that, for all f : P → Q, f′ : P′ → Q′ in P_{R}, we have:

Kos^{n}(f ⊕ f′) ≅ ⊕_{n_{1}+n_{2}=n} Kos^{n_{1}}(f) ⊗ Kos^{n_{2}}(f′).

(The tensor product of two complexes K. and L. is a priori only a double complex. In this paper, K. ⊗ L. denotes both the double complex and the total complex of this double complex.) Using Proposition 1.2, we obtain the following isomorphism for all i ≥ 0 and V_{1}, …, V_{i} ∈ P_{R/I} (see also Example 1.4(a)):

cr_{i}(F)(V_{1}, …, V_{i}) ≅ ⊕_{{(n_{1},... ,n_{i})∈\{1,... ,n\}^{i} \atop n_{1}+...+n_{i}=n}} H_{k} (Kos^{n_{1}}(f_{V_{1}}) ⊗ … ⊗ Kos^{n_{i}}(f_{V_{i}})) .

In particular, F is of degree ≤ n and, for V_{1} = … = V_{i} = R/I, we obtain

cr_{i}(F)(R/I, …, R/I) ≅ ⊕_{{(n_{1},... ,n_{i})∈\{1,... ,n\}^{i} \atop n_{1}+...+n_{i}=n}} H_{k} ((I ⟶^{can} R)^{⊗i})

since Kos^{m}(I → R) ≅ (I → R) for all m ≥ 1. The complex (I ⟶^{can} R)^{⊗i} has the shape

0 → I^{⊗i} → \begin{aligned}( & {i \atop i−1} & ) \\ & ⊕ \end{aligned} I^{⊗(i−1)} → … → \begin{matrix}( & {i \atop 1} & ) \\ ⊕ \end{matrix} I → R → 0;

here, the differential ⊕^{\binom{i}{k}}I^{⊗k} → ⊕^{\binom{i}{k−1}}I^{⊗(k−1)} is the composition of the differential ⊕^{\binom{i}{k}}I^{⊗k} → ⊕^{\binom{i}{k−1}}I^{⊗k} in the complex (R →^{id} R)^{⊗i} ⊗ I^{⊗k} with the canonical inclusion ⊕^{\binom{i}{k−1}}I^{⊗k} → ⊕^{\binom{i}{k−1}}I^{⊗(k−1)}. Hence, the module Z_{k} ((I → R)^{⊗i}) of kcycles is isomorphic to Z_{k} ((R → R)^{⊗i}) ⊗ I^{⊗k}, and the module B_{k} ((I → R)^{⊗i}) of k-boundaries is isomorphic to B_{k} ((R → R)^{⊗i}) ⊗ I^{⊗(k+1)}. Since (R → R)^{⊗i} is exact, we obtain:

H_{k} ((I → R)^{⊗i}) ≅ B_{k} ((R → R)^{⊗i}) ⊗ (I/I^{2})^{⊗k}.

Similarly, for the functor G, we obtain:

cr_{i}(G)(V_{1}, …, V_{i})
≅ ⊕_{{(n_{1},... ,n_{i})∈\{1,... ,n\}^{i} \atop n_{1}+...+n_{i}=n}} B_{k} (Kos^{n_{1}}(id_{V_{1}}) ⊗ … ⊗ Kos^{n_{i}}(id_{V_{i}})) ⊗ (I/I^{2})^{⊗k}.

In particular, G is of degree ≤ n and we have for V_{1} = … = V_{i} = R/I:

cr_{i}(G)(R/I, …, R/I) ≅ ⊕_{{(n_{1},... ,n_{i})∈\{1,... ,n\}^{i} \atop n_{1}+...+n_{i}=n}} B_{k} ((R/I →^{id} R/I)^{⊗i}) ⊗ (I/I^{2})^{⊗k}.

Thus, for all i ≥ 1, we obtain an isomorphism

α_{i}(R/I, …, R/I) : cr_{i}(F)(R/I, …, R/I) ˜→ cr_{i}(G)(R/I, …, R/I).

The multiplication with r̄ ∈ R/I in the l-th component of cr_{i}(F)(R/I, …, R/I) obviously corresponds to the endomorphism

⊕_{(n_{1},... ,n_{i})} r^{n_{l}} of ⊕_{(n_{1},... ,n_{i})} B_{k} ((R → R)^{⊗i}) ⊗ (I/I^{2})^{⊗k}.

The analogous statement also holds for G. Thus, the maps α_{i}(R/I, …, R/I), i = 1, …, n, are compatible with the action of R/I in each component. For any ε ∈ {1, …, n}^{i} and V_{1}, …, V_{i} ∈ P_{R/I}, the diagonal map

Δ_{ε} : cr_{i}(F)(V_{1}, …, V_{i}) → cr_{|ε|}(F)(V_{1}, …, V_{1}, …, V_{i}, …, V_{i})

is induced by an obvious complex homomorphism

Δ_{ε} : ⊕_{(n_{1},... ,n_{i})} Kos^{n_{1}}(f_{V_{1}}) ⊗ … ⊗ Kos^{n_{i}}(f_{V_{i}})

→ ⊕_{(n_{1},... ,n_{|ε|})} Kos^{n_{1}}(f_{V_{1}}) ⊗ … ⊗ Kos^{n_{|ε|}}(f_{V_{i}}).

Similarly, the diagonal map

Δ_{ε} : cr_{i}(G)(V_{1}, …, V_{i}) → cr_{|ε|}(G)(V_{1}, …, V_{1}, …, V_{i}, …, V_{i})

is induced by an obvious complex homomorphism

Δ_{ε} : ⊕_{(n_{1},... ,n_{i})} Kos^{n_{1}}(id_{V_{1}}) ⊗ … ⊗ Kos^{n_{i}}(id_{V_{i}})

→ ⊕_{(n_{1},... ,n_{|ε|})} Kos^{n_{1}}(id_{V_{1}}) ⊗ … ⊗ Kos^{n_{|ε|}}(id_{V_{i}}).

Using these facts, one easily sees that the isomorphisms α_{i}(R/I, …, R/I), i ≥ 1, are compatible with the diagonal maps. In the same way one can prove that these isomorphisms are compatible with the plus maps. Hence, by Theorem 1.5, the functors F and G are isomorphic, and Theorem 3.2 is proved. □

In the next proposition, we explicitly describe the isomorphism of Theorem 3.2. Proposition 3.4. The isomorphism

L_{k}^{n}(V) ⊗ I^{k}/I^{k+1} ˜→ H_{k}(Kos^{n}(f))

constructed in the proof of Theorem 3.2 maps the element

d_{k+1}(v_{1} ∧ … ∧ v_{k+1} ⊗ v_{k+2} ⋯ v_{n}) ⊗ \overline{r_{1} ⋯ r_{k}}

to the homology class of the element

∑_{i=1}^{k+1}(−1)^{k+1−i}(r_{1}q_{1}) ∧ … ∧ q̂_{i} ∧ … ∧ (r_{k}q_{k+1}) ⊗ q_{i}q_{k+2} ⋯ q_{n};

here, q_{i} is a preimage of v_{i} under the given surjective map Q → V (for i = 1, …, n) and, for any r ∈ I and q ∈ Q, the element rq of Q is considered as an element of P.

