A Modular Compactification

of the General Linear Group

Ivan Kausz

Received: June 2, 2000

Communicated by Peter Schneider

Abstract. We define a certain compactifiction of the general linear group and give a modular description for its points with values in arbitrary schemes. This is a first step in the construction of a higher rank generalization of Gieseker’s degeneration of moduli spaces of vector bundles over a curve. We show that our compactification has similar properties as the “wonderful compactification” of algebraic groups of adjoint type as studied by de Concini and Procesi. As a byproduct we obtain a modular description of the points of the wonderful compactification of PGl_{n}.

1991 Mathematics Subject Classification: 14H60 14M15 20G Keywords and Phrases: moduli of vector bundles on curves, modular compactification, general linear group

1. Introduction

In this paper we give a modular description of a certain compactification KGl_{n} of the general linear group Gl_{n}. The variety KGl_{n} is constructed as follows: First one embeds Gl_{n} in the obvious way in the projective space which contains the affine space of n × n matrices as a standard open set. Then one successively blows up the closed subschemes defined by the vanishing of the r × r subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with the hyperplane at infinity. We were led to the problem of finding a modular description of KGl_{n} in the course of our research on the degeneration of moduli spaces of vector bundles. Let me explain in some detail the relevance of compactifications of Gl_{n} in this context. Let B be a regular integral one-dimensional base scheme and b_{0} ∈ B a closed point. Let C → B be a proper flat familly of curves over B which is smooth outside b_{0} and whose fibre C_{0} at b_{0} is irreducible with one ordinary double point p_{0} ∈ C_{0}. Let C̃_{0} → C_{0} be the normalization of C_{0} and let p_{1}, p_{2} ∈ C̃_{0} the two points lying above the singular point p_{0}. Thus the situation may be depicted as follows:

\xymatrix{ & (C̃_{0}, p_{1}, p_{2}) \ar[dl] \ar[dr] \\ C̃_{0} &  & C_{0} \ar[d] \ar[r] & C \ar[d] \\  &  & b_{0} \ar[r] & B}

where the left arrow means “forgetting the points p_{1}, p_{2}”. There is a corresponding diagram of moduli-functors of vector bundles (v.b.) of rank n:

[figure: \{ \begin{matrix}v.b. E on C̃_{0} \\ together with an \\ isomorphism E[p_{1}] →^{∼} E[p_{2}] \end{matrix} \}; f_{1}; f_{2}; ≅; _{\{ v.b. on ; C̃_{0} \}}; _{\{ \begin{matrix}v.b. on \\ C_{0} \end{matrix} \}}; _{\{ \begin{matrix}v.b. on \\ C/B \end{matrix} \}}; b_{0}; B]

where E[p_{i}] denotes the fibre of E at the point p_{i} (cf. section 3 below). The morphism f_{1} is “forgetting the isomorphism between the fibres” and f_{2} is “glueing together the fibres at p_{1} and p_{2} along the given isomorphism”. The square on the right is the inclusion of the special fibre. It is clear that f is a locally trivial fibration with fibre Gl_{n}. Consequently, f_{1} is not proper and thus {v.b. on C/B} is not proper over B. It is desirable to have a diagram (∗):

[figure: \{ \begin{matrix}generalized \\ v.b.-data on \\ (C̃_{0}, p_{1}, p_{2}) \end{matrix} \}; \overline{f_{1}}; \overline{f_{2}}; _{\{ v.b. on ; C̃_{0} \}}; _{\{ \begin{matrix}generalized \\ v.b. on \\ C_{0} \end{matrix} \}}; _{\{ \begin{matrix}generalized \\ v.b. on \\ C/B \end{matrix} \}}; b_{0}; B]

where the functors of “generalized” objects contain the original ones as open subfunctors and where {generalized v.b. on C/B} is proper over B or at least satisfies the existence part of the valuative criterion for properness. The motivation is that such a diagram may help to calculate cohomological invariants of {v.b. on Y } (Y a smooth projective curve) by induction on the genus of Y (notice that the genus of C̃_{0} is one less than the genus of the generic fibre of C/B). In the current literature there exist two different approaches for the construction of diagram (∗). In the first approach the “generalized v.b.” on C_{0} are torsion-free sheaves (cf. [S1], [F], [NR], [Sun]). The second approach is by Gieseker who considered only the rank-two case (cf. [G]). Here the “generalized v.b.” on C_{0} are certain vector bundles on C_{0}, C_{1} or C_{2}, where C_{i} is built from C_{0} by inserting a chain of i copies of the projective line at p_{0}. (Cf. also [Tei] for a discussion of the two approaches). Of course, this is only a very rough picture of what is going on in these papers since I do not mention concepts of stability for the various objects nor the representability of the functors by varieties or by algebraic stacks. In both approaches the morphism \overline{f_{2}} is the normalization morphism (at least on the complement of a set of small dimension) and \overline{f_{1}} is a locally trivial fibration with fibre a compactification of Gl_{n}. In the torsion-free sheaves approach this compactification is Gr(2n, n), the grassmanian of n-dimensional subspaces of a 2n-dimensional vector space. In Gieseker’s construction the relevant compactification of Gl_{2} is KGl_{2}. An advantage of Gieseker’s construction is that in contrast to the torsion-free sheaves approach, the space {generalized v.b. on C/B} is regular and its special fibre over b_{0} is a divisor with normal crossings. Very recently, Nagaraj and Seshadri have generalized Gieseker’s construction of the right part of diagram (∗), i.e. the diagram

\xymatrix{\{ \begin{matrix}generalized \\ v.b. on \\ C_{0} \end{matrix} \} \ar[d] \ar[r] & \{ \begin{matrix}generalized \\ v.b. on \\ C/B \end{matrix} \} \ar[d] \\ b_{0} \ar[r] & B}

to arbitrary rank n (cf. [NS], [S2]). Nagaraj’s and Seshadri’s “generalized vector bundles” on C_{0} are certain equivalence classes of vector bundles on one of the curves C_{0}, …, C_{n}, whose push-forward to C_{0} are stable torsion free sheaves. Without worrying about stability I have recently (and independently from Nagaraj and Seshadri) constructed the full diagram (∗) at least at the level of functors (details will appear in a forthcoming paper) and I have reasons to believe that the fibres of the corresponding morphism \overline{f_{1}} should be represented by KGl_{n}. The present paper is the first step in the proof of this fact. The compactification KGl_{n} of Gl_{n} has properties similar to those of the “wonderful compactification” of algebraic groups of adjoint type as studied by De Concini and Procesi (cf. [CP]). Namely:

1. The group Gl_{n} × Gl_{n} acts on KGl_{n}, extending the operation of Gl_{n} × Gl_{n} on Gl_{n} induced by right and left multiplication (cf. 5.6).

2. The complement of Gl_{n} in KGl_{n} is a divisor with normal crossings with irreducible components Y_{i}, Z_{j} (i, j ∈ {0, …, n − 1}) (cf. 4.2).

3. The orbit closures of the operation of Gl_{n} × Gl_{n} on KGl_{n} are precisely the intersections Y_{I} ∩ Z_{J}, where I, J are subsets of {0, …, n − 1} with min(I) + min(J) ≥ n and where Y_{I} := ∩_{i∈I}Y_{i}, Z_{J} := ∩_{j∈J}Z_{j} (cf. 9.4).

4. For each I, J as above there exists a natural mapping from Y_{I} ∩Z_{J} to the product of two flag varieties. This mapping is a locally trivial fibration with standard fibre a product of copies of \overline{PGl}_{n_{k}} (the wonderful compactification of PGl_{n_{k}}) for some n_{k} ≥ 1 and of one copy of KGl_{m} for some m ≥ 0 (cf. 9.3).

Our main theorem 5.5 says that KGl_{n} parametrizes what we call “generalized isomorphisms” from the trivial bundle of rank n to itself. A generalized isomorphism between vector bundles E and F is by definition a diagram

[figure: E; =; E_{0}; ⊗; E_{1}; ⊗; E_{2}; …; E_{n−1}; ⊗; E_{n}; ∼; F_{n}; F_{n−1}; ⊗; …; F_{2}; F_{1}; ⊗; F_{0}; ⊗; =; F]

with certain properties, where the E_{i} and F_{j} are vector bundles of the same rank as E and F and where the arrow [figure: ⊗]indicates a morphims of the source into the target tensored with a line bundle to be specified. Cf. 5.2 for a precise definition. The wonderful compactification \overline{PGl}_{n} of PGl_{n} is contained as an orbit closure in KGl_{n}, in fact Y_{0} ≅ \overline{PGl}_{n}. Therefore theorem 5.5 implies a modular description of \overline{PGl}_{n}. One of the reasons why I decided to publish the present paper separately from my investigations on the degeneration of moduli spaces of vector bundles on curves is the fact that \overline{PGl}_{n} has been quite extensively studied in the past (cf. [Lak1] for a historical overview and also the recent paper [Tha2]). Although some efford has been made to find a modular description for it, up to now only partial results in this direction have been obtained (cf. [V], [Lak2], [TK]). In section 8 we explain the connection of these results with ours. Recently Lafforgue has used \overline{PGl}_{n} to compactify the stack of Drinfeld’s shtukas (cf. [Laf1], [Laf2]). Sections 4 and 5 contain the main definitions: In section 4 we give the construction of KGl_{n} and in section 5 we define the notion of generalized isomorphisms. At the end of section 5 we state our main theorem 5.5. Its proof is given in sections 6 and 7. In section 8 we define complete collineations and compare our notion with the one given by previous authors, in section 9 we study the orbit closures of the operation of Gl_{n} × Gl_{n} on KGl_{n} and in section 10 we define an equivariant morphism of KGl_{n} onto the Grassmannian compactification of Gl_{n} and compute its fibres. My interest in degeneration of moduli spaces of bundles on curves has been greatly stimulated by a workshop on conformal blocks and the Verlinde formula, organized in March 1997 by the physicists Jürgen Fuchs and Christoph Schweigert at the Mathematisches Forschungsinstitut in Oberwolfach. Part of this work has been prepared during a stay at the Mathematical Institute of the University of Oxford. Its hospitality is gratefully acknowledged. Thanks are due to Daniel Huybrechts for mentioning to me the work of Thaddeus, to M. Thaddeus himself for sending me a copy of part of his thesis and to M.

Rapoport for drawing my attention to the work of Laksov and Lafforgue. I would also like to thank Uwe Jannsen for his constant encouragement.

2. An elementary example

This section is not strictly necessary for the comprehension of what follows. But since the rest of the paper is a bit technical, I felt that a simple example might facilitate its understanding. Let A be a discrete valuation ring, K its field of fractions, 𝔪 its maximal ideal, t ∈ 𝔪 a local parameter and k := A/𝔪 the residue class field of A. Let E and F be two free A-modules of rank n and let φ_{K} : E_{K} →^{∼} F_{K} be an isomorphism between the generic fibers E_{K} := E ⊗_{A} K and F_{K} := F ⊗_{A} K of E and F. We can choose A-bases of E and F such that φ_{K} has the matrix presentation diag(t^{m_{1}}, …, t^{m_{n}}) with respect to these bases, where m_{i} ∈ ℤ and m_{1} ≤ ⋯ ≤ m_{n}. Now let a_{0} := 0 =: b_{0} and for 1 ≤ i ≤ n set a_{i} := − min(0, m_{n+1−i}) and b_{i} := max(0, m_{i}). Note that we have

0 = a_{0} = ⋯ = a_{n−l} ≤ a_{n−l+1} ≤ ⋯ ≤ a_{n}
and  0 = b_{0} = ⋯ = b_{l} ≤ b_{l+1} ≤ ⋯ ≤ b_{n}

for some l ∈ {0, …, n}. Let

E_{n} ⊆ ⋯ ⊆ E_{1} ⊆ E_{0} := E and F_{n} ⊆ ⋯ ⊆ F_{1} ⊆ F_{0} := F

be the A-submodules defined by

E_{i+1} := [ t^{a_{i+1}−a_{i}}𝕀_{n−i} 0 0  𝕀_{i} ] E_{i} , F_{i+1} := [ 𝕀_{i}  0 0 t^{b_{i+1}−b_{i}}𝕀_{n−i} ] F_{i} ,

where 𝕀_{i} denotes the i × i unit matrix. Then φ_{K} induces an isomorphism φ : E_{n} →^{∼} F_{n} and we have the natural injections

E_{i} ↪ 𝔪^{a_{i}−a_{i+1}}E_{i+1} , E_{i} ↩ E_{i+1}
F_{i+1} ↪ F_{i} , 𝔪^{b_{i}−b_{i+1}}F_{i+1} ↩ F_{i}  .