Proof. The isomorphism G ˜→ F constructed in the proof of Theorem 3.2 induces an isomorphism cr_{n}(G) ˜→ cr_{n}(F) of n-functors. Obviously, the diagrams

[figure: cr_{n} (Λ^{k+1} ⊗ Sym^{n−k−1}) (V, …, V) ⊗ I^{k}/I^{k+1}; +_{n}⊗id; cr_{n}(G)(V, …, V); +_{n}; Λ^{k+1}(V) ⊗ Sym^{n−k−1}(V) ⊗ I^{k}/I^{k+1}; G(V)]

(where both horizontal maps are induced by d_{k+1}) and

\xymatrix{cr_{n}(G)(V, …, V) \ar[r]^{∼} \ar[d]^{+_{n}} & cr_{n}(F)(V, …, V) \ar[d]^{+_{n}} \\ G(V) \ar[r]^{∼} & F (V)}

commute for all V ∈ P_{R/I}. The R/I-module cr_{n} (Λ^{k+1} ⊗ Sym^{n−k−1}) (V, …, V)

is isomorphic to a direct sum of \binom{n}{k+1} copies of V^{⊗n}. We index these summands by the subsets T of {1, …, n} with k + 1 elements in the canonical way. For v_{1} ⊗ … ⊗ v_{n} ∈ V^{⊗n} and T = {t_{1} < … < t_{k+1}}, let (v_{1} ⊗ … ⊗ v_{n})[T] denote the corresponding element of cr_{n} (Λ^{k+1} ⊗ Sym^{n−k−1}) (V, …, V). We obviously have

+_{n}((v_{1} ⊗ … ⊗ v_{n})[T])
= v_{t_{1}} ∧ … ∧ v_{t_{k+1}} ⊗ ∏_{t∈{1,... ,n}\T} v_{t} in Λ^{k+1}(V) ⊗ Sym^{n−k−1}(V).

In particular, the restriction of +_{n} to the direct summand which corresponds e.g. to {1, …, k + 1} is surjective. Thus it suffices to show that the composition of the upper horizontal maps of the above diagrams with the right vertical map in the lower diagram maps the element (v_{1} ⊗ … ⊗ v_{n})[\{1, …, k + 1\}] ⊗ \overline{r_{1} ⋯ r_{k}} to the homology class of the element given in Proposition 3.4. We obviously have

cr_{n}(d_{k+1})(V, …, V)((v_{1} ⊗ … ⊗ v_{n})[{1, …, k + 1}])

= ∑_{i=1}^{k+1}(−1)^{k+1−i}(v_{1} ⊗ … ⊗ v_{n})[{1, …, î, …, k + 1}]

in cr_{n} (Λ^{k} ⊗ Sym^{n−k}) (V, …, V). The isomorphism

cr_{n}(G)(V, …, V) ˜→ cr_{n}(F)(V, …, V)

is equal to the composition

cr_{n}(G)(V, …, V) ≅ V^{⊗n} ⊗ cr_{n}(G)(R/I, …, R/I)
≅ V^{⊗n} ⊗ cr_{n}(F)(R/I, …, R/I) ≅ cr_{n}(F)(V, …, V)

where the middle isomorphism has been explicitly constructed in the proof of Theorem 3.2 and the exterior isomorphisms have been introduced in Example 1.7. The composition

Im (cr_{n}(d_{k+1})(R/I, …, R/I)) ⊗ I^{k}/I^{k+1}
≅ cr_{n}(G)(R/I, …, R/I) ≅ cr_{n}(F)(R/I, …, R/I)

maps the element cr_{n}(d_{k+1})(R/I, …, R/I)(1[\{1, …, k + 1\}]) ⊗ \overline{r_{1} ⋯ r_{k}} to the homology class of the element

∑_{i=1}^{k+1}(−1)^{k+1−i}(r_{1} ⊗…⊗ r_{i−1} ⊗ 1 ⊗ r_{i} ⊗…⊗ r_{k} ⊗ 1 ⊗…⊗ 1)[{1,…, î,…, k + 1}]

in the complex (I → R)^{⊗n} (see the proof of Theorem 3.2). Furthermore, the isomorphism V^{⊗n} ⊗ cr_{n}(F)(R/I, …, R/I) ≅ cr_{n}(F)(V, …, V) maps the element v_{1} ⊗ … ⊗ v_{n} ⊗ x to the image of x under the map

cr_{n}(F)(v_{1}, …, v_{n}) : cr_{n}(F)(R/I, …, R/I) → cr_{n}(F)(V, …, V).

(This easily follows from Example 1.7). Hence, the composition of the upper horizontal maps in the above diagrams maps the element

(v_{1} ⊗ … ⊗ v_{n})[\{1, …, k + 1\}] ⊗ \overline{r_{1} ⋯ r_{k}}

to the homology class of the element

∑_{i=1}^{k+1}(−1)^{k+1−i}((r_{1}q_{1}) ⊗ … ⊗ (r_{i−1}q_{i−1}) ⊗ q_{i}

⊗(r_{i}q_{i+1}) ⊗ … ⊗ (r_{k}q_{k+1}) ⊗ q_{k+2} ⊗ … ⊗ q_{n})[{1, …, î, …, k + 1}]

in the complex (P → Q)^{⊗n}. The plus map +_{n} : cr_{n}(F)(V, …, V) → F (V) maps the latter homology class to the homology class stated in Proposition 3.4. Thus, Proposition 3.4 is proved. □

In the following remark, we sketch an alternative proof of Theorem 3.2. Remark 3.5. n easy, but tedious calculation shows directly that the association described in Proposition 3.4 yields a well-defined homomorphism

G(V) = L_{k}^{n}(V) ⊗ I^{k}/I^{k+1} → H_{k}(Kos^{n}(f)) = F (V)

for all V ∈ P_{R/I}. Furthermore, one can verify that the induced homomorphisms cr_{i}(G)(R/I, …, R/I) → cr_{i}(F)(R/I, …, R/I), i ≥ 1, are bijective. By Proposition 1.2, this implies that the morphism G → F of functors is an isomorphism. So, in this proof, the abstract argument Theorem 1.5 has been replaced by more explicit computations which, from the combinatorial point of view, are more complicated. Remark 3.6. Analogously to Theorem 3.2, one can prove the following statement for the Koszul complex K̃os^{n}(f) introduced in Remark 2.7: Under the assumptions of Theorem 3.2, we have:

H_{k}(K̃os^{n}(f)) ≅ L̃_{k}^{n}(V) ⊗ (I/I^{2})^{⊗k} for all k ≥ 0;

here, L̃_{k}^{n}(V) denotes the coSchur module of type (k + 1, 1, …, 1) which can be defined analogously to 3.1 (see also Definition II 1.3 on p. 220 in [ABW]).

4. A New Proof of the Adams-Riemann-Roch Formula for Regular Closed Immersions

In this section, we deduce the Adams-Riemann-Roch theorem for regular closed immersions of codimension 1 from the main result of the previous section. Using blowing up and the excess intersection formula, the Adams-Riemann-Roch theorem for regular closed immersions of arbitrary codimension then follows from this as in [SGA 6]. In particular, we obtain an easy, elementary and natural proof of the Adams-Riemann-Roch theorem which does not use the deformation to the normal cone any longer.

First, we recall the following definitions and facts of the theory of λ-rings (see Expose V in [SGA 6], section 4.5 in [S], or Chapter I in [FL]). Let K be a (special) λ-ring, C ∈ K of (finite) λ-degree d, and x ∈ K arbitrary. Furthermore, let μ be a polynomial in the λ-operations λ_{1}, λ_{2}, … without constant term. Then, there is a unique element μ(C, x) ∈ K which is functorial in (K, C, x) and which has the following property:

μ(x · λ_{−1}(C)) = μ(C, x) · λ_{−1}(C)

(here, λ_{−1}(C) := ∑_{i=0}^{d}(−1)^{i}λ_{i}(C)). This immediately follows from the following fact which is easy to prove (cf. Lemma 5.2 in Expose V of [SGA 6]): If the elements λ_{1}(C), …, λ_{d}(C) and λ_{1}(x), λ_{2}(x), … are considered as indeterminates and if the polynomial ring ℤ[λ_{1}(C), …, λ_{d}(C), λ_{1}(x), λ_{2}(x), …] is equipped with the obvious λ-structure, then the element μ(x · λ_{−1}(C)) is divisible by λ_{−1}(C). For n ≥ 1, let ψ_{n} denote the n-th Adams operation, i.e., ψ_{n} = N_{n}(λ_{1}, …, λ_{n}) where N_{n} denotes the n-th Newton polynomial. Furthermore, let θ^{n}(C) denote the n-th Bott element of C (see §4 in [K 1] or p. 24 in [FL]). The n-th symmetric power operation is inductively defined by σ_{0} ≡ 1 and σ_{n} = ∑_{i=1}^{n}(−1)^{i−1}λ_{i}σ_{n−i} for n ≥ 1. For 0 ≤ k ≤ n, the Schur operation s_{k}^{n} of type (k + 1, 1, …, 1) is defined by s_{k}^{n} = ∑_{i=k+1}^{n}(−1)^{i−k−1}λ_{i}σ_{n−i}.

Lemma 4.1.

(a) Let μ be the product of two polynomials μ_{1} and μ_{2}. Then we have:

μ(C, x) = μ_{1}(C, x) · μ_{2}(C, x) · λ_{−1}(C).

(b) For all n ≥ 0 and x, y ∈ K, we have:

σ_{n}(C, x + y) = σ_{n}(C, x) + ∑_{i=1}^{n−1} σ_{i}(C, x) · σ_{n−i}(C, y) · λ_{−1}(C) + σ_{n}(C, y).

(c) For all n ≥ 1 we have: ψ_{n}(C, x) = θ^{n}(C) · ψ_{n}(x).

(d) Let d = 1. Then we have for all n ≥ 1:

σ_{n}(C, x) = ∑_{k=0}^{n−1}(−1)^{k}s_{k}^{n}(x)C^{k}.

In particular, we have σ_{n}(1, x) = ψ_{n}(x).

Proof. The assertions (a) and (b) immediately follow from the definition. The assertion (c) follows from the multiplicativity of ψ_{n}. If d = 1, then we have:

σ_{n}(x · λ_{−1}(C)) = σ_{n}(x · (1 − C)) = ∑_{i=0}^{n} σ_{n−i}(x) · σ_{i}(−xC)

= ∑_{i=0}^{n}(−1)^{i}σ_{n−i}(x) · λ_{i}(xC) (since σ_{i}(−x) = (−1)^{i}λ_{i}(x))

= ∑_{i=0}^{n}(−1)^{i}σ_{n−i}(x) · λ_{i}(x) · C^{i} (since C is of λ-degree 1)

= ∑_{i=0}^{n}(−1)^{i}σ_{n−i}(x) · λ_{i}(x) · (C^{i} − 1) (since ∑_{i=0}^{n}(−1)^{i}σ_{n−i}(x) · λ_{i}(x) = 0)

= ∑_{i=0}^{n}(−1)^{i+1}σ_{n−i}(x) · λ_{i}(x) · (1 + C + … + C^{i−1}) · λ_{−1}(C) (geom. sum).

Hence we have:

σ_{n}(C, x) = ∑_{i=1}^{n}(−1)^{i+1}σ_{n−i}(x) · λ_{i}(x) · (1 + C + … + C^{i−1})

= ∑_{k=0}^{n−1} ( ∑_{i=k+1}^{n}(−1)^{i+1}σ_{n−i}(x) · λ_{i}(x)) C^{k} = ∑_{k=0}^{n−1}(−1)^{k}s_{k}^{n}(x)C^{k}.

In particular, we have σ_{n}(1, x) = ∑_{k=0}^{n−1}(−1)^{k}s_{k}^{n}(x) = ψ_{n}(x) by section 3 in [G]. □

Now, let X be a noetherian scheme with the property that each coherent O_{X}- module is a quotient of a locally free O_{X}-module (of finite type). Let K_{0}(X) denote the Grothendieck group of all locally free O_{X}-modules and K_{0}^{∞}(X) the Grothendieck group of all coherent O_{X}-modules which have a finite resolution by locally free O_{X}-modules. By Proposition 4.1 on p. 126 in [FL], the canonical homomorphism K_{0}(X) → K_{0}^{∞}(X) is bijective. Let i : Y ↪ X be a regular closed immersion of codimension d with conormal sheaf C. Then, for any locally free O_{Y} - module V, the direct image i_{∗}(V) is a coherent O_{X}-module which has a finite locally free resolution (see p. 127 in [FL]). Thus, the association [V] ↦ [i_{∗}(V)] induces a well-defined homomorphism

i_{∗} : K_{0}(Y) → K_{0}^{∞}(X) ≅ K_{0}(X).

The following formula describes the behavior of products with respect to this homomorphism. It follows from the self intersection formula and it will be used in our proof of the Adams-Riemann-Roch formula. Proposition 4.2. For all x, y ∈ K_{0}(Y), we have:

i_{∗}(x) · i_{∗}(y) = i_{∗}(x · y · λ_{−1}([C])) in K_{0}(X).

Proof. See Corollaire 2.8 on p. 436 in [SGA 6]. □

The assertion (a) of the following theorem has already been proved in Théorème 4.3 on p. 449 in [SGA 6], however only modulo torsion. The claimed version without denominators has been proved in Théorème 2.1 on p. 24 in [J] using the deformation to the normal cone. The Adams-Riemann-Roch formula given in (b) follows also from Théorème 2.1 on p. 24 in [J] and it has been proved in Theorem 6.3 on p. 142 in [FL]. Theorem 4.3. Let n ≥ 1.

(a) (Riemann-Roch formula for σ_{n}) For all y ∈ K_{0}(Y) we have:

σ_{n}(i_{∗}(y)) = i_{∗}(σ_{n}([C], y)) in K_{0}(X).

(b) (Adams-Riemann-Roch formula) For all y ∈ K_{0}(Y) we have:

ψ_{n}(i_{∗}(y)) = i_{∗}(θ^{n}([C]) · ψ_{n}(y)) in K_{0}(X).

Proof.

(a) First, let d = 1. By Lemma 4.1(b) and Proposition 4.2, the elements y of K_{0}(Y) for which the Riemann-Roch formula for σ_{n} holds for all n ≥ 1 is a subgroup of K_{0}(Y). Thus, it suffices to prove the Riemann-Roch formula for y = [V] where V is a locally free O_{Y} -module. We choose a locally free resolution 0 → P →^{f} Q → i_{∗}(V) → 0 of i_{∗}(V) on X. By gluing the isomorphisms of Theorem 3.2, we obtain isomorphisms

H_{k}(Kos^{n}(f)) ≅ i_{∗}(L_{k}^{n}(V) ⊗ C^{⊗k}), k = 0, …, n.