Observe that the compositions E_{i+1} ↪ E_{i} ↪ 𝔪^{a_{i}−a_{i+1}}E_{i+1} and E_{i} ↪ 𝔪^{a_{i}−a_{i+1}}E_{i+1} ↪ 𝔪^{a_{i}−a_{i+1}}E_{i} are both the injections induced by the inclusion A ↪ 𝔪^{a_{i}−a_{i+1}}. Furthermore, if a_{i} − a_{i+1} < 0 then the morphism of k-vectorspaces E_{i+1} ⊗ k → E_{i} ⊗ k is of rank i and the sequence

E_{i+1} ⊗ k → E_{i} ⊗ k → (𝔪^{a_{i}−a_{i+1}}E_{i+1}) ⊗ k → (𝔪^{a_{i}−a_{i+1}}E_{i}) ⊗ k

is exact. This shows that the tupel

(𝔪^{a_{i}−a_{i+1}}, 1 ∈ 𝔪^{a_{i}−a_{i+1}} , E_{i+1} ↪ E_{i} , 𝔪^{a_{i}−a_{i+1}}E_{i+1} ↩ E_{i} , i)

is what we call a “bf-morphism” of rank i (cf. definition 5.1). Observe now that if a_{i} − a_{i+1} < 0 and (f, g) is one of the following two pairs of morphisms:

E ⊗ k ⟶^{f} (𝔪^{−a_{i}}E_{i}) ⊗ k ⟶^{g} (𝔪^{−a_{i+1}}E_{i+1}) ⊗ k ,

E_{i} ⊗ k ⟵^{g} E_{i+1} ⊗ k ⟵^{f} E_{n} ⊗ k ,

then im(g ∘ f) = im(g). The above statements hold true also if we replace the E_{i}-s by the F_{i}-s and the a_{i}-s by the b_{i}-s. Observe finally that in the diagram

\xymatrix{ &  & 0 \ar[d] \\  &  & ker(E_{n} ⊗ k → E_{0} ⊗ k) \ar[d] \ar@{-->}[dr] \\ 0 \ar[r] & ker(F_{n} ⊗ k → F_{0} ⊗ k) \ar[r] \ar@{-->}[dr] & E_{n} ⊗ k ≅ F_{n} ⊗ k \ar[d] \ar[r] & im(F_{n} ⊗ k → F_{0} ⊗ k) \ar[r] & 0 \\  &  & im(E_{n} ⊗ k → E_{0} ⊗ k) \ar[d] \\  &  & 0}

the oblique arrows are injections. All these properties are summed up in the statement that the tupel

Φ := ((𝔪^{b_{i}−b_{i+1}}, 1), (𝔪^{a_{i}−a_{i+1}}, 1), E_{i} ↪ 𝔪^{a_{i}−a_{i+1}}E_{i+1}, E_{i} ↩ E_{i+1},

F_{i+1} ↪ F_{i}, 𝔪^{b_{i}−b_{i+1}}F_{i+1} ↩ F_{i} (0 ≤ i ≤ n − 1), φ : E_{n} →^{∼} F_{n})

is a generalized isomorphism from E to F in the sense of definition 5.2, where for a ≤ 0 we consider 𝔪^{a} as an invertible A-module with global section 1 ∈ 𝔪^{a}. Observe that Φ does not depend on our choice of the bases for E and F. Indeed, it is well-known that the sequence (m_{1}, …, m_{n}) is independent of such a choice and it is easy to see that E_{n} = φ_{K}^{−1}(F) ∩ E, F_{n} = φ_{K}(E_{n}) and

E_{i} = E_{n} + 𝔪^{a_{i}}E , F_{i} = F_{n} + 𝔪^{b_{i}}F

for 1 ≤ i ≤ n−1, where the +-sign means generation in E_{K} and F_{K} respectively. Observe furthermore that by pull-back the generalized isomorphism Φ induces a generalized isomorphism f^{∗}Φ on a scheme S for every morphism f : S → Spec (A). Of course the morphisms f^{∗}E_{i+1} → f^{∗}E_{i} etc. will be in general no longer injective, but this is not required in the definition.

3. Notations

We collect some less common notations, which we will use freely in this paper:

• For two integers a ≤ b we sometimes denote by [a, b] the set {c ∈ ℤ | a ≤ c ≤ b}.

• For a n × n-matrix with entries a_{ij} in some ring, and for two subsets A and B of cardinality r of {1, …, n}, we will denote by det_{AB}(a_{ij}) the determinant of the r × r-matrix (a_{ij})_{i∈A,j∈B}.

• For a scheme X we will denote by K_{X} the sheaf of total quotient rings of O_{X}.

• For a scheme X, a coherent sheaf E on X and a point x ∈ X, we denote by E[x] the fibre E ⊗_{O_{X}} κ(x) of E at x.

• For n ∈ ℕ, the symbol S_{n} denotes the symmetric group of permutations of the set {1, …, n}.

4. Construction of the compactification

Let X^{(0)} := Proj ℤ[x_{00}, x_{ij} (1 ≤ i, j ≤ n)]. We define closed subschemes

[figure: Y_{0}^{(0)}; Y_{1}^{(0)}; …; Y_{n−1}^{(0)}; Z_{n−1}^{(0)}; …; Z_{1}^{(0)}; Z_{0}^{(0)}]

of X^{(0)}, by setting Y_{r}^{(0)} := V^{+}(I_{r}^{(0)}), Z_{r}^{(0)} := V^{+}(J_{r}^{(0)}), where I_{r}^{(0)} is the homogenous ideal in ℤ[x_{00}, x_{ij} (1 ≤ i, j ≤ n)], generated by all (r+1)×(r+1)- subdeterminants of the matrix (x_{ij})_{1≤i,j≤n}, and where J_{r}^{(0)} = (x_{00}) + I_{n−r}^{(0)} for 0 ≤ r ≤ n − 1. For 1 ≤ k ≤ n let the scheme X^{(k)} together with closed subschemes Y_{r}^{(k)}, Z_{r}^{(k)} ⊂ X^{(k)} (0 ≤ r ≤ n − 1) be inductively defined as follows: X^{(k)} → X^{(k−1)} is the blowing up of X^{(k−1)} along the closed subscheme Y_{k−1}^{(k−1)} ∪ Z_{n−k}^{(k−1)}. The subscheme Y_{k−1}^{(k)} ⊂ X^{(k)} (respectively Z_{n−k}^{(k)} ⊂ X^{(k)}) is the inverse image of Y_{k−1}^{(k−1)} (respectively of Z_{n−k}^{(k−1)}) under the morphism X^{(k)} → X^{(k−1)}, and for r ≠ k − 1 (respectively r ≠ n − k) the subscheme Y_{r}^{(k)} ⊂ X^{(k)} (respectively of Z_{r}^{(k)} ⊂ X^{(k)}) is the complete transform of Y_{r}^{(k−1)} ⊂ X^{(k−1)} (respectively Z_{r}^{(k−1)} ⊂ X^{(k−1)}). We set

KGl_{n} := X^{(n)}  and  Y_{r} := Y_{r}^{(n)} , Z_{r} := Z_{r}^{(n)} (0 ≤ r ≤ n − 1) .

We are interested in finding a modular description for the compactification KGl_{n} of Gl_{n} = Spec ℤ[x_{ij}/x_{00} (1 ≤ i, j ≤ n), det(x_{ij}/x_{00})^{−1}]. Let (α, β) ∈ S_{n} × S_{n} and set

x_{ij}^{(0)}(α, β) := \frac{x_{α(i),β(j)}}{x_{00}} (1 ≤ i, j ≤ n) .

For 1 ≤ k ≤ n we define elements

y_{ji}(α, β) , z_{ij}(α, β) (1 ≤ i ≤ k, i < j ≤ n)
x_{ij}^{(k)}(α, β)  (k + 1 ≤ i, j ≤ n)

of the function field ℚ(X^{(0)}) = ℚ(x_{ij}/x_{00} (1 ≤ i, j ≤ n)) of X^{(0)} inductively as follows:

y_{ik}(α, β) := \frac{x_{ik}^{(k−1)}(α, β)}{x_{kk}^{(k−1)}(α, β)}  (k + 1 ≤ i ≤ n) ,

z_{kj}(α, β) := \frac{x_{kj}^{(k−1)}(α, β)}{x_{kk}^{(k−1)}(α, β)}  (k + 1 ≤ j ≤ n) ,

x_{ij}^{(k)}(α, β) := \frac{x_{ij}^{(k−1)}(α, β)}{x_{kk}^{(k−1)}(α, β)} − y_{ik}(α, β) z_{kj}(α, β)  (k + 1 ≤ i, j ≤ n) .

Finally, we set t_{0}(α, β) := t_{0} := x_{00} and

t_{i}(α, β) := t_{0} · ∏_{j=1}^{i} x_{jj}^{(j−1)}(α, β) (1 ≤ i ≤ n) .

Observe, that for each k ∈ {0, …, n}, we have the following decomposition of the matrix [x_{ij}/x_{00}]:

[\frac{x_{ij}}{x_{00}} ] = n_{α} [ [figure: 1; 0; 1; y_{ij} (α, β); 𝕀_{n−k}]] [ _{\frac{t_{1}(α,β)}{t_{0}}}  _{0}  _{0}_{0}  _{\frac{t_{k}(α,β)}{t_{0}}}_{0}  _{\frac{t_{k}(α,β)}{t_{0}} [x_{ij}^{(k)}(α, β)]} ] [ [figure: 1; z_{ij} (α, β); 1; 0; 𝕀_{n−k}]] n_{β}^{−1}

Here, n_{α} is the permutation matrix associated to α, i.e. the matrix, whose entry in the i-th row and j-th column is δ_{i,α(j)}. For convenience, we define for

each l ∈ {0, …, n} a bijection ι_{l} : {1, …, n + 1} →^{∼} {0, …, n}, by setting

ι_{l}(i) = \{ \begin{matrix}i & if 1 ≤ i ≤ l \\ 0 & if i = l + 1 \\ i − 1 & if l + 2 ≤ i ≤ n + 1 \end{matrix}

for 1 ≤ i ≤ n + 1. With this notaton, we define for each triple (α, β, l) ∈ S_{n} × S_{n} × [0, n] polynomial subalgebras R(α, β, l) of ℚ(KGl_{n}) = ℚ(X^{(0)}) together with ideals I_{r}(α, β, l) and J_{r}(α, β, l) (0 ≤ r ≤ n − 1) of R(α, β, l) as follows:

R(α, β, l) := ℤ [ \frac{t_{ι_{l}(i+1)}(α, β)}{t_{ι_{l}(i)}(α, β)} (1 ≤ i ≤ n), y_{ji}(α, β), z_{ij}(α, β) (1 ≤ i < j ≤ n)] ,
I_{r}(α, β, l) := ( \frac{t_{ι_{l}(r+2)}(α, β)}{t_{ι_{l}(r+1)}(α, β)} ) if l ≤ r ≤ n − 1 and I_{r}(α, β, l) := (1) else,
J_{r}(α, β, l) := ( \frac{t_{ι_{l}(n−r+1)}(α, β)}{t_{ι_{l}(n−r)}(α, β)} ) if n − l ≤ r ≤ n − 1 and J_{r}(α, β, l) := (1) else.

Proposition 4.1. There is a covering of KGl_{n} by open affine pieces X(α, β, l) ((α, β, l) ∈ S_{n} × S_{n} × [0, n]), such that Γ(X(α, β, l), O) = R(α, β, l) (equality as subrings of the function field ℚ(KGl_{n})). Furthermore, for 0 ≤ r ≤ n − 1 the ideals I_{r}(α, β, l) and J_{r}(α, β, l) of R(α, β, l) are the defining ideals for the closed subschemes Y_{r}(α, β, l) := Y_{r} ∩X(α, β, l) and Z_{r}(α, β, l) := Z_{r} ∩X(α, β, l) respectively.

Proof. We make the blowing-up procedure explicit, in terms of open affine coverings. For each k ∈ {0, …, n} we define a finite index set P_{k}, consisting of all pairs

(p, q) = ([ \begin{matrix}p_{0} \\ : \\ p_{k} \end{matrix} ] , [ \begin{matrix}q_{0} \\ : \\ q_{k} \end{matrix} ]) ∈ \{0, …, n\}^{k+1} × \{0, …, n\}^{k+1}

with the property that p_{i} ≠ p_{j} and q_{i} ≠ q_{j} for i ≠ j and that p_{i} = 0 for some i, if and only if q_{i} = 0. Observe that for each k ∈ {0, …, n} there is a surjection S_{n} × S_{n} × {0, …, n} → P_{k}, which maps the triple (α, β, l) to the element

(p, q) = ([ \begin{matrix}α(ι_{l}(1)) \\ : \\ α(ι_{l}(k + 1)) \end{matrix} ] , [ \begin{matrix}β(ι_{l}(1)) \\ : \\ β(ι_{l}(k + 1)) \end{matrix} ])

of P_{k}. (Here we have used the convention that α(0) := 0 for any permutation α ∈ S_{n}). Furthermore, this surjection is in fact a bijection in the case of k = n. Let (p, q) ∈ P_{k} and chose an element (α, β, l) in its preimage under the surjection S_{n} × S_{n} × {0, …, n} → P_{k}. We define subrings R^{(k)}(p, q) of ℚ(x_{ij}/x_{00} (1 ≤ i, j ≤ n)) together with ideals I_{r}^{(k)}(p, q), J_{r}^{(k)}(p, q) (0 ≤ r ≤ n), distinguishing three cases.