Hence, we have in K_{0}^{∞}(X):

∑_{k=0}^{n}(−1)^{k}[Λ^{k}(P) ⊗ Sym^{n−k}(Q)] = i_{∗} (∑_{k=0}^{n−1}(−1)^{k}[L_{k}^{n}(V) ⊗ C^{⊗k}]) .

Thus, we obtain:

σ_{n}(i_{∗}([V])) = σ_{n}([Q] − [P]) = ∑_{k=0}^{k}(−1)^{k}[Λ^{k}(P) ⊗ Sym^{n−k}(Q)]

= i_{∗} (∑_{k=0}^{n−1}(−1)^{k}s_{k}^{n}([V]) · [C]^{k}) = i_{∗}(σ_{n}([C], [V])) in K_{0}(X);

here, we have used the exactness of the Koszul complex Kos^{n}(id_{V}) and Lemma 4.1(d). This proves assertion (a) in the case d = 1. The general case can be deduced from this as on p. 449 and p. 450 in [SGA 6] using blowing up and the excess intersection formula.

(b) Let ψ_{n} = ∑_{ν} a_{ν}σ_{1}^{ν_{1}} ⋯ σ_{n}^{ν_{n}} be the representation of ψ_{n} as a polynomial in the symmetric power operations σ_{1}, …, σ_{n}. Then we have in K_{0}(X):

ψ_{n}(i_{∗}(y)) = ∑_{ν} a_{ν}σ_{1}(i_{∗}(y))^{ν_{1}} ⋯ σ_{n}(i_{∗}(y))^{ν_{n}}

= ∑_{ν} a_{ν} i_{∗}(σ_{1}([C], y))^{ν_{1}} ⋯ i_{∗}(σ_{n}([C], y))^{ν_{n}} (Theorem 4.3(a))

= i_{∗} (∑_{ν} a_{ν} σ_{1}([C], y)^{ν_{1}} ⋯ σ_{n}([C], y)^{ν_{n}} · λ_{−1}([C])^{ν_{1}+...+ν_{n}−1}) (Prop. 4.2)

= i_{∗}(ψ_{n}([C], y)) (Lemma 4.1(a)) = i_{∗}(θ^{n}([C]) · ψ_{n}(y)) (Lemma 4.1(c)).

This ends the proof of Theorem 4.3. □

5. Riemann-Roch for Tensor Powers

In the paper [K 2], we have established Riemann-Roch formulas for tensor power operations. In the case of regular closed immersions, we have used the deformation to the normal cone to prove them. The aim of this section is to prove these formulas by using the method developed in the previous sections. In contrast to the previous sections, this method can here be applied to regular closed immersions not only of codimension 1 but of arbitrary codimension.

Let R be a noetherian ring and n ∈ ℕ. Let Σ_{n} denote the n-th symmetric group, i.e., the group of permutations of I_{n} := {1, …, n}. We write R[I_{n}] for the direct sum of n copies of R together with the obvious action of Σ_{n}. The Σ_{n}-module K_{n} is defined by the short exact sequence

0 ⟶ K_{n} ⟶ R[I_{n}] ⟶^{sum} R ⟶ 0.

Let I be an ideal in R which is locally generated by a regular sequence of length d. Furthermore, let P. → V be an R-projective resolution of a f. g. projective R/I-module V. We view the total complex P.^{⊗n} of the n-th tensor power of the complex P. as a complex of Σ_{n}-modules in the canonical way. In particular, the homology modules H_{k}(P.^{⊗n}), k ≥ 0, are equipped with a canonical Σ_{n}-action. Theorem 5.1. For all k ≥ 0, there is a canonical Σ_{n}-isomorphism

H_{k}(P.^{⊗n}) ≅ V^{⊗n} ⊗ Λ^{k}(I/I^{2} ⊗ K_{n});

here, V^{⊗n} is considered as a Σ_{n}-module with the natural action and I/I^{2} as a Σ_{n}- module with the trivial action. In particular, H_{k}(P.^{⊗n}) is a projective R/I-module.

Proof. We fix k ∈ {0, …, n}. A homotopy between two complex homomorphisms \xymatrix{f, g : P. \ar[r] \ar[r] & Q.} induces a (non-equivariant) homotopy between the complex homomorphisms \xymatrix{f^{⊗n}, g^{⊗n} : P.^{⊗n} \ar[r] \ar[r] & Q.^{⊗n}}. In particular, the Σ_{n}-homomorphisms H_{k}(f^{⊗n}) and H_{k}(g^{⊗n}) coincide. Therefore, we obtain a well-defined functor

F : P_{R/I} → (R[Σ_{n}]-modules), V ↦ H_{k}(P.^{⊗n}),

as in the proof of Theorem 3.2. Let G denote the functor

G : P_{R/I} → (R[Σ_{n}]-modules), V ↦ V^{⊗n} ⊗ Λ^{k}(I/I^{2} ⊗ K_{n}).

We have to show that F (V) ≅ G(V) for all V ∈ P_{R/I}. Again, we verify the assumptions of Theorem 1.5. First, we suppose that the ideal I is globally generated by a regular sequence. This means that there is a homomorphism ε : F → R from a free R-module F of rank d to R such that the Koszul complex Kos^{d}(ε) is a resolution of R/I (by virtue of the canonical map Kos^{d}(ε)_{0} ≅ R⟶R/I). One easily checks that the complex (Kos^{d}(ε))^{⊗n} is Σ_{n}-isomorphic to the Koszul complex Kos^{nd}(F ⊗R[I_{n}] ⟶^{sum} F ⟶^{ε} R). By Lemma 3.3 in [K 2], the homology module H_{k}(Kos^{nd}(F ⊗R[I_{n}] → F → R)) is Σ_{n}-isomorphic to the k-th exterior power of

ker(F/IF ⊗ R[I_{n}] ⟶^{sum} F/IF ⟶^{ε̄} I/I^{2}) ≅ F/IF ⊗ K_{n} ≅ I/I^{2} ⊗ K_{n}.

Thus, we have established a Σ_{n}-isomorphism F (R/I) ≅ Λ^{k}(I/I^{2} ⊗ K_{n}). It does obviously not depend on the chosen homomorphism ε. Hence, we can glue these isomorphisms and obtain a canonical Σ_{n}-isomorphism F (R/I) ≅ Λ^{k}(I/I^{2} ⊗ K_{n}) also in the case the ideal I is only locally generated by a regular sequence. For any two complexes P. and Q., we have:

(P. ⊕ Q.)^{⊗n} ≅ ⊕_{n_{1}+n_{2}=n} Ind_{Σ_{n_{1}} ×Σ_{n_{2}}}^{Σ_{n}} (P.^{⊗n_{1}} ⊗ Q.^{⊗n_{2}}).

Hence, as in the proof of Theorem 3.2, we obtain for all i ≥ 0:

cr_{i}(F)(R/I, …, R/I) ≅ ⊕_{{(n_{1},... ,n_{i})∈\{1,... ,n\}^{i} \atop n_{1}+...+n_{i}=n}} Ind_{Σ_{n_{1}} ×...×Σ_{n_{i}}}^{Σ_{n}} Res_{Σ_{n_{1}} ×...×Σ_{n_{i}}}^{Σ_{n}} F (R/I)

≅ ( ⊕_{(n_{1},... ,n_{i})} Ind_{Σ_{n_{1}} ×...×Σ_{n_{i}}}^{Σ_{n}} (R)) ⊗ Λ^{k}(I/I^{2} ⊗ K_{n})
≅ cr_{i}(G)(R/I, …, R/I).