First case: 0 ≤ l ≤ k − 1

R^{(k)}(p, q) := ℤ [ \frac{t_{ιl(i+1)}(α, β)}{t_{ιl(i)}(α, β)} (1 ≤ i ≤ k), y_{ji}(α, β), z_{ij}(α, β) ( \begin{matrix}1 ≤ i ≤ k, \\ i < j ≤ n \end{matrix} ) ,
x_{ij}^{(k)}(α, β) (k + 1 ≤ i, j ≤ n)]

I_{r}^{(k)}(p, q) := \{ \begin{matrix}(1) & if r ∈ [0, l − 1] \\ ( \frac{tιl(r+2)(α,β)}{tιl(r+1)(α,β)} ) & if r ∈ [l, k − 1] \\ (det_{AB}(x_{ij}^{(k)}(α, β)) ( \begin{matrix}A, B ⊆ \{k + 1, …, n\} \\ ♯A = ♯B = r + 1 − k \end{matrix} ) ) & if r ∈ [k, n − 1] \end{matrix}

J_{r}^{(k)}(p, q) := \{ \begin{matrix}(1) & if r ∈ [0, n − l − 1] \\ ( \frac{tιl(n−r+1)(α,β)}{tιl(n−r)(α,β)} ) & if r ∈ [n − l, n − 1] \end{matrix}

Second case: l = k

R^{(k)}(p, q) := ℤ [ \frac{t_{ιl(i+1)}(α, β)}{t_{ιl(i)}(α, β)} (1 ≤ i ≤ k), y_{ji}(α, β), z_{ij}(α, β) ( \begin{matrix}1 ≤ i ≤ k, \\ i < j ≤ n \end{matrix} ) ,

\frac{t_{k}(α, β)}{t_{0}} x_{ij}^{(k)}(α, β) (k + 1 ≤ i, j ≤ n)]

I_{r}^{(k)}(p, q) := \{ \begin{matrix}(1) & if r ∈ [0, l − 1] \\ (det_{AB} ( \frac{tk(α,β)}{t0} x_{ij}^{(k)}(α, β)) ( \begin{matrix}A, B ⊆ \{k + 1, …, n\} \\ ♯A = ♯B = r + 1 − k \end{matrix} ) ) & if r ∈ [l, n − 1] \end{matrix}

J_{r}^{(k)}(p, q) := \{ \begin{matrix}(1) & if r ∈ [0, n − l − 1] \\ ( \frac{tιl(n−r+1)(α,β)}{tιl(n−r)(α,β)} ) & if r ∈ [n − l, n − 1] \end{matrix}

Third case: k + 1 ≤ l ≤ n

R^{(k)}(p, q) := ℤ [ \frac{t_{ιl(i+1)}(α, β)}{t_{ιl(i)}(α, β)} (1 ≤ i ≤ k), \frac{t_{0}}{t_{k+1}(α, β)} ,

y_{ji}(α, β), z_{ij}(α, β) ( \begin{matrix}1 ≤ i ≤ k + 1, \\ i < j ≤ n \end{matrix} ) , x_{ij}^{(k+1)}(α, β) (k + 2 ≤ i, j ≤ n)]

I_{r}^{(k)}(p, q) := \{ \begin{matrix}(1) & if r ∈ [0, k] \\ (det_{AB} (x_{ij}^{(k+1)}(α, β)) ( \begin{matrix}A, B ⊆ \{k + 2, …, n\} \\ ♯A = ♯B = r − k \end{matrix} ) ) & if r ∈ [k + 1, n − 1] \end{matrix}

J_{r}^{(k)}(p, q) := \{ \begin{matrix}( \frac{t0}{tk+1(α,β)} , det_{AB}(x_{ij}^{(k+1)}(α, β)) & ( \begin{matrix}A, B ⊆ \{k + 2, …, n\} \\ ♯A = ♯B = n − r − k \end{matrix} ) ) \\ & if r ∈ [0, n − k − 1] \\ ( \frac{tιl(n−r+1)(α,β)}{tιl(n−r)(α,β)} ) & if r ∈ [n − k, n − 1] \end{matrix}

Observe that the objects R^{(k)}(p, q), I_{r}^{(k)}(p, q), J_{r}^{(k)}(p, q) thus defined, depend indeed only on (p, q) and not on the chosen element (α, β, l). By induction on k one shows that X^{(k)} is covered by open affine pieces X^{(k)}(p, q) ((p, q) ∈ P_{k}), such that Γ(X^{(k)}(p, q), O) = R^{(k)}(p, q) (equality as subrings of the function field ℚ(X^{(k)})), and such that the ideals I_{r}^{(k)}(α, β) and J_{r}^{(k)}(α, β) are the defining ideals of the closed subschemes Y_{r}^{(k)} ∩ X^{(k)}(p, q) and Z_{r}^{(k)} ∩ X^{(k)}(p, q) respectively.

Corollary 4.2. The scheme KGl_{n} is smooth and projective over Spec ℤ and contains Gl_{n} as a dense open subset. The complement of Gl_{n} in KGl_{n} is the union of the closed subschemes Y_{i}, Z_{i} (0 ≤ i ≤ n − 1), which is a divisor with normal crossings. Furthermore, we have Y_{i} ∩ Z_{j} = ∅ for i + j < n.

Proof. This is immediate from the local description given in 4.1.

We will now define a certain toric scheme, which will play an important role in the sequel. Let M := ℤ^{n}, with canonical basis e_{1}, …, e_{n}. For m ∈ M we denote by t^{m} the corresponding monomial in the ring ℤ[M]. Furthermore, we write t_{i}/t_{0} for the canonical generator t^{e_{i}} of ℤ[M]. Let N := M^{∨} be the dual of M with the dual basis e_{1}^{∨}, …, e_{n}^{∨}. For 0 ≤ l ≤ n let σ_{l} ⊂ N_{ℚ} := N ⊗ ℚ be the cone generated by the elements − ∑_{j=1}^{i} e_{j}^{∨} (1 ≤ i ≤ l) and the elements ∑_{j=i}^{n} e_{j}^{∨} (l + 1 ≤ i ≤ n). In other words:

σ_{l} = ∑_{i=1}^{l} ℚ_{+} · (− ∑_{j=1}^{i} e_{j}^{∨}) + ∑_{i=l+1}^{n} ℚ_{+} · (∑_{j=i}^{n} e_{j}^{∨}) .

Let Σ be the fan generated by all σ_{l} (0 ≤ l ≤ n) and let T̃ := X_{Σ} the associated toric scheme (over ℤ). See e.g. [Da] for definitions. T̃ is covered by the open sets T̃_{l} := X_{σ_{l}^{∨}} = Spec ℤ[t^{m} (m ∈ σ_{l}^{∨} ∩ M)] = Spec ℤ[t_{ι_{l}(i+1)}/t_{ι_{l}(i)} (1 ≤ i ≤ n)]. Observe that there are Cartier divisors Y_{r,T̃}, Z_{r,T̃} (0 ≤ r ≤ n − 1) on T̃, such that for each l ∈ {0, …, n} over the open part T̃_{l} ⊂ T̃,

Y_{r,T̃} is given by the equation \{ \begin{matrix}1 & if 0 ≤ r ≤ l − 1 \\ t_{ι_{l}(r+2)}/t_{ι_{l}(r+1)} & if l ≤ r ≤ n − 1 \end{matrix}
Z_{r,T̃} is given by the equation \{ \begin{matrix}1 & if 0 ≤ r ≤ n − l − 1 \\ t_{ι_{l}(n−r+1)}/t_{ι_{l}(n−r)} & if n − l ≤ r ≤ n − 1 \end{matrix}

Observe furthermore that Y_{i,T̃} ∩ Z_{j,T̃} = ∅ for i + j < n and that for each r ∈ {1, …, n}, multiplication by t_{r}/t_{0} establishes an isomorphism

OT̃ (− ∑_{i=0}^{n−r} Z_{i,T̃}) ⟶^{∼} OT̃ (− ∑_{i=0}^{r−1} Y_{i,T̃}) .

Lemma 4.3. The toric scheme T̃ together with the “universal” tupel

(OT̃(Y_{i,T̃}), 1_{OT̃(Y_{i,T̃})}, OT̃(Z_{i,T̃}), 1_{OT̃(Z_{i,T̃})} (0 ≤ i ≤ n − 1), t_{r}/t_{0} (1 ≤ r ≤ n))

represents the functor, which to each scheme S associates the set of equivalence classes of tupels

(L_{i}, λ_{i}, M_{i}, μ_{i} (0 ≤ i ≤ n − 1), φ_{r} (1 ≤ r ≤ n)) ,

where the L_{i} and M_{i} are invertible O_{S}-modules with global sectons λ_{i} and μ_{i} respectively, such that for i + j < n the zero-sets of λ_{i} and μ_{j} do not intersect, and where the φ_{r} are isomorphisms

⨂_{i=0}^{n−r} M_{i}^{∨} ⟶^{∼} ⨂_{i=0}^{r−1} L_{i}^{∨} .

Here two tupels (L_{i}, λ_{i}, M_{i}, μ_{i} (0 ≤ i ≤ n − 1), φ_{r} (1 ≤ r ≤ n)) and (L′_{i}, λ′_{i}, M′_{i}, μ′_{i} (0 ≤ i ≤ n − 1), φ′_{r} (1 ≤ r ≤ n)) are called equivalent, if there exist isomorphisms L_{i} →^{∼} L′_{i} and M_{i} →^{∼} M′_{i} for 0 ≤ i ≤ n − 1, such that all the obvious diagrams commute.

Proof. Let S be a scheme and (L_{i}, λ_{i}, M_{i}, μ_{i} (0 ≤ i ≤ n−1), φ_{r} (1 ≤ r ≤ n)) a tupel defined over S, which has the properties stated in the lemma. Let us first consider the case, where all the sheaves L_{i}, M_{i} are trivial and where there exists an l ∈ {0, …, n}, such that λ_{i} and μ_{j} is nowhere vanishing for 0 ≤ i < l and 0 ≤ j < n − l respectively. Observe that under theses conditions there exists a unique set of trivializations L_{i} →^{∼} O_{S}, M_{i} →^{∼} O_{S}, (0 ≤ i ≤ n) such that λ_{i} ↦ 1 for 0 ≤ i < l, μ_{j} ↦ 1 for 0 ≤ j < n − l, and such that the diagrams

\xymatrix{⨂_{i=0}^{n−r} M_{i}^{∨} \ar@{-->}[dr]^{∼} \ar[rr]^{φ_{r}}_{∼} &  & ⨂_{i=0}^{r−1} L_{i}^{∨} \ar[dl]_{∼} \\  & O_{S}}

commute for 1 ≤ r ≤ n. Let a_{ν} ∈ Γ(S, O_{S}) (1 ≤ ν ≤ n) be defined by λ_{i} ↦ a_{i+1} for l ≤ i ≤ n − 1 and μ_{j} ↦ a_{n−j} for n − l ≤ j ≤ n − 1, and let f_{l} : S → T̃_{l} be the morphism defined by f_{l}^{∗}(t_{ι_{l}(ν+1)}/t_{ι_{l}(ν)}) = a_{ν} (1 ≤ ν ≤ n). It is straightforward to check that the induced morphism f : S → T̃ does not depend on the chosen number l and that it is unique with the property that the pull-back under f of the universal tupel is equivalent to the given one on S. Returning to the general case, observe that there is an open covering S = ∪_{k}U_{k}, such that for each k there exists an l with the property that over U_{k} all the L_{i}, M_{i} are trivial and that λ_{i} and μ_{j} is nowhere vanishing over U_{k} for 0 ≤ i < l and 0 ≤ j < n − l. The above construction shows that there exists a unique morphism f : S → T̃ such that for each k the restriction to U_{k} of the pullback under f of the universal tupel is equivalent to the restriction to U_{k} of the given one. Thus it remains only to show that the isomorphisms defining the equivalences over the U_{k} glue together to give a global equivalence of the pull-back of the universal tupel with the given one. However, this is clear, since it is easy to see that there exists at most one set of isomorphisms L_{i} →^{∼} L′_{i}, M_{i} →^{∼} M′_{i} establishing an equvalence between two tuples (L_{i}, λ_{i}, M_{i}, μ_{i}, φ_{r}) and (L′_{i}, λ′_{i}, M′_{i}, μ′_{i}, φ′_{r}).

For each pair (α, β) ∈ S_{n} × S_{n} we define the open subset X(α, β) ⊆ KGl_{n} as the union of the open affines X(α, β, l) (0 ≤ l ≤ n). Let

U^{−} := Spec ℤ[y_{ji} (1 ≤ i < j ≤ n)] ,
U^{+} := Spec ℤ[z_{ij} (1 ≤ i < j ≤ n)] .

Let y : X(α, β) → U^{−} (respectively z : X(α, β) → U^{+}) be the morphism defined by the property that y^{∗}(y_{ji}) = y_{ji}(α, β) (respectively z^{∗}(z_{ij}) = z_{ij}(α, β)) for 1 ≤ i, j ≤ n. Observe that just as in the case of T̃, multiplication by the rational function t_{r}(α, β)/t_{0} provides an isomorphism

O_{X(α,β)} (− ∑_{i=0}^{n−r} Z_{i}(α, β)) ⟶^{∼} O_{X(α,β)} (− ∑_{i=0}^{r−1} Y_{i}(α, β))

for 1 ≤ r ≤ n, where Y_{i}(α, β) (respectively Z_{i}(α, β)) denotes the restriction of Y_{i} (respectively Z_{i}) to the open set X(α, β). Thus, by lemma 4.3, the tupel

(O(Y_{i}(α, β)), 1, O(Z_{i}(α, β)), 1 (i ∈ [0, n − 1]), t_{r}(α, β)/t_{0} (r ∈ [1, n]))

defines a morphism t : X(α, β) → T̃.

Lemma 4.4. The morphism (y, t, z) : X(α, β) → U^{−} × T̃ × U^{+} is an isomorphism.

Proof. Let Ω(α, β) ⊂ X(α, β) be the preimage of Gl_{n} under the morphism X(α, β) ↪ KGl_{n} → X^{(0)}. By definition of KGl_{n}, we have for all l ∈ {0, …, n}:

Ω(α, β) = X(α, β, l) \ ⋃_{i=0}^{n−1} (Y_{i}(α, β, l) ∪ Z_{i}(α, β, l))
= Spec ℤ[y_{ji}(α, β), z_{ij}(α, β) (1 ≤ i < j ≤ n),

(t_{i}(α, β)/t_{0})^{±1} (1 ≤ i ≤ n)] .

Let T := Spec ℤ[(t_{i}/t_{0})^{±1}] ⊂ T̃ be the Torus in T̃. We have an isomorphism

Ω(α, β) →^{∼} U^{−} × T × U^{+} defined by y_{ji} ↦ y_{ji}(α, β), z_{ij} ↦ z_{ij}(α, β), t_{i}/t_{0} ↦

t_{i}(α, β)/t_{0}, and a commutative quadrangle

X(α, β) ⟶^{(y,t,z)} U^{−} × T̃ × U^{+}
↑  ↑
Ω(α, β) ⟶^{∼} U^{−} × T × U^{+} ,

where the vertical arrows are the natural inclusions. Furthermore, the map (y, t, z) induces an isomorphism X(α, β, l) →^{∼} U^{−} × T̃_{l} × U^{+} for 0 ≤ l ≤ n. Using the fact that X(α, β) is separated and that Ω(α, β) dense in X(α, β), the lemma now follows easily.

5. bf-morphisms and generalized isomorpisms

Definition 5.1. Let S be a scheme, E and F two localy free O_{S}-modules and r a nonnegative integer. A bf-morphism of rank r from E to F is a tupel

g = (M, μ, E → F, M ⊗ E ← F, r) ,

where M is an invertible O_{S}-module and μ a global section of M such that the following holds:

1. The composed morphisms E → F → M ⊗ E and F → M ⊗ E → M ⊗ F are both induced by the morphism μ : O_{S} → M.

2. For every point x ∈ S with μ(x) = 0, the complex

E[x] → F[x] → (M ⊗ E)[x] → (M ⊗ F)[x]

is exact and the rank of the morphism E[x] → F[x] equals r.

The letters “bf” stand for “back and forth”. As a matter of notation, we will sometimes write g^{♯} for the morphism E → F and g^{♭} for the morphism F → M ⊗ E occuring in the bf-morphism g. Note that in case μ is nowhere vanishing, the number rk g := r cannot be deduced from the other ingredients of g. Sometimes we will use the following more suggestive notation for the bf-morphism g:

g = ( [figure: E; r; (M,μ); F; ⊗] ) .

In situations where it is clear, what (M, μ) and r are, we will sometimes omit these data from our notation:

g = ( [figure: E; F; ⊗] ) .

Definition 5.2. Let S be a scheme, E and F two locally free O_{S}-modules of rank n. A generalized isomorphism from E to F is a tupel

Φ = (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) ,

where E = E_{0}, E_{1}, …, E_{n}, F_{n}, …, F_{1}, F_{0} = F, are localy free O_{S}-modules of rank n and the tupels

(M_{i}, μ_{i}, E_{i+1} → E_{i}, M_{i} ⊗ E_{i+1} ← E_{i}, i)
and  (L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i}, i)

are bf-morphisms of rank i for 0 ≤ i ≤ n − 1, such that for each x ∈ S the following holds:

1. If μ_{i}(x) = 0 and (f, g) is one of the following two pairs of morphisms:

E[x] ⟶^{f} ((⊗_{j=0}^{i−1}M_{j}) ⊗ E_{i})[x] ⟶^{g} ((⊗_{j=0}^{i}M_{j}) ⊗ E_{i+1})[x] ,

E_{i}[x] ⟵^{g} E_{i+1}[x] ⟵^{f} E_{n}[x] ,

then im(g ∘ f) = im(g). Likewise, if λ_{i}(x) = 0 and (f, g) is one of the following two pairs of morphisms:

F_{n}[x] ⟶^{f} F_{i+1}[x] ⟶^{g} F_{i}[x] ,

((⊗_{j=0}^{i}L_{j}) ⊗ F_{i+1})[x] ⟵^{g} ((⊗_{j=0}^{i−1}L_{j}) ⊗ F_{i})[x] ⟵^{f} F[x] ,

then im(g ∘ f) = im(g).

2. In the diagram:

\xymatrix{ &  & 0 \ar[d] \\  &  & ker(E_{n}[x] → E_{0}[x]) \ar[d] \ar@{-->}[dr] \\ 0 \ar[r] & ker(F_{n}[x] → F_{0}[x]) \ar[r] \ar@{-->}[dr] & E_{n}[x] ≅ F_{n}[x] \ar[d] \ar[r] & im(F_{n}[x] → F_{0}[x]) \ar[r] & 0 \\  &  & im(E_{n}[x] → E_{0}[x]) \ar[d] \\  &  & 0}

the oblique arrows are injections.

Definition 5.3. A quasi-equivalence between two generalized isomorphisms

Φ = (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) ,

Φ′ = (L′_{i}, λ′_{i}, M′_{i}, μ′_{i}, E′_{i} → M′_{i} ⊗ E′_{i+1}, E′_{i} ← E′_{i+1},

F′_{i+1} → F′_{i}, L′_{i} ⊗ F′_{i+1} ← F′_{i} (0 ≤ i ≤ n − 1), E′_{n} →^{∼} F′_{n})

from E to F consists in isomorphisms L_{i} →^{∼} L′_{i} and M_{i} →^{∼} M′_{i} for 0 ≤ i ≤ n−1, and isomorphisms E_{i} →^{∼} E′_{i} and F_{i} →^{∼} F′_{i} for 0 ≤ i ≤ n, such that all the obvious diagrams are commutative. A quasi-equivalence between Φ and Φ′ is called an equivalence, if the isomorphisms E_{0} →^{∼} E′_{0} and F_{0} →^{∼} F′_{0} are in fact the identity on E and F respectively.

After these general definitions, we now return to our scheme KGl_{n}. The notations are as in the previous section. From the matrix-decomposition on page 560 (for k = n) we see that the matrix [x_{ij}/x_{00}]_{1≤i,j≤n} has entries in the subspace Γ(KGl_{n}, O(∑_{i=0}^{n−1} Z_{i})) of the function field ℚ(KGl_{n}) of KGl_{n}. Therefore it defines a morphism

x : E_{0} ⟶ O (∑_{i=0}^{n−1} Z_{i}) · F_{0} ,

where E_{0} = F_{0} = ⊕^{n}O_{KGl_{n}}. Let E_{n} ⊂ E_{0} be the preimage under x of F_{0} ⊂ O(∑_{i=0}^{n−1} Z_{i})·F_{0} and let F_{n} ⊂ F_{0} be the image under x of E_{n}. Thus x induces a morphism

E_{n} ⟶ F_{n} ,

which we again denote by x. For 1 ≤ i ≤ n − 1 we define O_{KGl_{n}}-submodules E_{i} and F_{i} of ⊕^{n}K_{KGl_{n}} as follows:

E_{i} := E_{n} + O (− ∑_{j=0}^{i−1} Z_{j}) · E_{0}

F_{i} := F_{n} + O (− ∑_{j=0}^{i−1} Y_{j}) · F_{0}

(the plus-sign means generation in ⊕^{n}K_{KGl_{n}}). Observe that for 0 ≤ i ≤ n − 1 we have the following natural injections:

E_{i} ↪ O(Z_{i}) · E_{i+1} , E_{i} ↩ E_{i+1}
F_{i+1} ↪ F_{i} , O(Y_{i}) · F_{i+1} ↩ F_{i}

Proposition 5.4. The tupel

Φ_{univ} := (O(Y_{i}), 1_{O(Y_{i})}, O(Z_{i}), 1_{O(Z_{i})}, E_{i} ↪ O(Z_{i}) · E_{i+1}, E_{i} ↩ E_{i+1},

F_{i+1} ↪ F_{i}, O(Y_{i}) · F_{i+1} ↩ F_{i} (0 ≤ i ≤ n − 1), x : E_{n} → F_{n}) is a generalized isomorphism from ⊕^{n}O_{KGl_{n}} to itself.

Proof. It suffices to show that for each (α, β) ∈ S_{n} × S_{n} the restriction of Φ_{univ} to the open set X(α, β) is a generalized isomorphism from ⊕^{n}O_{X(α,β)} to itself. Let z(α, β) (y(α, β)) be the upper (lower) triangular n × n matrix with 1 on the diagonal and entries z_{ij}(α, β) (y_{ji}(α, β)) over (under) the diagonal (1 ≤ i < j ≤ n). For 0 ≤ i ≤ n we define

E_{i}(α, β) := z(α, β) · n_{β}^{−1} · E_{i}|_{X(α,β)} ,
F_{i}(α, β) := y(α, β)^{−1} · n_{α}^{−1} · F_{i}|_{X(α,β)} .

Here we interprete the matrices z(α, β) · n_{β}^{−1} and y(α, β)^{−1} · n_{α}^{−1} as automorphisms of ⊕^{n}K_{X(α,β)}. Accordingly we view the sheaves E_{i}(α, β) and F_{i}(α, β) as subsheaves of ⊕^{n}K_{X(α,β)}. We have to show that the tupel

Φ(α, β) := (O(Y_{i}(α, β)) , 1_{O(Y_{i}(α,β))} , O(Z_{i}(α, β)) , 1_{O(Z_{i}(α,β))} ,

E_{i}(α, β) ↪ O(Z_{i}(α, β)) · E_{i+1}(α, β) , E_{i}(α, β) ↩ E_{i+1}(α, β), F_{i+1}(α, β) ↪ F_{i}(α, β) , O(Y_{i}(α, β)) · F_{i+1}(α, β) ↩ F_{i}(α, β) (0 ≤ i ≤ n − 1) ,

y(α, β)^{−1}n_{α}^{−1}xn_{β}z(α, β)^{−1} : E_{n}(α, β) →^{∼} F_{n}(α, β))

is a generalized isomorphism from ⊕^{n}O_{X(α,β)} to itself. We have for 0 ≤ i ≤ n the following equality of subsheaves of ⊕^{n}K_{X(α,β)}:

E_{i}(α, β) = ⨁_{j=1}^{n−i} O (− ∑_{ν=0}^{i−1} Z_{ν}(α, β)) ⊕ ⨁_{j=n−i+1}^{n} O (− ∑_{ν=0}^{n−j} Z_{ν}(α, β)) ,

F_{i}(α, β) = ⨁_{j=1}^{i} O (− ∑_{ν=0}^{j−1} Y_{ν}(α, β)) ⊕ ⨁_{j=i+1}^{n} O (− ∑_{ν=0}^{i−1} Y_{ν}(α, β)) .

This is easily checked by restricting both sides of the equations to the open subsets X(α, β, l), (0 ≤ l ≤ n) of X(α, β) and using 4.1. Observe that the morphisms

E_{i}(α, β) ↪ O(Z_{i}(α, β)) · E_{i+1}(α, β) , E_{i}(α, β) ↩ E_{i+1}(α, β)

are described by the matrices

[ \begin{matrix}𝕀_{n−i} & 0 \\ 0 & μ_{i}𝕀_{i} \end{matrix} ] and [ \begin{matrix}μ_{i}𝕀_{n−i} & 0 \\ 0 & 𝕀_{i} \end{matrix} ] ,

and the morphisms

F_{i+1}(α, β) ↪ F_{i}(α, β) , O(Y_{i}(α, β)) · F_{i+1}(α, β) ↩ F_{i}(α, β)

by the matrices

[ \begin{matrix}𝕀_{i} & 0 \\ 0 & λ_{i}𝕀_{n−i} \end{matrix} ] and [ \begin{matrix}λ_{i}𝕀_{i} & 0 \\ 0 & 𝕀_{n−i} \end{matrix} ]

respectively, where we have abbreviated 1_{O(Y_{i}(α,β))} by λ_{i}, and 1_{O(Z_{i}(α,β))} by μ_{i}. Furthermore the matrix-decomposition on page 560 (for k = n) shows that y(α, β)^{−1}n_{α}^{−1}xn_{β}z(α, β)^{−1} is the diagonal matrix with entries (t_{1}(α, β)/t_{0}, …, t_{n}(α, β)/t_{0}). With this information at hand, it is easy to see that Φ(α, β) is indeed a generalized isomorphism from ⊕^{n}O_{X(α,β)} to itself.

Theorem 5.5. Let S be a scheme and Φ a generalized isomorphism from ⊕^{n}O_{S} to itself. Then there is a unique morphism f : S → KGl_{n} such that f^{∗}Φ_{univ} is equivalent to Φ. In other words, the scheme KGl_{n} together with Φ_{univ} represents the functor, which to each scheme S associates the set of equivalence classes of generalized isomorphisms from ⊕^{n}O_{S} to itself.

The proof of the theorem will be given in section 7.

Corollary 5.6. There is a (left) action of Gl_{n} × Gl_{n} on KGl_{n}, which extends the action ((φ, ψ), Φ) ↦ ψΦφ^{−1} of Gl_{n} × Gl_{n} on Gl_{n}. The divisors Z_{i} and Y_{i} are invariant under this action.

Proof. The the morphism (Gl_{n} × Gl_{n}) × KGl_{n} → KGl_{n} defining the action is given on S-valued points by

((φ, ψ), Φ) ↦ Φ′ ,

where Φ is a generalized isomorphism as in definition 5.2 from E_{0} = ⊕^{n}O_{S} to F_{0} = ⊕^{n}O_{S} and Φ′ is the generalized isomorphism where for 2 ≤ i ≤ n the bf-morphisms from E_{i} to E_{i−1}, the ones from F_{i} to F_{i−1} and the isomorphism E_{n} →^{∼} F_{n} are the same as in the tupel Φ, and where the bf-morphisms

(M_{0}, μ_{0}, E_{1} → E_{0}, M_{0} ⊗ E_{0} ← E_{1}, 0)
and  (L_{0}, λ_{0}, F_{1} → F_{0}, L_{0} ⊗ F_{0} ← F_{1}, 0)

in the tupel Φ are replaced by the bf-morphisms

(M_{0}, μ_{0}, E_{1} → E_{0} →^{φ} E_{0}, M_{0} ⊗ E_{1} ← E_{0} ←^{φ^{−1}} E_{0}, 0)

and  (L_{0}, λ_{0}, F_{1} → F_{0} →^{ψ} F_{0}, L_{0} ⊗ F_{1} ← F_{0} ←^{ψ^{−1}} F_{0}, 0)

respectively. The invariance of the divisors Z_{i} and Y_{i} is clear, since they are defined by the vanishing of μ_{i} and λ_{i} respectively.