One easily sees that these Σ_{n}-isomorphisms are compatible with the action of R/I in each component and with the diagonal and plus maps. Thus, Theorem 5.1 follows from Theorem 1.5. □

Example 5.2. Let n = 2. Then Theorem 5.1 yields the well known Σ_{2}-isomorphism

Tor_{k}^{R}(V, V) ≅ V^{⊗2} ⊗ Λ^{k}((I/I^{2})_{sgn})

where the transposition in Σ_{2} acts by −1 on (I/I^{2})_{sgn}.

Now, let X be a noetherian scheme with the property that each coherent O_{X}- module is a quotient of a locally free O_{X}-module. Let K_{0}(Σ_{n}, X) denote the Grothendieck group of all locally free Σ_{n}-modules on X. Using the “binomial theorem”

(F ⊕ G)^{⊗n} ≅ ⊕_{n_{1}+n_{2}=n} Ind_{Σ_{n_{1}} ×Σ_{n_{2}}}^{Σ_{n}} (F^{⊗n_{1}} ⊗ G^{⊗n_{2}}),

the association [F] ↦ [F^{⊗n}] can be canonically extended to a map

τ_{n} : K_{0}(X) → K_{0}(Σ_{n}, X)

(see section 1 in [K 2]). It is called the n-th tensor power operation. Let i : Y ↪ X be a regular closed immersion of codimension d with conormal sheaf C. Let i_{∗} denote both the homomorphism from K_{0}(Y) to K_{0}(X) introduced in section 4 and the homomorphism from K_{0}(Σ_{n}, Y) to K_{0}(Σ_{n}, X) defined analogously. The following theorem is Theorem 4.1 of [K 2].

Theorem 5.3 (Riemann-Roch formula for τ_{n}). For all y ∈ K_{0}(Y), we have:

τ_{n}(i_{∗}(y)) = i_{∗}(λ_{−1}([C ⊗ K_{n}]) · τ_{n}(y)) in K_{0}(Σ_{n}, X).

Proof. If Theorem 5.3 is true for y_{1}, y_{2} ∈ K_{0}(Y) and for all n ≥ 1, then it is also true for y_{1} + y_{2}. This follows from the following calculation (Σ_{n_{1},n_{2}} := Σ_{n_{1}} × Σ_{n_{2}}):

τ_{n}(i_{∗}(y_{1} + y_{2})) = ∑_{n_{1}+n_{2}=n} Ind_{Σ_{n_{1},n_{2}}}^{Σ_{n}} (τ_{n_{1}}(i_{∗}(y_{1})) · τ_{n_{2}}(i_{∗}(y_{2})))

= ∑ Ind_{Σ_{n_{1},n_{2}}}^{Σ_{n}} (i_{∗}(λ_{−1}([C ⊗ K_{n_{1}}]) · τ_{n_{1}}(y_{1})) · i_{∗}(λ_{−1}([C ⊗ K_{n_{2}}]) · τ_{n_{2}}(y_{2})))

= ∑ Ind_{Σ_{n_{1},n_{2}}}^{Σ_{n}} (i_{∗}(λ_{−1}([C ⊗ K_{n_{1}}])λ_{−1}([C ⊗ K_{n_{2}}])λ_{−1}([C])τ_{n_{1}}(y_{1})τ_{n_{2}}(y_{2})))

= ∑ Ind_{Σ_{n_{1},n_{2}}}^{Σ_{n}} (i_{∗}(Res_{Σ_{n_{1},n_{2}}}^{Σ_{n}} λ_{−1}([C ⊗ K_{n}]) · τ_{n_{1}}(y_{1}) · τ_{n_{2}}(y_{2})))

= i_{∗} (λ_{−1}([C ⊗ K_{n}]) · ∑_{n_{1}+n_{2}=n} Ind_{Σ_{n_{1},n_{2}}}^{Σ_{n}} τ_{n_{1}}(y_{1}) · τ_{n_{2}}(y_{2}))

= i_{∗} (λ_{−1}([C ⊗ K_{n}]) · τ_{n}(y_{1} + y_{2})) .

(Here, we have used Proposition 4.2 and Frobenius reciprocity.) Thus, it suffices to prove Theorem 5.3 for y = [V] where V is a locally free O_{Y} -module. Let P. → i_{∗}(V) be a locally free resolution of i_{∗}(V) on X. By gluing the isomorphisms constructed in Theorem 5.1, we obtain Σ_{n}-isomorphisms

H_{k}(P.^{⊗n}) ≅ i_{∗}(V^{⊗n} ⊗ Λ^{k}(C ⊗ K_{n})), k ≥ 0.

Hence, we have in K_{0}(Σ_{n}, X):

τ_{n}(i_{∗}([V])) = τ_{n} (∑_{i≥0}(−1)^{k}[P_{k}]) = ∑_{k≥0}(−1)^{k} [(P.^{⊗n})_{k}]

= ∑_{k≥0}(−1)^{k}i_{∗} ([V^{⊗n} ⊗ Λ^{k}(C ⊗ K_{n})]) = i_{∗} (λ_{−1}([C ⊗ K_{n}]) · τ_{n}([V])) .

This proves Theorem 5.3. □

6. Some Computations for Regular Immersions of Higher Codimension

Let X be a noetherian scheme with the property that each coherent O_{X}-module is a quotient of a locally free O_{X}-module. Let i : Y ↪ X be a regular closed immersion of codimension d with conormal sheaf C and n ∈ ℕ. In this section, we study the question whether, in the case d ≥ 2, the Riemann-Roch formula for σ_{n} and i_{∗} (cf. Theorem 4.3(a)) can be proved directly with the methods developed in the previous sections (i.e., without using blowing up). More precisely, for any locally free O_{Y} -module V and any locally free resolution P. → i_{∗}(V) of i_{∗}(V) on X, we are looking for a complex K.^{n} of locally free O_{X}-modules such that we have

σ_{n}(i_{∗}([V])) =^{def} σ_{n}(∑_{k≥0}(−1)^{k}[P_{k}]) = ∑_{k≥0}(−1)^{k}[K_{k}^{n}] in K_{0}(X)

and

σ_{n}([C], [V]) = ∑_{k≥0}(−1)^{k}[H_{k}(K.^{n})] in K_{0}(Y)