6. Exterior powers

Lemma 6.1. Let S be a scheme and E, F two locally free O_{S}-modules of rank n. Let

g = (M, μ, E → F, M ⊗ E ← F, r)

be a bf-morphism of rank r from E to F. Then each point x ∈ S has an open neighbourhood U such that over U there exist local frames (e_{1}, …, e_{n}) and (f_{1}, …, f_{n}) for E and F respectively with the property that the matrices for the morphisms

E ⟶ F and M ⊗ E ⟵ F

with respect to these frames are

[ \begin{matrix}𝕀_{r} & 0 \\ 0 & μ/m𝕀_{n−r} \end{matrix} ] and [ \begin{matrix}μ𝕀_{r} & 0 \\ 0 & m𝕀_{n−r} \end{matrix} ]

respectively, where m is a nowhere vanishing section of M over U.

Proof. Restricting to a neighbourhood of x, we may assume that the sheaves M, E, F are free. Let (ẽ_{1}, …, ẽ_{n}) and (f̃_{1}, …, f̃_{n}) be global frames for E and F respectively. After permutation of their elements, and restricting to a possibly smaller neighbourhood of x, we may further assume that the morphisms

⟨ẽ_{1}, …, ẽ_{r}⟩ ⟶ F/⟨f̃_{r+1}, …, f̃_{n}⟩
⟨f̃_{r+1}, …, f̃_{n}⟩ ⟶ M ⊗ E/⟨ẽ_{1}, …, ẽ_{r}⟩

induced by g^{♯} and g^{♭} respectively, are isomorphisms. Let

Ẽ:= ⟨ẽ_{1}, …, ẽ_{r}⟩ , Ẽ′ := ker(E → M ⊗ F/⟨f̃_{r+1}, …, f̃_{n}⟩)
F̃ := ker(F → E/⟨ẽ_{1}, …, ẽ_{r}⟩)  ,  F̃′ := ⟨f̃_{r+1}, …, f̃_{n}⟩ .

Then we have direct-sum decompositions E = Ẽ ⊕ Ẽ′ , F = F̃ ⊕ F̃′ , which are respected by g^{♯} and g^{♭}. Let m be a nowhere vanishing section of M. The frames (e_{1}, …, e_{n}), (f_{1}, …, f_{n}) of E, F, defined by setting e_{i} := ẽ_{i}, f_{i} := g^{♯}(ẽ_{i}) for 1 ≤ i ≤ r and e_{i} := (1/m)g^{♭}(f̃_{i}), f_{i} := f̃_{i} for r + 1 ≤ i ≤ n, have the desired property.

Proposition 6.2. Let S be a scheme and E, F two locally free O_{S}-modules of rank n. Let

g = (M, μ, E → F, M ⊗ E ← F, i)

be a bf-morphism of rank i from E to F and let 1 ≤ r ≤ n.

1. There exists a unique morphism

∧^{r}g : ∧^{r}E ⟶ (M^{∨})^{⊗ max(0,r−i)} ⊗ ∧^{r}F

with the following property: If (e_{1}, …, e_{n}) and (f_{1}, …, f_{n}) are local frames for E and F respectively over an open set U ⊆ S, such that the matrices for the morphisms

E ⟶ F and M ⊗ E ⟵ F

with respect to these frames are

[ \begin{matrix}𝕀_{i} & 0 \\ 0 & μ/m𝕀_{n−i} \end{matrix} ] and [ \begin{matrix}μ𝕀_{i} & 0 \\ 0 & m𝕀_{n−i} \end{matrix} ]

respectively (m being a nowhere vanishing section of M over U), then (∧^{r}g)|_{U} takes the form

e_{I} ∧ e_{J} ↦ m^{p−r} ⊗ μ^{min(i,r)−p} ⊗ f_{I} ∧ f_{J} ,

where I ⊆ {1, …, i}, J ⊆ {i+1, …, n} with ♯I = p, ♯J = r−p and where e_{I} = e_{i_{1}} ∧ ⋯ ∧ e_{i_{p}}, if I = {i_{1}, …, i_{p}} and i_{1} < ⋯ < i_{p}. The e_{J}, f_{I}, f_{J} are defined analoguosly.

2. Similarly, there exists a unique morphism

∧^{−r}g : ∧^{r}F ⟶ M^{⊗ min(r,n−i)} ⊗ ∧^{r}E

with the following property: If (e_{1}, …, e_{n}) and (f_{1}, …, f_{n}) are local frames for E and F respectively over an open set U ⊆ S, such that the matrices for the morphisms

F ⟶ M ⊗ E and E ⟵ F

with respect to these frames are

[ \begin{matrix}m𝕀_{n−i} & 0 \\ 0 & μ𝕀_{i} \end{matrix} ] and [ \begin{matrix}μ/m𝕀_{n−i} & 0 \\ 0 & 𝕀_{i} \end{matrix} ]

respectively (m being a nowhere vanishing section of M over U), then (∧^{−r}g)|_{U} takes the form

f_{I} ∧ f_{J} ↦ m^{p} ⊗ μ^{min(r,n−i)−p} ⊗ e_{I} ∧ e_{J} ,

where I ⊆ {1, …, n − i}, J ⊆ {n − i + 1, …, n} with ♯I = p, ♯J = r − p.

Proof. 1. An easy calculation shows that the morphism given by the prescription

e_{I} ∧ e_{J} ↦ m^{p−r} ⊗ μ^{min(i,r)−p} ⊗ f_{I} ∧ f_{J} ,

does not depend on the chosen local frames (e_{1}, …, e_{n}), (f_{1}, …, f_{n}). Therefore using 6.1, we may define ∧^{r}g by this local prescription.

2. This can be proven along the same lines as 1. Alternatively, it follows by applying 1. to the bf-morphism (M, μ, E′ → F′, M ⊗ E′ ← F′, n − r) obtained from g by setting E′ := F and F′ := M ⊗ E.

In the situation of the above proposition 6.2, assume that E = E_{1} ⊕ E_{2} and F = F_{1} ⊕ F_{2}, where rk E_{i} = rk F_{i} =: n_{i} for i = 1, 2. Assume furthermore that the morphisms E → F and F → M ⊗ E both respect these direct-sum decompositions and that there are i_{1}, i_{2} ≥ 0 with i_{1} + i_{2} = i, such that the tupels

g_{1} := (M, μ, E_{1} → F_{1}, M ⊗ E_{1} ← F_{1}, i_{1})
and g_{2} := (M, μ, E_{2} → F_{2}, M ⊗ E_{2} ← F_{2}, i_{2}) ,

induced by g, are also bf-morphisms. We write g = g_{1} ⊕ g_{2}. The following lemma says that exterior powers of bf-morphisms are compatible with direct sums whenever this makes sense.

Lemma 6.3. Let 1 ≤ r ≤ n and r = r_{1} + r_{2} for some r_{1}, r_{2} ≥ 0.

1. If max(0, r −i) = max(0, r_{1} −i_{1})+max(0, r_{2} −i_{2}), then we have for every ϵ_{1} ∈ Γ(S, ∧^{r_{1}}E_{1}), ϵ_{2} ∈ Γ(S, ∧^{r_{2}}E_{2}) the following equality:

(∧^{r}g)(ϵ_{1} ∧ ϵ_{2}) = (∧^{r_{1}}g_{1})(ϵ_{1}) ∧ (∧^{r_{2}}g_{2})(ϵ_{2}) .

2. If min(i, r) = min(i_{1}, r_{1}) + min(i_{2}, r_{2}), then we have for every ω_{1} ∈ Γ(S, ∧^{r_{1}}F_{1}), ω_{2} ∈ Γ(S, ∧^{r_{2}}F_{2}) the following equality:

(∧^{−r}g)(ω_{1} ∧ ω_{2}) = (∧^{−r_{1}}g_{1})(ω_{1}) ∧ (∧^{−r_{2}}g_{2})(ω_{2}) .

Proof. This follows immediately from the local description of ∧^{r}g and ∧^{−r}g respectively.

Definition 6.4. Let S be a scheme, E and F two localy free O_{S}-modules of rank n and

Φ = (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) ,

a generalized isomorphism from E = E_{0} to F = F_{0}. For 1 ≤ r ≤ n we define the r-th exterior power

∧^{r}Φ : ⋀^{r} E ⟶ ⨂_{ν=1}^{r} (⨂_{i=0}^{ν−1} L_{i}^{∨} ⊗ ⨂_{i=0}^{n−ν} M_{i}) ⊗ ⋀^{r} F

of Φ as the composition

∧^{r}Φ := (∧^{−r}g_{0}) ∘ (∧^{−r}g_{1}) ∘ … ∘ (∧^{−r}g_{n−1}) ∘ (∧^{r}h_{n}) ∘ (∧^{r}h_{n−1}) ∘ … ∘ (∧^{r}h_{0}) ,

where g_{i} and h_{i} are the bf-morphisms

(M_{i}, μ_{i}, E_{i+1} → E_{i}, M_{i} ⊗ E_{i+1} ← E_{i}, i)
and  (L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i}, i)

respectively for 0 ≤ i ≤ n − 1, and where h_{n} is the isomorphism E_{n} →^{∼} F_{n}.

In the situation of the above definition consider especially the case, where E = ⊕^{n}O_{S} and F = ⊕^{n}L for some invertible O_{S}-module L. Then we have natural direct sum decompositions

⋀^{r} E = ⨁_{B} O_{S} and ⋀^{r} F = ⨁_{A} L^{r} ,

where A and B run through all subsets of cardinality r of {1, …, n}. For two such subsets A and B, we denote by π_{A} (respectively by ι_{B}) the projection ∧^{r}F → L^{r} onto the A-th component (respectively the inclusion O_{S} ↪ ∧^{r}E of the B-th component). Now we define

det_{A,B}Φ := π_{A} ∘ (∧^{r}Φ) ∘ ι_{B} : O_{S} ⟶ ⨂_{ν=1}^{r} (⨂_{i=0}^{ν−1} L_{i}^{∨} ⊗ ⨂_{i=0}^{n−ν} M_{i}) ⊗ L^{r} .

Lemma 6.5. Let (α, β) ∈ S_{n} × S_{n} and let X(α, β) be the open set of KGl_{n} defined in section 4. Let Φ_{univ} be the generalized isomorphism defined in 5.4. Then the sections det_{α[1,r],β[1,r]}Φ_{univ} are nowhere vanishing on X(α, β) for 1 ≤ r ≤ n.

Proof. From the proof of 5.4 it follows readily that the restriction of det_{α[1,r],β[1,r]}Φ_{univ} to X(α, β) is ∏_{ν=1}^{r}(t_{ν}(α, β)/t_{0}) as an element of

Γ (X(α, β), O (∑_{ν=1}^{r} (∑_{i=0}^{n−ν} Z_{i} − ∑_{i=0}^{ν−1} Y_{i}))) ⊂ Γ(X(α, β), K_{KGl_{n}}) .

On the other hand, 4.1 tells us that ∏_{ν=1}^{r}(t_{ν}(α, β)/t_{0}) is a generator of O(∑_{ν=1}^{r}(∑_{i=0}^{n−ν} Z_{i} − ∑_{i=0}^{ν−1} Y_{i})) over X(α, β).

7. Proof of theorem 5.5

Let S be a scheme, L an invertible O_{S}-module. For 0 ≤ i ≤ n − 1 let (L_{i}, λ_{i}), (M_{i}, μ_{i}) be invertible O_{S}-modules together with global sections, such that the zero sets of λ_{i} and μ_{j} do not intersect for i + j < n. Given these data, we associate to every tupel (φ_{1}, …, φ_{n}) of isomorphisms

φ_{r} : ⨂_{i=0}^{n−r} M_{i}^{∨} ⟶^{∼} ⨂_{i=0}^{r−1} L_{i}^{∨} ⊗ L (1 ≤ r ≤ n)

the following generalized isomorphism from ⊕^{n}O_{S} to ⊕^{n}L :

Φ(φ_{1}, …, φ_{n}) := (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) ,

where the locally free modules E_{i} and F_{i} are defined as

E_{i} := ⨁_{j=1}^{n−i} (⨂_{ν=0}^{i−1} M_{ν}^{∨}) ⊕ ⨁_{j=n−i+1}^{n} (⨂_{ν=0}^{n−j} M_{ν}^{∨}) ,

F_{i} := (⨁_{j=1}^{i} (⨂_{ν=0}^{j−1} L_{ν}^{∨}) ⊕ ⨁_{j=i+1}^{n} (⨂_{ν=0}^{i−1} L_{ν}^{∨})) ⊗ L ,

the morphisms

E_{i} ⟶ M_{i} ⊗ E_{i+1} and E_{i} ⟵ E_{i+1}

are described by the matrices

[ \begin{matrix}𝕀_{n−i} & 0 \\ 0 & μ_{i}𝕀_{i} \end{matrix} ] and [ \begin{matrix}μ_{i}𝕀_{n−i} & 0 \\ 0 & 𝕀_{i} \end{matrix} ] ,

the morphisms

F_{i+1} ⟶ F_{i} and L_{i} ⊗ F_{i+1} ⟵ F_{i}

by the matrices

[ \begin{matrix}𝕀_{i} & 0 \\ 0 & λ_{i}𝕀_{n−i} \end{matrix} ] and [ \begin{matrix}λ_{i}𝕀_{i} & 0 \\ 0 & 𝕀_{n−i} \end{matrix} ]

respectively, and the isomorphism E_{n} →^{∼} F_{n} is given by the diagonal matrix with entries (φ_{1}, …, φ_{n}).

Definition 7.1. Let S be a scheme, L an invertible O_{S}-module and

Φ := (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n})

an arbitrary generalized isomorphism from E_{0} = ⊕^{n}O_{S} to F_{0} = ⊕^{n}L. A diagonalization of Φ with respect to a pair (α, β) ∈ S_{n} × S_{n} of permutations is a tupel (u_{i},v_{i} (0 ≤ i ≤ n), (φ_{1}, …, φ_{n})) of isomorphisms

u_{i} :  E_{i} ⟶^{∼} ⨁_{j=1}^{n−i} (⨂_{ν=0}^{i−1} M_{ν}^{∨}) ⊕ ⨁_{j=n−i+1}^{n} (⨂_{ν=0}^{n−j} M_{ν}^{∨})  (0 ≤ i ≤ n)

v_{i} :  F_{i} ⟶^{∼} (⨁_{j=1}^{i} (⨂_{ν=0}^{j−1} L_{ν}^{∨}) ⊕ ⨁_{j=i+1}^{n} (⨂_{ν=0}^{i−1} L_{ν}^{∨})) ⊗ L  (0 ≤ i ≤ n)

φ_{r} :  ⨂_{i=0}^{n−r} M_{i}^{∨} ⟶^{∼} ⨂_{i=0}^{r−1} L_{i}^{∨} ⊗ L  (1 ≤ r ≤ n)

such that (u_{i}, v_{i} (0 ≤ i ≤ n)) establishes a quasi-equivalence between Φ and Φ(φ_{1}, …, φ_{n}) and such that

u_{n} · n_{β} : E_{0} = ⊕^{n}O_{S} ⟶^{∼} ⊕^{n}O_{S} and v_{n} · n_{α} : F_{0} = ⊕^{n}L ⟶^{∼} ⊕^{n}L

are described by upper and lower triangular matrices respectively, with unit diagonal entries.

Definition 7.2. As in the above definition let Φ be a generalized isomorphism from ⊕^{n}O_{S} to ⊕^{n}L. A pair (α, β) ∈ S_{n} × S_{n} of permutations is called admissible, if for all 1 ≤ r ≤ n the global sections det_{α[1,r],β[1,r]}Φ of

⨂_{ν=1}^{r} (⨂_{i=0}^{n−ν} M_{i} ⊗ ⨂_{i=0}^{ν−1} L_{i}^{∨}) ⊗ L^{r}

are nowhere vanishing on S.

Proposition 7.3. Let S be a scheme, L an invertible O_{S}-module and Φ a generalized isomorphism from ⊕^{n}O_{S} to ⊕^{n}L. Then:

1. For (α, β) ∈ S_{n} × S_{n} the following are equivalent:

(a) there exists a diagonalization of Φ with respect to (α, β)

(b) (α, β) is admissible for Φ

2. Every point of S has an open neighbourhood U, such that there is a diagonalization of Φ|_{U} with respect to some pair (α, β) ∈ S_{n} × S_{n}.

3. For a given pair (α, β) ∈ S_{n} × S_{n} there is at most one diagonalization of Φ with respect to (α, β).

Proof. Let

Φ := (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) ,

be a generalized isomorphism from E_{0} = ⊕^{n}O_{S} to F_{0} = ⊕^{n}L. For 0 ≤ i ≤ n−1 denote bf-morphisms as follows:

g_{i} := (M_{i}, μ_{i}, E_{i+1} → E_{i}, M_{i} ⊗ E_{i+1} ← E_{i}, i) ,
h_{i} := (L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i}, i) ,

and let h_{n} the isomorphism E_{n} →^{∼} F_{n}.

1. We may assume that α = β = id. We show by induction on n that admissibility of (id, id) for Φ implies the diagonalizability of Φ with respect to (id, id). The case n = 1 is trivial, so assume n ≥ 2. By assumption, the morphism

det_{{1},{1}}Φ : O_{S} ⟶ L_{0}^{∨} ⊗ ⨂_{i=0}^{n−1} M_{i} ⊗ L

is an isomorphism. Let

Ẽ′_{i} := ker(π_{1} ∘ (∧^{1}h_{0}) ∘ … ∘ (∧^{−1}g_{i}) : E_{i} ⟶ L_{0}^{∨} ⊗ ⨂_{j=i}^{n−1} M_{j} ⊗ L)
(i ∈ [0, n])
F̃′_{i} := ker(π_{1} ∘ (∧^{1}h_{0}) ∘ … ∘ (∧^{1}h_{i−1}) : F_{i} ⟶ L_{0}^{∨} ⊗ L)  (i ∈ [1, n])
F̃′_{0} := ker(π_{1} : ⊕^{n}L ⟶ L) = ⊕^{n−1}L

and

Ẽ_{i} := im((∧^{−1}g_{i−1}) ∘ … ∘ (∧^{−1}g_{0}) ∘ ι_{1} : ⨂_{j=0}^{i−1} M_{j}^{∨} ⟶ E_{i})  (i ∈ [0, n])

F̃_{i} := im((∧^{1}h_{i}) ∘ … ∘ (∧^{−1}g_{0}) ∘ ι_{1} : ⨂_{j=0}^{n−1} M_{j}^{∨} ⟶ F_{i})  (i ∈ [1, n])

F̃_{0} := im((∧^{1}h_{0}) ∘ … ∘ (∧^{−1}g_{0}) ∘ ι_{1} : L_{0}^{∨} ⊗ ⨂_{j=0}^{n−1} M_{j}^{∨} ⟶ F_{0})

Then we have natural direct sum decompositions

E_{i} = Ẽ_{i} ⊕ Ẽ′_{i}  (0 ≤ i ≤ n) ,
F_{i} = F̃_{i} ⊕ F̃′_{i}  (0 ≤ i ≤ n) .

Since the bf-morphisms g_{i} and h_{i} respect these decompositions, we can write g_{i} = g̃_{i} ⊕ g̃′_{i} and h_{i} = h̃_{i} ⊕ h̃′_{i} where g̃_{i} (respectively g̃′_{i}, h̃_{i}, h̃′_{i}) is a bf-morphism from Ẽ_{i+1} to Ẽ_{i} (respectively from Ẽ′_{i+1} to Ẽ′_{i}, from F̃_{i+1} to F̃_{i}, from F̃′_{i+1} to F̃′_{i}) for 0 ≤ i ≤ n − 1. By the same reason, we can write h_{n} = h̃_{n} ⊕ h̃′_{n}, where h̃_{n} : Ẽ_{n} →^{∼} F̃ and h̃′_{n} : Ẽ′_{n} →^{∼} F̃′. Observe that rk g̃_{i} = 0 and rk g̃′_{i} = i for 0 ≤ i ≤ n − 1 and that

rk h̃_{i} = \{ \begin{matrix}0 , & if i = 0 \\ 1 , & if i > 0 \end{matrix} , rk h̃′_{i} = \{ \begin{matrix}0 , & if i = 0 \\ i − 1 , & if i > 0 \end{matrix} .

Now we define

L′ := L ⊗ L_{0}^{∨}
L′_{i} := L_{i+1} , λ′_{i} := λ_{i+1}  (0 ≤ i ≤ n − 2)
M′_{i} := M_{i} , μ′_{i} := μ_{i}  (0 ≤ i ≤ n − 2)
E′_{i} := Ẽ′_{i}  (0 ≤ i ≤ n − 1)
F′_{i} := F̃′_{i+1}  (0 ≤ i ≤ n − 1)

where we identify E′_{0} with ⊕^{n−1}O_{S} via the isomorphism

E′_{0} = Ẽ′_{0} ⟶^{inclusion} E_{0} = ⊕^{n}O_{S} ⟶^{π_{[2,n]}} ⊕^{n−1}O_{S} ,

and F′_{0} with ⊕^{n−1}L′ via the isomorphism

F′_{0} = F̃′_{1} ⟶^{∧^{1}h̃′_{0}} L_{0}^{∨} ⊗ ⊕^{n−1}L = ⊕^{n−1}L′ .

Let

Φ′ := (L′_{i}, λ′_{i}, M′_{i}, μ′_{i}, E′_{i} → M′_{i} ⊗ E′_{i+1}, E′_{i} ← E′_{i+1},

F′_{i+1} → F′_{i}, L′_{i} ⊗ F′_{i+1} ← F′_{i} (0 ≤ i ≤ n − 2), E′_{n−1} →^{∼} F′_{n−1}) ,

where E′_{n−1} →^{∼} F′_{n−1} is the composition

E′_{n−1} = Ẽ′_{n−1} ⟶^{∧^{−1}g̃′_{n−1}} Ẽ′_{n} ⟶^{h̃′_{n}} F̃′_{n} = F′_{n−1} ,

and where the other morphisms are the ones from the g̃′_{i} and the h̃′_{i}. It is easy to see that Φ′ is a generalized isomorphism from ⊕^{n−1}O_{S} to ⊕^{n−1}L′. Furthermore, it follows from 6.3 that

(∧^{r}Φ)(e_{1} ∧ ⋯ ∧ e_{r}) = (∧^{1}Φ)(e_{1}) ∧ (∧^{r−1}Φ′)(e′_{1} ∧ ⋯ ∧ e′_{r−1}) (2 ≤ r ≤ n) ,

where (e_{1}, …, e_{n}) ⊂ Γ(S, E_{0}) and (e′_{1}, …, e′_{n−1}) ⊂ Γ(S, E′_{0}) are the canonical global frames of ⊕^{n}O_{S} and ⊕^{n−1}O_{S} respectively. Therefore we have

det_{[1.r][1,r]}Φ = det_{{1}{1}}Φ ⊗ det_{[1,r−1][1,r−1]}Φ′ (2 ≤ r ≤ n) .

Since, by assumption, the sections det_{[1.r][1,r]}Φ are nowhere vanishing, the above equation implies that the same is true also for the sections det_{[1,r−1][1,r−1]}Φ′ (2 ≤ r ≤ n). In other words, (id, id) is admissible for Φ′. By induction-hypothesis, we conclude that there exists a diagonalization (u′_{i}, v′_{i}, (0 ≤ i ≤ n − 1), (φ′_{1}, …, φ′_{n−1})) of Φ′ with respect to (id, id). Let

ũ′_{i} := u′_{i}  (0 ≤ i ≤ n − 1) ,
ṽ′_{i} := v′_{i−1}  (1 ≤ i ≤ n) ,

and

ũ′_{n} : Ẽ′_{n} ⟶_{∼}^{g̃_{n−1}^{♯}} Ẽ′_{n−1} = E′_{n−1} ⟶_{∼}^{u′_{n−1}} ⨁_{j=2}^{n} (⨂_{ν=0}^{n−j} M^{∨}) ,
ṽ′_{0} : F̃′_{0} = ⨁^{n−1} L = L_{0} ⊗ F′_{0} ⟶_{∼}^{v′_{0}} ⨁^{n−1} L .

Observe that there are natural isomorphisms

ũ_{i} : Ẽ_{i} ⟶^{∼} ⨂_{j=0}^{i−1} M_{j}^{∨}  (0 ≤ i ≤ n)
ṽ_{i} : F̃_{i} ⟶^{∼} ⨂_{j=0}^{n−1} M_{j}^{∨} ⟶^{det_{{1}{1}}Φ} L ⊗ L_{0}^{∨}  (1 ≤ i ≤ n)
ṽ_{0} : F̃_{0} ⟶^{∼} L_{0} ⊗ ⨂_{j=0}^{n−1} M_{j}^{∨} ⟶^{det_{{1}{1}}Φ} L

We set u_{i} := ũ_{i} ⊕ ũ′_{i} , v_{i} := ṽ_{i} ⊕ ṽ′_{i} for 0 ≤ i ≤ n, and φ_{r} := φ′_{r−1} for 2 ≤ r ≤ n. Finally, we let φ_{1} : ⨂_{i=0}^{n−1} M_{i}^{∨} →^{∼} L ⊗ L_{0}^{∨} be the isomorphism induced by det_{{1}{1}}Φ. It is now easy to see that the tupel (u_{i}, v_{i} (0 ≤ i ≤ n), (φ_{1}, …, φ_{n})) is a diagonalization of Φ with respect to (id, id). Conversely, assume that there exists a diagonalization (u_{i}, v_{i} (0 ≤ i ≤ n), (φ_{1}, …, φ_{n})) of Φ with respect to (id, id). Observe that the diagram

\xymatrix{ & ∧^{r}E_{0} \ar[dd]^{∧^{r}u_{0}} \ar[r]^{∧^{r}Φ} & N_{r} ⊗ (∧^{r}F_{0}) \ar[dd]^{∧^{r}v_{0}} \ar@{-->}[dr]^{π_{[1,r]}} \\ O_{S} \ar[ur]^{ι_{[1,r]}} \ar[dr]_{ι_{[1,r]}} &  &  & N_{r} ⊗ L^{⊗r} \\  & ∧^{r}(⊕^{n}O_{S}) \ar[r]^{∧^{r}Φ(φ_{1},...,φ_{n})} & N_{r} ⊗ (∧^{r}(⊕^{n}L)) \ar@{-->}[ur]_{π_{[1,r]}}}

where N_{r} := ⨂_{ν=1}^{r}(⨂_{i=0}^{n−ν} M_{i} ⊗ ⨂_{i=0}^{ν−1} L_{i}^{∨}), is commutative for 1 ≤ r ≤ n. Therefore we may assume that Φ = Φ(φ_{1}, …, φ_{n}). But then det_{[1,r][1,r]}Φ is the section induced by the isomorphism

φ_{1} ⊗ … ⊗ φ_{r} : ⨂_{ν=1}^{r} ⨂_{i=0}^{n−ν} M_{i}^{∨} ⟶^{∼} ⨂_{ν=1}^{r} ⨂_{i=0}^{ν−1} L_{i}^{∨}

for 1 ≤ r ≤ n. In particular the det_{[1,r][1,r]}Φ are nowhere vanishing on S, which is precisely what is required for the admissibility of (id, id) for Φ.