(or at least in the relative K_{0}-group of Y in X, see [TT]). The existence of such a complex would immediately imply the Riemann-Roch formula for σ_{n} (see the proof of Theorem 4.3(a)). In the case d = 1, we have used the complex Kos^{n}(P_{1} → P_{0}) (see section 4) which is quasi-isomorphic to the complex NSym^{n}Γ(P_{1} → P_{0}) by Proposition 2.4. In section 5, we have used the complex P.^{⊗n} which is quasiisomorphic to the complex N(Γ(P.)^{⊗n}) by the Eilenberg-Zilber theorem (see Application 8.5.3 on p. 277 in [W]). Thus, a natural candidate for the above complex K.^{n} is the complex K.^{n} := NSym^{n}Γ(P.). We prove in Lemma 6.1 that this complex satisfies at least the first condition σ_{n}(i_{∗}([V])) = ∑_{k≥0}(−1)^{k}[K_{k}^{n}]. In Lemma 6.2, we give a representation of σ_{2}([C], [V]) as an alternating sum which suggests the following conjecture: H_{k}(K.^{2}) ≅ Sym^{2}(V) ⊗ Λ^{k}(C) for k even and H_{k}(K.^{2}) ≅ Λ^{2}(V) ⊗ Λ^{k}(C) for k odd. In fact, this conjecture is true if 2 is invertible on X (see Corollary 6.3). Theorem 6.4 states that, in the case d = 2, we have H_{0}(K.^{2}) ≅ Sym^{2}(V), H_{1}(K.^{2}) ≅ Λ^{2}(V) ⊗ C, however H_{2}(K.^{2}) ≅ D^{2}(V) ⊗ Λ^{2}(V), i.e., the conjecture is false for k = 2. Nevertheless, this theorem implies the second condition for the complex K.^{2} in the case d = 2 since [D^{2}(V)] = [Sym^{2}(V)] in K_{0}(Y). In the case d > 2, H_{k}(K.^{2}) probably has non-trivial 2-torsion for k ≥ 2 (see Remark 6.5). In particular, it is not clear how to compute H_{k}(K.^{2}) in general and to check the second condition for the complex K.^{2}. On the other hand, it might be possible to compute H_{k}(K.^{n}) in the case d = 2 even for all n ≥ 2 and to verify the second condition for K.^{n} in this case (see Example 6.6). I hope to say more on this in a future paper.

For any finite complex P. of locally free O_{X}-modules let [P.] denote the Euler characteristic ∑_{i≥0}(−1)^{i}[P_{i}] ∈ K_{0}(X).

Lemma 6.1. Let P. be a finite complex of locally free O_{X}-modules. Then we have for all n ≥ 0:

[NSym^{n}Γ(P.)] = σ_{n}([P.]) in K_{0}(X).

Proof. One easily checks that the complex NSym^{n}Γ(P.) has finite length. Furthermore, the O_{X}-modules NSym^{n}Γ(P.)_{k}, k ≥ 0, are locally free (see section 2 in [JM]). Therefore, the expression [NSym^{n}Γ(P.)] ∈ K_{0}(X) is well-defined. First, we suppose that P. is the complex P[−l] where P is a locally free O_{X}-module and l ∈ ℕ_{0}. We have to show that σ_{n}((−1)^{l}[P]) = [NSym^{n}Γ(P[−l])] in K_{0}(X). We prove this by induction on l. For l = 0, 1 this follows for example from Proposition 2.4. We write C(P[−l]) for the mapping cone of the identity of the complex P[−l]. The obvious short exact sequence

0 → P[−l] → C(P[−l]) → P[−l − 1] → 0

of complexes induces a natural filtration of the complex NSym^{n}Γ(C(P[−l])) whose successive quotients are the complexes N(Sym^{i}Γ(P[−l])⊗Sym^{n−i}Γ(P[−l−1])), i = 0, …, n. The complex NSym^{n}Γ(C(P[−l])) is homotopic to the zero complex (cf. section 2). For all i = 0, …, n, the complex N(Sym^{i}Γ(P[−l])⊗Sym^{n−i}Γ(P[−l−1])) is quasi-isomorphic to the complex NSym^{i}Γ(P[−l]) ⊗ NSym^{n−i}Γ(P[−l − 1]) by the Eilenberg-Zilber theorem (see Application 8.5.3 on p. 277 in [W]). Hence, we have for all n ≥ 1:

∑_{i=0}^{n}[NSym^{i}Γ(P[−l])] · [NSym^{n−i}Γ(P[−l − 1])]

= ∑_{i=0}^{n}[NSym^{i}Γ(P[−l]) ⊗ NSym^{n−i}Γ(P[−l − 1])]

= ∑_{i=0}^{n}[N(Sym^{i}Γ(P[−l]) ⊗ Sym^{n−i}Γ(P[−l − 1]))]
= NSym^{n}Γ(C(P[−l])) = 0;

i.e., the power series ∑_{i≥0}[NSym^{i}Γ(P[−l − 1])]t^{i} is the inverse of the power series ∑_{i≥0}[NSym^{i}Γ(P[−l])]t^{i}. Using the induction hypothesis, we obtain:

[NSym^{n}Γ(P[−l − 1])] = σ_{n}(−[P[−l]]) = σ_{n}((−1)^{l+1}[P])

as was to be shown. (Alternatively, this can be deduced from Proposition 4.3.2.1(i) on p. 111 in [I 1].) Now, let P. be an arbitrary complex of length l. We prove Lemma 6.1 by induction on l. Let σ_{≤l−1}(P.) denote the brutal truncation of P.. The obvious short exact sequence

0 → σ_{≤l−1}(P.) → P. → P_{l}[−l] → 0

of complexes yields the equality

[NSym^{n}Γ(P.)] = ∑_{i=0}^{n}[NSym^{i}Γ(σ_{≤l−1}(P.))] · [NSym^{n−i}Γ(P_{l}[−l])] in K_{0}(X)

as above. Using the induction hypothesis and the formula already proved, we finally obtain

[NSym^{n}Γ(P.)] = ∑_{i=0}^{n} σ_{i} (∑_{i=0}^{l−1}(−1)^{i}[P_{i}]) · σ_{n−i}((−1)^{l}[P_{l}]) = σ_{n}([P.])

as was to be shown. □

Lemma 6.2. Let K be a λ-ring, C ∈ K of finite λ-degree d and x ∈ K arbitrary. Then we have:

σ_{2}(C, x) = σ_{2}(x)λ_{0}(C) − λ_{2}(x)λ_{1}(C) + σ_{2}(x)λ_{2}(C) − λ_{2}(x)λ_{3}(C) + … in K.

Proof. We prove Lemma 6.2 by induction on d. The case d = 1 is a special case of Lemma 4.1(d). So, we may assume that d ≥ 2 and that Lemma 6.2 holds for d − 1. By the splitting principle (see Theorem 6.1 on p. 266 in [AT]), we may assume that C = C′ + L where C′ ∈ K is of λ-degree d − 1 and L ∈ K is of λ-degree 1. Using the induction hypothesis, we obtain:

σ_{2}(λ_{−1}(C)x) = σ_{2}(λ_{−1}(C′)(1 − L)x) = = (σ_{2}((1 − L)x) − λ_{2}((1 − L)x)C′ + σ_{2}((1 − L)x)λ_{2}(C′) − …)λ_{−1}(C′).

Since σ_{2}((1−L)x) = (σ_{2}(x)−λ_{2}(x)L)(1−L) and λ_{2}((1−L)x) = (λ_{2}(x)−σ_{2}(x)L)(1− L), this implies:

σ_{2}(C, x) = σ_{2}(x) − λ_{2}(x)L − λ_{2}(x)C′ + σ_{2}(x)LC′ + σ_{2}(x)λ_{2}(C′) − λ_{2}(x)Lλ_{2}(C′) − …

= σ_{2}(x) − λ_{2}(x)C + σ_{2}(x)λ_{2}(C) − λ_{2}(x)λ_{3}(C) + …

as was to be shown. □

Now, let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. For any f. g. projective R/I-module V, we choose an R-projective resolution P.(V) of V. The following statement is a consequence of Theorem 5.1. Corollary 6.3. Let 2 be invertible in R. Then we have:

H_{k}NSym^{2}Γ(P.(V)) ≅ \{ \begin{matrix}Sym^{2}(V) ⊗ Λ^{k}(I/I^{2}) & for k even \\ Λ^{2}(V) ⊗ Λ^{k}(I/I^{2}) & for k odd. \end{matrix}

Proof. By the Eilenberg-Zilber theorem, we have:

H_{k}NSym^{2}Γ(P.(V)) ≅ H_{k}N(Γ(P.(V))^{⊗2})^{Σ_{2}} ≅ H_{k}(P.(V)^{⊗2})^{Σ_{2}}.