2. By 1, it suffices to show that in the case S = Spec k (k a field) there exists a pair (α, β) ∈ S_{n} × S_{n} which is admissible for Φ. We apply induction on n, the case n = 1 being trivial. It is an easy exercise in linear algebra, to show that the morphism

∧^{1}Φ = (h_{0}^{♭})^{−1} ∘ h_{1}^{♯} … ∘ h_{n−1}^{♯} ∘ h_{n} ∘ g_{n−1}^{♭} ∘ … ∘ g_{0}^{♭}

has at least rank one. Consequently there exist indices i_{1}, j_{1} ∈ {1, …, n}, such that the composition det_{{i_{1}}{j_{1}}}Φ = π_{i_{1}} ∘(∧^{1}Φ)∘ι_{j_{1}} is an isomorphism. Let the sheaves Ẽ′_{i}, F̃′_{i} (0 ≤ i ≤ n) be defined as on page 575, with π_{1} replaced by π_{i_{1}}, and using these sheaves, let Φ′ be defined as on page 576. This is a generalized isomophism from k^{n−1} to ⊕^{n−1}(L_{0}^{∨} ⊗ L). By induction-hypothesis there exists a pair (α′, β′) ∈ S_{n−1} ×S_{n−1}, which is admissible for Φ′. Let α ∈ S_{n} be defined by

α(r) := \{ \begin{matrix}i_{1} , & if r = 1 \\ α′(r − 1) + 1 , & if 2 ≤ r ≤ n \end{matrix} ,

and let β ∈ S_{n} be defined analogously. As on page 576 we have

det_{α[1.r],β[1,r]}Φ = det_{{i_{1}}{j_{1}}}Φ ⊗ det_{α′[1,r−1],β′[1,r−1]}Φ′ (2 ≤ r ≤ n)

for 2 ≤ r ≤ n, i.e. the pair (α, β) is admissible for Φ.

3. This follows from the proof of 1, since it is clear that the construction of the diagonalization there is unique.

Proposition 7.4. Let S be a scheme and Φ a generalized isomorphism from ⊕^{n}O_{S} to itself.

1. If (α, β) ∈ S_{n} × S_{n} is admissible for Φ, then there exists a unique morphism S → X(α, β), such that the pull-back of Φ_{univ} to S by this morphism is equivalent to Φ.

2. We have the following description of X(α, β) as an open subset of KGl_{n}:

X(α, β) = {x ∈ KGl_{n} | (det_{α[1,r],β[1,r]}Φ_{univ})(x) ≠ 0 for 1 ≤ r ≤ n} .

3. If (α′, β′) ∈ S_{n}×S_{n} is a further admissible pair for Φ, then the above morphism S → X(α, β) factorizes over the inclusion X(α, β) ∩ X(α′, β′) ↪ X(α, β).

Proof. 1. Let

Φ = (L_{i}, λ_{i}, M_{i}, μ_{i}, E_{i} → M_{i} ⊗ E_{i+1}, E_{i} ← E_{i+1},

F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1), E_{n} →^{∼} F_{n}) .

By proposition 7.3 there exists a diagonalization (u_{i}, v_{i} (0 ≤ i ≤ n), (φ_{1}, …, φ_{n})) of Φ with respect to (α, β). Let a_{ji} ∈ Γ(S, O_{S}) (respectively b_{ij} ∈ Γ(S, O_{S})) (1 ≤ i < j ≤ n) be the nontrivial entries of the lower (respectively upper) triangular matrix (v_{0} · n_{α})^{−1} (respectively u_{0} · n_{β}). Let a : S → U^{−} and b : S → U^{+} be the morphisms defined by a^{∗}(y_{ji}) = a_{ji} and b^{∗}(z_{ij}) = b_{ij} respectively. Furthermore, let φ : S → T̃ be the morphism induced by the tupel

(L_{i}, λ_{i}, M_{i}, μ_{i} (0 ≤ i ≤ n − 1), φ_{r} (1 ≤ r ≤ n)) ,

(cf lemma 4.3). Thus we have a morphism

f : S ⟶^{(a,φ,b)} U^{−} × T̃ × U^{+} ⟶^{∼} X(α, β) ,

where the right isomorphism is the inverse of the one in lemma 4.4. It is clear that

f^{∗}O(Y_{i}(α, β)) ≅ L_{i} , f^{∗}1_{O(Y_{i}(α,β))} = λ_{i}
f^{∗}O(Z_{i}(α, β)) ≅ M_{i} , f^{∗}1_{O(Z_{i}(α,β))} = μ_{i}  (0 ≤ i ≤ n − 1) .

Denote by (u′_{i}, v′_{i} (0 ≤ i ≤ n), (φ′_{1}, …, φ′_{n})) the pull-back under f of the diagonalization of Φ_{univ}|_{X(α,β)}, which exists by 6.5 and 7.3. By the uniquness of diagonalizations (cf 7.3), we have u_{0} = u′_{0}, v_{0} = v′_{0} and (φ_{1}, …, φ_{n}) = (φ′_{1}, …, φ′_{n}). Therefore the isomorphisms

(u′_{i})^{−1} ∘ u_{i} : E_{i} ⟶^{∼} f^{∗}E_{i}
(v′_{i})^{−1} ∘ v_{i} : F_{i} ⟶^{∼} f^{∗}F_{i}

induce an equivalence between Φ and f^{∗}Φ_{univ}. This proves the existence part of the proposition. For uniqueness, assume that f̃ is a further morphism from S to X(α, β), such that Φ is equivalent to f̃^{∗}Φ_{univ}. Let \overline{u}_{i} : E_{i} →^{∼} f̃^{∗}E_{i}, \overline{v}_{i} : F_{i} →^{∼} f̃^{∗}F_{i}, (0 ≤ i ≤ n) be an equivalence. Note that by definition \overline{u}_{0} = id_{⊕^{n}O_{S}} = \overline{u}_{0}. Let (ũ_{i}, ṽ_{i} (0 ≤ i ≤ n), (φ̃_{1}, …, φ̃_{n})) be the pull-back under f̃ of the diagonalization with respect to (α, β) of Φ_{univ}|_{X(α,β)}. Then (ũ_{i} ∘ \overline{u}_{i}, ṽ_{i} ∘ \overline{v}_{i} (0 ≤ i ≤ n), (φ̃_{1}, …, φ̃_{n})) is a diagonalization of Φ with respect to (α, β). By 7.3.3 we conclude that ũ_{0} = ũ_{0} ∘ \overline{u}_{0} = u_{0}, ṽ_{0} = ṽ_{0} ∘ \overline{v}_{0} = v_{0} and (φ̃_{1}, …, φ̃_{n}) = (φ_{1}, …, φ_{n}). But this implies that the composite morphism

S ⟶^{f̃} X(α, β) ⟶^{∼} U^{−} × T̃ × U^{+}

equals (a, φ, b) and thus that f̃ = f.

2. Denote for a moment by U the open subset of KGl_{n}, defined by the nonvanishing of det_{α[1,r],β[1,r]}Φ_{univ} for 1 ≤ r ≤ n. We have already seen in 6.5 that X(α, β) is contained in U. Let x ∈ U. Since X(α, β) is dense in U, there exists a generalization y ∈ X(α, β) of x. Then there exists a morphism f : S → U, where S is the Spec of a valuation ring, such that the special point of S is mapped to x and its generic point to y. By definition of U, the pair (α, β) is admissible for the generalized isomorphism f^{∗}Φ_{univ}. Therefore 1 tells us that there exists a morphism f′ : S → X(α, β), which coincides with f at the generic point of S. Since KGl_{n} is separabel, it follows that f = f′ and thus that x ∈ X(α, β).

3. This follows immediatelly from 2.

Proof. (Of theorem 5.5). Let S be a scheme and Φ a generalized isomorphism from ⊕^{n}O_{S} to itself. By proposition 7.3, there is a covering of S by open sets U_{i} (i ∈ I), and for every index i ∈ I a pair (α_{i}, β_{i}) ∈ S_{n} × S_{n}, which is admissible for Φ|_{U_{i}}. Proposition 7.4.1 now tells us that there exists for each i ∈ I a unique morhpism f_{i} : U_{i} → X(α_{i}, β_{i}) with the property that there is an equivalence, say u_{i}, from Φ|_{U_{i}} to f_{i}^{∗}Φ_{univ}. By proposition 7.4.2, the f_{i} glue together, to give a morphism f : S → KGl_{n}. It remains to show that also the u_{i} glue together, to give an overall equivalence from Φ to f^{∗}Φ_{univ}. For this, it suffices to show that for two generalized isomorphisms Φ and Φ′ from ⊕^{n}O_{S} to itself there exists at most one equivalence from Φ to Φ′. The question being local, we may assume by proposition 7.3.2 that Φ′ is diagonalizable with respect to some pair (α, β) ∈ S_{n} × S_{n}. Composing the diagonalization of Φ′ with any equivalence from Φ to Φ′ gives a diagonalization of Φ with respect to (α, β). Since different equivalences from Φ to Φ′ would yield different diagonalizations of Φ, proposition 7.3.3 tells us that there exists at most one equivalence.

8. Complete collineations

In this section we prove a modular property for the compactification \overline{PGl}_{n} of PGl_{n} and compare it with the results of other authors. The scheme \overline{PGl}_{n} together with closed subschemes \overline{Δ}_{r} (1 ≤ r ≤ n−1) is defined by successive blow ups as follows. Let \overline{Ω}^{(0)} := Proj (ℤ[x_{i,j} (1 ≤ i, j ≤ n)]) and let \overline{Δ}_{r}^{(0)} := V^{+}((det_{AB}(x_{ij}) | A, B ⊆ \{1, …, n\}, ♯A = ♯B = r + 1)) (1 ≤ r ≤ n − 1). Inductively, define \overline{Ω}^{(ν)} as the blowing up of \overline{Ω}^{(ν−1)} along \overline{Δ}_{ν}^{(ν−1)}. The closed subscheme \overline{Δ}_{r}^{(ν)} ⊂ \overline{Ω}^{(ν)} is by definition the strict (resp. total) transform of \overline{Δ}_{r}^{(ν−1)} for r ≠ ν (resp. r = ν). By definition, \overline{PGl}_{n} := \overline{Ω}^{(n−1)} and \overline{Δ}_{r} := \overline{Δ}_{r}^{(n−1)} for 1 ≤ r ≤ n − 1. The variety \overline{PGl}_{n}×Spec (ℂ) is the so-called “wonderful compactification” of the homogenuos space PGl_{n,ℂ} = (PGl_{n,ℂ}×PGl_{n,ℂ})/PGl_{n,ℂ} (cf. [CP]). Vainsencher [V], Laksov [Lak2] and Thorup-Kleiman [TK] have given a modular description for (some of) the S-valued points of \overline{PGl}_{n}. We will give a brief account of their results. Let R ⊆ [1, n − 1] and let S be a scheme. Following the terminology of Vainsencher, an S-valued complete collineation of type R from a rank-n vector bundle E to a rank-n vector bundle F is a collection of morphisms

v_{i} : E_{i} → N_{i} ⊗ F_{i} (0 ≤ i ≤ k),

where R = {r_{1}, …, r_{k}}, 0 =: r_{0} < r_{1} < ⋯ < r_{k} < r_{k+1} := n, the N_{i} are line bundles, the E_{i}, F_{i} are vector bundles on S and v_{i} has overall rank r_{i+1} − r_{i}; furthermore it is required that E_{0} = E, F_{0} = F, and E_{i} = ker(v_{i−1}), F_{i} = N_{i−1}^{∨} ⊗ coker(v_{i−1}) for 1 ≤ i ≤ k. Vainsencher proves that the locally closed subscheme (∩_{r∈R}\overline{Δ}_{r}) \ ∪_{r∉R}\overline{Δ}_{r} of \overline{PGl}_{n} represents the functor which to each scheme S associates the set of isomorphism classes of S-valued complete collineations of type R from ⊕^{n}O_{S} to itself. Laksov went further. He succeeded to give a modular description for those S-valued points of \overline{Δ}(R) := ∩_{r∈R}\overline{Δ}_{r} for which the pull-back of the divisor ∑_{r∉R} \overline{Δ}_{r}|_{\overline{Δ}(R)} on \overline{Δ}(R) is a well-defined divisor on S. We refer the reader to [Lak2] for more details. Finally, Thorup and Kleiman gave the following description for all S-valued points of \overline{PGl}_{n}. A morphism u from (⊕^{n}O_{S}) ⊗ (⊕^{n}O_{S})^{∨} to a line bundle L is called a divisorial form, if for each i ∈ [1, n] the image M_{i}(u) of the induced map ∧^{i}(⊕^{n}O_{S}) ⊗ ∧^{i}(⊕^{n}O_{S})^{∨} → L^{⊗i} is an invertible sheaf. In this case denote by u^{i} the induced surjection ∧^{i}(⊕^{n}O_{S}) ⊗ ∧^{i}(⊕^{n}O_{S})^{∨} → M_{i}(u). Following Thorup and Kleiman, we define a projectively complete bilinear form as a tupel u = (u_{1}, …, u_{n}), where u_{i} : ∧^{i}(⊕^{n}O_{S}) ⊗ ∧^{i}(⊕^{n}O_{S})^{∨} → M_{i} is a surjection onto an invertible sheaf M_{i} for 1 ≤ i ≤ n, such that u is “locally the pull-back of a divisorial form”. The last requirement means the following: For each point x ∈ S, there exists an open neighborhood U of x, a morphism from U to some scheme S′ and a divisorial form u : (⊕^{n}O_{S′}) ⊗ (⊕^{n}O_{S′})^{∨} → L′ on S′ such that the restriction of u to U is isomorphic to the pull-back of (u^{1}, …, u^{n}). Thorup and Kleiman show that \overline{PGl}_{n} represents the functor that to each scheme S associates the set of isomorphism classes of projectively complete bilinear forms on S. None of these descriptions is completely satisfactory: Those of Vainsencher and Laksov deal only with special S-valued points and the description of Thorup-Kleiman is not explicit and is not truely modular, since the condition “to be locally pull-back of a divisorial form” makes reference to the existence of morphisms between schemes. The terminology in the following definition will be justified by the corollary 8.2 below.