Therefore, Corollary 6.3 follows from Theorem 5.1. □

Theorem 6.4. Let d = 2. Then we have:

H_{k}NSym^{2}Γ(P.(V)) ≅ \{ \begin{matrix}Sym^{2}(V) & for k = 0 \\ Λ^{2}(V) ⊗ I/I^{2} & for k = 1 \\ D^{2}(V) ⊗ Λ^{2}(I/I^{2}) & for k = 2 \\ 0 & for k ≥ 3. \end{matrix}

Proof. Let F_{k} denote the functor

F_{k} : P_{R/I} → (R-modules), V ↦ H_{k}NSym^{2}Γ(P.(V)).

Obviously, the functor F_{k} is of degree ≤ 2. By the Eilenberg-Zilber theorem (see Application 8.5.3 on p. 277 in [W]), we have for all V, W ∈ P_{R/I}:

cr_{2}(F_{k})(V, W) ≅ H_{k}N(Γ(P.(V)) ⊗ Γ(P.(W)))
≅ H_{k}Tot(P.(V) ⊗ P.(W)) ≅ Tor_{k}^{R}(V, W).

Furthermore, we have Tor_{k}^{R}(R/I, R/I) ≅ Λ^{k}(I/I^{2}) by Example 5.2. Therefore, by Proposition 1.8, Corollary 1.6, Proposition 1.9, and Example 1.7, it suffices to show: The plus map +_{2} : cr_{2}(F_{0})(R/I, R/I) → F_{0}(R/I) is bijective, F_{1}(R/I) vanishes, the diagonal map Δ_{2} : F_{2}(R/I) → cr_{2}(R/I, R/I) is bijective, and F_{k}(R/I) vanishes for k ≥ 3. For this, we may assume that the ideal I is globally generated by a regular sequence, say I = (f, g). Let K. denote the complex … → 0 → R →^{f} R and L. the complex … → 0 → R →^{g} R. Then,

Tot(K. ⊗ L.) ≅ Kos^{2}(R ⊕ R ⟶^{(f,g)} R)

is a resolution of R/I. For any bisimplicial R-module M.., let ΔM.. denote the simplicial R-module Δ^{op} → (R-modules), [n] ↦ M_{nn}. The Dold-Kan correspondence (see Proposition 2.1) and the Eilenberg-Zilber theorem yield natural isomorphisms

F_{k}(R/I) ≅ H_{k}NSym^{2}ΓTot(K. ⊗ L.) ≅ H_{k}NSym^{2}ΓTot(NΓ(K.) ⊗ NΓ(L.))

≅ H_{k}NSym^{2}ΓNΔ(Γ(K.) ⊗ Γ(L.)) ≅ H_{k}NSym^{2}Δ(Γ(K.) ⊗ Γ(L.))
≅ H_{k}NΔSym^{2}(Γ(K.) ⊗ Γ(L.)).

For any two f. g. projective R-modules P, Q, the following sequence is exact:

0 → Λ^{2}(P) ⊗ Λ^{2}(Q) → Sym^{2}(P ⊗ Q) → Sym^{2}(P) ⊗ Sym^{2}(Q) → 0

(see p. 242 in [ABW]); here, the first map is given by

p_{1} ∧ p_{2} ⊗ q_{1} ∧ q_{2} ↦ (p_{1} ⊗ q_{1})(p_{2} ⊗ q_{2}) − (p_{1} ⊗ q_{2})(p_{2} ⊗ q_{1}),

and the second map is given by (p_{1} ⊗ q_{1})(p_{2} ⊗ q_{2}) ↦ p_{1}p_{2} ⊗ q_{1}q_{2}. Thus, we obtain the following short exact sequence of bisimplicial R-modules:

\xymatrix{Λ^{2}Γ(K.) ⊗ Λ^{2}Γ(L.) \ar@{^{(}->}[r] & Sym^{2}(Γ(K.) ⊗ Γ(L.)) \ar@{->>}[r] & Sym^{2}Γ(K.) ⊗ Sym^{2}Γ(L.);}

applying NΔ to this sequence then yields a short exact sequence of complexes. By the Eilenberg-Zilber theorem and Proposition 2.4, we have:

H_{k}NΔ(Sym^{2}Γ(K.) ⊗ Sym^{2}Γ(L.))
≅ H_{k}Tot(NSym^{2}Γ(K.) ⊗ NSym^{2}Γ(L.))
≅ H_{k}Tot(Kos^{2}(f) ⊗ Kos^{2}(g))
≅ H_{k}Tot(K. ⊗ L.)
≅ \{ \begin{matrix}R/I & for k = 0 \\ 0 & for k ≥ 1 \end{matrix}

Similarly, we obtain (cf. Remark 2.7):

H_{k}NΔ(Λ^{2}Γ(K.) ⊗ Λ^{2}Γ(L.)) ≅ \{ \begin{matrix}R/I & for k = 2 \\ 0 & for k ≠ 2. \end{matrix}

This finally implies:

F_{k}(R/I) ≅ \{ \begin{matrix}R/I & for k = 0, 2 \\ 0 & for k = 1 or k ≥ 3. \end{matrix}

In particular, this finishes the proof of Theorem 6.4 in the cases k = 1 and k ≥ 3. Next, we prove that the diagonal map Δ_{2} : F_{2}(R/I) → cr_{2}(F_{2})(R/I, R/I) is bijective. As above, we have an isomorphism

cr_{2}(F_{2})(R/I, R/I) ≅ H_{2}N (Γ(Tot(K. ⊗ L.))^{⊗2}) ≅ H_{2}NΔ ((Γ(K.) ⊗ Γ(L.))^{⊗2})

such that the following diagram commutes:

[figure: F_{2}(R/I); Δ_{2}; cr_{2}(F_{2})(R/I, R/I); H_{2}NΔSym^{2}(Γ(K.) ⊗ Γ(L.)); H_{2}NΔ(Δ_{2}); H_{2}NΔ ((Γ(K.) ⊗ Γ(L.))^{⊗2}) .]

Furthermore, we have the commutative diagram of bisimplicial R-modules

\xymatrix{Λ^{2}Γ(K.) ⊗ Λ^{2}Γ(L.) \ar@{^{(}->}[r] \ar[d]^{id⊗Δ_{2}} & Sym^{2}(Γ(K.) ⊗ Γ(L.)) \ar@{->>}[r] \ar[d]^{Δ_{2}} & Sym^{2}Γ(K.) ⊗ Sym^{2}Γ(L.) \ar[d]^{id⊗Δ_{2}} \\ Λ^{2}Γ(K.) ⊗ Γ(L.)^{⊗2} \ar@{^{(}->}[r] & (Γ(K.) ⊗ Γ(L.))^{⊗2} \ar@{->>}[r] & Sym^{2}Γ(K.) ⊗ Γ(L.)^{⊗2}}

where the upper row is the exact sequence already constructed above and the lower row is induced by the natural exact sequence

0 → Λ^{2}Γ(K.) → Γ(K.)^{⊗2} → Sym^{2}Γ(K.) → 0.

Since both cr_{2}(F_{2})(R/I, R/I) and F_{2}(R/I) have rank 1, it therefore suffices to show that the map

H_{2}NΔ(id ⊗ Δ_{2}) : H_{2}NΔ(Λ^{2}Γ(K.) ⊗ Λ^{2}Γ(L.)) → H_{2}NΔ(Λ^{2}Γ(K.) ⊗ Γ(L.)^{⊗2})

is surjective and that H_{2}NΔ(Sym^{2}Γ(K.) ⊗ Γ(L.)^{⊗2}) ≅ 0. As above, we have an isomorphism

H_{2}NΔ(Λ^{2}Γ(K.) ⊗ Γ(L.)^{⊗2}) ≅ H_{2}Tot(K̃os^{2}(K.) ⊗ L.^{⊗2})

≅ H_{2}Tot(K.[−1] ⊗ L.^{⊗2}) ≅ H_{1}((R/(f) →^{ḡ} R/(f))^{⊗2}) ≅ (ḡ)/(ḡ)^{2} ≅ R/I.

(Note that ḡ is a non-zero-divisor in R/(f).) Similarly, we get

H_{2}NΔ(Sym^{2}Γ(K.) ⊗ Γ(L.)^{⊗2}) ≅ 0.

The homology class of the element (1 ⊗ 1) ⊗ (1 ⊗ 1) ∈ (K_{1} ⊗ K_{0}) ⊗ (L_{1} ⊗ L_{0}) ≅ Tot(K̃os^{2}(f) ⊗ K̃os^{2}(g))_{2} obviously generates H_{2}Tot(K̃os^{2}(f) ⊗ K̃os^{2}(g)). The map id ⊗ Δ_{2} maps this element to the element (1 ⊗ 1) ⊗ (1 ⊗ 1, −1 ⊗ 1) of

(K_{1} ⊗ K_{0}) ⊗ (L_{1} ⊗ L_{0} ⊕ L_{0} ⊗ L_{1})
≅ K̃os^{2}(K.)_{1} ⊗ (L.^{⊗2})_{1} ⊆ Tot(K̃os^{2}(K.) ⊗ L.^{⊗2})_{2}.

The homology class of (1⊗1)⊗(1⊗1, −1⊗1) obviously generates H_{2}Tot(K̃os^{2}(K.)⊗ L.^{⊗2}). This finishes the proof of Theorem 6.4 in the case k = 2. Similarly, one can show that the plus map +_{2} : cr_{2}(F_{0})(R/I, R/I) → F_{0}(R/I) is bijective. Then, the proof of Theorem 6.4 is complete. However, this final case can also be proved directly as follows:

H_{0}NSym^{2}Γ(P.(V)) ≅ coker(\xymatrix{Sym^{2}(P_{1}) ⊕ P_{1} ⊗ P_{0} \ar[r]^{d_{0}−d_{1}} & Sym^{2}(P_{0})})

≅ coker(\xymatrix{P_{1} ⊗ P_{0} \ar[r]^{can} & Sym^{2}(P_{0})}) ≅ Sym^{2}(V)

(see also Remark 3.3(b)). □

Remark 6.5. For any functor F : P_{R} → P_{R} (with F (0) = 0) which commutes with localization, the support of the homology modules H_{k}NF Γ(P.(V)), k ≥ 0, is contained in V (I) = Spec(R/I) since the complex NF Γ(P.(V))|_{Spec(R)\V (I)} is homotopic to the zero complex (cf. section 2). In contrast to Theorem 3.2, Theorem 5.1, and Theorem 6.4, these homology modules need not to be projective R/Imodules, they even need not to be annihilated by I in general. For instance, if d = 1, the natural short exact sequence of complexes

0 → ND^{2}Γ(P.(V)) → N(Γ(P.(V))^{⊗2}) → NΛ^{2}Γ(P.(V)) → 0

together with Remark 3.6 and Example 5.2 yield the long exact sequence

0 → H_{1}ND^{2}Γ(P.(V)) → V^{⊗2} ⊗ I/I^{2} → D^{2}(V) ⊗ I/I^{2}
→ H_{0}ND^{2}Γ(P.(V)) → V^{⊗2} → Λ^{2}(V) → 0,

and one easily sees that H_{0}ND^{2}Γ(P.(R/I)) ≅ R/I^{2} if char(R) = 2. If d = 2, the natural short exact sequence of complexes

0 → NΛ^{2}Γ(P.(V)) → N(Γ(P.(V))^{⊗2}) → NSym^{2}Γ(P.(V)) → 0

together with Theorem 6.4 and Example 5.2 yield the long exact sequence

0 → H_{2}NΛ^{2}Γ(P.(V)) → V^{⊗2} ⊗ Λ^{2}(I/I^{2}) → D^{2}(V) ⊗ Λ^{2}(I/I^{2})
→ H_{1}NΛ^{2}Γ(P.(V)) → V^{⊗2} ⊗ I/I^{2} → Λ^{2}(V) ⊗ I/I^{2}
→ Λ^{2}(V) → V^{⊗2} → Sym^{2}(V) → 0.

One can show that the map V^{⊗2} ⊗Λ^{2}(I/I^{2}) → D^{2}(V)⊗Λ^{2}(I/I^{2}) is +_{2} ⊗id. Hence, H_{k}NΛ^{2}Γ(P.(V)) has non-trivial 2-torsion for k = 1, 2. Similarly, one may expect that H_{k}NSym^{2}Γ(P.(V)) can have non-trivial 2-torsion if d ≥ 3 and k ≥ 2. This is also suggested by the calculation of the stable derived functors of Sym^{2} (e.g., see Theorem 4.7 in [JM]). Example 6.6. Let K be a λ-ring, C ∈ K of λ-degree 2, and x ∈ K, arbitrary. An easy, but somewhat tedious calculation shows the following representation of σ_{3}(C, x):

σ_{3}(C, x) = σ_{3}(x) − s_{1}^{3}(x)C + (σ_{2}(x)xλ_{2}(C) + λ_{3}(x)σ_{2}(C)) − s_{1}^{3}(x)Cλ_{2}(C) + σ_{3}(x)λ_{2}(C)^{2}.

In view of Theorem 5.1 and Theorem 6.4, this suggests the following conjecture: If d = 2, then we have:

H_{k}NSym^{3}Γ(P.(V)) ≅ \{ \begin{matrix}Sym^{3}(V) & for k = 0 \\ L_{1}^{3}(V) ⊗ I/I^{2} & for k = 1 \\ L̃_{1}^{3}(V) ⊗ I/I^{2} ⊗ Λ^{2}(I/I^{2}) & for k = 3 \\ D^{3}(V) ⊗ Λ^{2}(I/I^{2})^{⊗2} & for k = 4 \\ 0 & for k ≥ 5, \end{matrix}

and there is a short exact sequence

0 → D^{2}(V) ⊗ V ⊗ Λ^{2}(I/I^{2}) → H_{2}NSym^{3}Γ(P.(V)) → Λ^{3}(V) ⊗ Sym^{2}(I/I^{2}) → 0.

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Mathematisches Institut II, Universität Karlsruhe, 76128 Karlsruhe, Germany

   E-mail address: Bernhard.Koeck@math.uni-karlsruhe.de