Definition 8.1. Let S be a scheme and E, F two locally free O_{S}-modules of rank n. A complete collineation from E to F is a tupel

Ψ = (L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1)) ,

where E = F_{n}, F_{n−1}, …, F_{1}, F_{0} = F are locally free O_{S}-modules of rank n, the tupels

(L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i}, i)

are bf-morphisms of rank i for 0 ≤ i ≤ n − 1 and λ_{0} = 0, such that for each point x ∈ S and index i ∈ {0, …, n − 1} with the property that λ_{i}(x) = 0, the following holds: If (f, g) is one of the following two pairs of morphisms:

F_{n}[x] ⟶^{f} F_{i+1}[x] ⟶^{g} F_{i}[x] ,

((⊗_{j=0}^{i}L_{j}) ⊗ F_{i+1})[x] ⟵^{g} ((⊗_{j=0}^{i−1}L_{j}) ⊗ F_{i})[x] ⟵^{f} F_{0}[x] ,

then im(g ∘ f) = im(g). Two complete collineations Ψ and Ψ′ from E to F are called equivalent, if there are isomorphisms L_{i} →^{∼} L′_{i}, F_{i} →^{∼} F′_{i}, such that all the obvious diagrams commute and such that F_{n} →^{∼} F′_{n} and F_{0} →^{∼} F′_{0} is the identity on E and F.

Corollary 8.2. On \overline{PGl}_{n} there exists a universal complete collineation Ψ_{univ} from ⊕^{n}O to itself, such that the pair (\overline{PGl}_{n}, Ψ_{univ}) represents the functor, which to every scheme S associates the set of equivalence classes of complete collineations from ⊕^{n}O_{S} to itself.

Proof. Observe that \overline{PGl}_{n} is naturally isomorphic to the closed subscheme Y_{0} of KGl_{n}. The restriction of Φ_{univ} to \overline{PGl}_{n} induces in an obvious way a complete collineation Ψ_{univ} of ⊕^{n}O to itself on \overline{PGl}_{n}. The corollary now follows from theorem 5.5.

We conclude this section by indicating how one can recover Vainsencher’s and Thorup-Kleiman’s description from corollary 8.2. Let S be a scheme and let

Ψ = (L_{i}, λ_{i}, F_{i+1} → F_{i}, L_{i} ⊗ F_{i+1} ← F_{i} (0 ≤ i ≤ n − 1))

be a complete collineation from ⊕^{n}O_{S} to itself in the sense of definition 8.1. First assume that there exists a subset R of [1, n − 1], such that the map S → \overline{PGl}_{n} corresponding to Ψ factors through (∩_{r∈R}\overline{Δ}_{r}) \ ∪_{r∉R}\overline{Δ}_{r}. This means that λ_{r} is zero for r ∈ R and is nowhere vanishing for r ∈ [1, n − 1] \ R. As above, let R = {r_{1}, …, r_{k}}, 0 =: r_{0} < r_{1} < ⋯ < r_{k} < r_{k+1} := n. For 0 ≤ i ≤ k let

E_{i} := ker(F_{n} → F_{n−1} → ⋯ → F_{r_{i}})
F_{i} := N_{i}^{∨} ⊗ ker(F_{r_{i+1}} → F_{r_{i+1}−1} → ⋯ → F_{r_{i}}) ,

where N_{i} := ⊗_{j=1}^{i}L_{r_{i}}^{∨}. Observe that the data in Ψ provide natural maps

v_{i} : E_{i} → N_{i} ⊗ F_{i}

of overall rank r_{i+1} − r_{i} for 0 ≤ i ≤ k. Furthermore we have natural isomorphisms E_{0} = F_{n} = ⊕^{n}O, F_{0} ≅ F_{0} = ⊕^{n}O, and for 1 ≤ i ≤ k:

ker(v_{i−1}) = E_{i}
coker(v_{i−1}) ≅ coker(F_{n} → F_{r_{i}}) = coker(F_{r_{i+1}} → F_{r_{i}}) ≅
≅ ker(L_{r_{i}} ⊗ F_{r_{i+1}} → L_{r_{i}} ⊗ F_{r_{i}}) ≅ N_{i−1} ⊗ F_{i} .

Thus, (v_{i})_{0≤i≤k} is a complete homomorphism of type R in the sense of Vainsencher. Now let Ψ be arbitrary. As in section 6, Ψ induces nowhere vanishing morphisms

∧^{r}Ψ : ∧^{r}F_{n} → (⨂_{ν=1}^{r} ⨂_{i=0}^{ν−1} L_{i}^{∨}) ⊗ ∧^{r}F_{0}

and thus surjections

u_{r} : ∧^{r}F_{n} ⊗ ∧^{r}F_{0}^{∨} → ⨂_{ν=1}^{r} ⨂_{i=0}^{ν−1} L_{i}^{∨}

for 1 ≤ r ≤ n. The tupel (u_{r})_{1≤r≤n} is a projectively complete bilinear form in the sense of Thorup-Kleiman. This follows from the fact that ∧^{1}Ψ_{univ} induces a divisorial form on \overline{PGl}_{n}.

9. Geometry of the strata

In this section we need relative versions of the varieties KGl_{n}, \overline{PGl}_{n} and \overline{O}_{I,J} := ∩_{i∈I}Z_{i} ∩ ∩_{j∈J}Y_{j}, where I and J are subsets of [0, n − 1]. They are defined in the following theorem.

Theorem 9.1. Let T be a scheme and let E and F be two locally free O_{T} - modules of rank n. For a T -scheme S we write E_{S} and F_{S} for the pull-back of E and F to S. Let I, J be two subsets of [0, n − 1] Consider the following contravariant functors from the category of T -schemes to the category of sets:

KGl(E, F) : S ↦ \{ \begin{matrix}equivalence classes of \\ generalized isomorphisms \\ from E_{S} to F_{S} \end{matrix} \}

\overline{PGl}(E, F) : S ↦ \{ \begin{matrix}equivalence classes of \\ complete collineations \\ from E_{S} to F_{S} \end{matrix} \}

\overline{O}_{I,J}(E, F) : S ↦ \{ \begin{matrix}equivalence classes of \\ generalized isomorphisms Φ \\ as in definition 5.2 from \\ E_{S} to F_{S}, with μ_{i} = 0 for i ∈ I \\ and λ_{j} = 0 for j ∈ J \end{matrix} \}

These functors are representable by smooth projective T -schemes, which we will call KGl(E, F), \overline{PGl}(E, F) and \overline{O}_{I,J}(E, F) respectively.

Proof. In the case of T = Spec ℤ and E = F = ⊕^{n}O_{Spec ℤ}, the theorem is a consequence of 5.5 and 8.2, where the representing objects are KGl_{n}, \overline{PGl}_{n} and \overline{O}_{I,J} respectively. Let T = ∪U_{i} an open covering such that there exist trivializations

ξ_{i} : E|_{U_{i}} →^{∼} ⊕^{n}O_{U_{i}}
ζ_{i} : F |_{U_{i}} →^{∼} ⊕^{n}O_{U_{i}} .

Let KGl_{U_{i}} := KGl_{n} ×_{Spec ℤ} U_{i} and π_{i} : KGl_{U_{i}} → U_{i} the projection. By corollary 5.6, over the intersections U_{i} ∩ U_{j} the pairs (ξ_{i}ξ_{j}^{−1}, ζ_{i}ζ_{j}^{−1}) induce isomorphisms π_{i}^{−1}(U_{i} ∩ U_{j}) →^{∼} π_{j}^{−1}(U_{i} ∩ U_{j}). These provide the data for the pieces KGl_{U_{i}} to glue together to define KGl(E, F). Using theorem 5.5 it is easy to check that KGl(E, F) has the required universal property. This proves the existence of KGl(E, F). The existence of \overline{PGl}(E, F) and of \overline{O}_{I,J}(E, F) is proved analogously.

Definition 9.2. Let T be a scheme and E a locally free O_{T} -module of rank n. Let d := (d_{0}, …, d_{t}), where 0 ≤ d_{0} ≤ ⋯ ≤ d_{t} ≤ n Let Fl^{d}(E) be the flag variety which represents the following contravariant functor from the category of T -schemes to the category of sets:

S ↦ \{ \begin{matrix}All filtrations F_{0}E ⊆ ⋯ ⊆ F_{t}E, where \\ F_{p}E is a subbundle of rank d_{p} of E_{S} \\ for 0 ≤ p ≤ t \end{matrix} \}

Here as usual, a subbundle of E_{S} means a locally free subsheaf of E_{S}, which is locally a direct summand.

After these preliminaries we can state the main result of this section, which descibes the structure of the schemes \overline{O}_{I,J} defined above.

Theorem 9.3. Let T be a scheme and let E and F be two locally free O_{T} - modules of rank n. Let I := {i_{1}, …, i_{r}} and J := {j_{1}, …, j_{s}}, where i_{1} + j_{1} ≥ n and 0 ≤ i_{1} < ⋯ < i_{r+1} := n, 0 ≤ j_{1} < ⋯ < j_{s+1} := n. Let d := (d_{0}, …, d_{r+s+1}) and δ := (δ_{0}, …, δ_{r+s+1}), where

d_{p} := \{ \begin{matrix}n − j_{s+1−p} for 0 ≤ p ≤ s \\ i_{p−s} for s + 1 ≤ p ≤ r + s + 1 \end{matrix}

and δ_{q} := n − d_{r+s+1−q} for 0 ≤ q ≤ r + s + 1. Let

0 = U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{r+s+1} = E_{Fl^{d}(E)×Fl^{δ}(F)}
and  0 = V_{0} ⊆ V_{1} ⊆ ⋯ ⊆ V_{r+s+1} = F_{Fl^{d}(E)×Fl^{δ}(F)}

be the pull back to Fl^{d}(E) × Fl^{δ}(F) of the universal flag on Fl^{d}(E) and Fl^{δ}(F) respectively. Then there is a natural isomorphism

\overline{O}_{I,J}(E, F) →^{∼} P_{1} ×_{Fl} … ×_{Fl} P_{r} ×_{Fl} Q_{s} ×_{Fl} … ×_{Fl} Q_{1} ×_{Fl} K′ ,

where Fl := Fl^{d}(E) × Fl^{δ}(F) and where

P_{p} := \overline{PGl}(V_{r−p+1}/V_{r−p}, U_{s+p+1}/U_{s+p}) (1 ≤ p ≤ r)
Q_{q} := \overline{PGl}(U_{s−q+1}/U_{s−q}, V_{r+q+1}/V_{r+q}) (1 ≤ q ≤ s)
K′ := KGl(U_{s+1}/U_{s}, V_{r+1}/V_{r}) .

Proof. The isomorphism

\overline{O}_{I,J} ≅ P_{1} ×_{Fl} … ×_{Fl} P_{r} ×_{Fl} Q_{s} ×_{Fl} … ×_{Fl} Q_{1} ×_{Fl} K′

is given on S-valued points by the bijectiv correspondence

Φ ⟷ ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) ,

where

Φ = ( [figure: E_{S}; ⊗; (M_{0},μ_{0}); E_{1}; 0; (M_{1},μ_{1}); ⊗; E_{2}; 1; …; E_{n−1}; (M_{n−1},μ_{n−1}); ⊗; E_{n}; n−1; ∼; F_{n}; n−1; F_{n−1}; ⊗; (L_{n−1},λ_{n−1}); …; F_{2}; 1; F_{1}; 0; (L_{1},λ_{1}); ⊗; F_{S}; (L_{0},λ_{0}); ⊗] )

is a generalized isomorphism from E_{S} to F_{S} with μ_{i} = λ_{j} = 0 for i ∈ I and j ∈ J,

F_{•}E = (0 = F_{0}E ⊆ ⋯ ⊆ F_{r+s+1}E = E_{S})
and  F_{•}F = (0 = F_{0}F ⊆ ⋯ ⊆ F_{r+s+1}F = F_{S})

are flags of type d and δ in E_{S} and F_{S} respectively,

φ_{p} = ( [figure: E_{0}^{(p)}; ⊗; E_{1}^{(p)}; 0; (M_{0}^{(p)},μ_{0}^{(p)}); …; E_{m_{p}−1}^{(p)}; ⊗; E_{m_{p}}^{(p)}; m_{p}; (M_{mp}^{(p)} ,μ_{mp}^{(p)} )] )

is a complete collineation from E_{m_{p}}^{(p)} = F_{r−p+1}F/F_{r−p}F to E_{0}^{(p)} = F_{s+p+1}E/F_{s+p}E for 1 ≤ p ≤ r,

ψ_{q} = ( [figure: F_{n_{q}}^{(q)}; n_{q}; (L_{nq}^{(q)},λ_{nq}^{(q)}); F_{n_{q}−1}^{(q)}; ⊗; …; F_{1}^{(q)}; 0; (L_{0}^{(q)},λ_{0}^{(q)}); F_{0}^{(p)}; ⊗] )

is a complete collineation from F_{n_{q}}^{(q)} = F_{s−q+1}F/F_{s−q}E to F_{0}^{(q)} = F_{r+q+1}F/F_{r+q}F for s ≥ q ≥ 1 and Φ′ =

( [figure: E′_{0}; ⊗; (M′_{0},μ′_{0}); E′_{1}; 0; (M′_{1},μ′_{1}); ⊗; E′_{2}; 1; …; E′_{n′−1}; (M′_{n′−1},μ′_{n′−1}); ⊗; E′_{n′}; n′−1; ∼; F′_{n′}; n′−1; F′_{n′−1}; ⊗; (L′_{n′−1},λ′_{n′−1}); …; F′_{2}; 1; F′_{1}; 0; (L′_{1},λ′_{1}); ⊗; F′_{0}; (L′_{0},λ′_{0}); ⊗] )

is a generalized isomorphism from E′_{0} = F_{s+1}E/F_{s}E to F′_{0} = F_{r+1}F/F_{r}F. The mapping

Φ ↦ ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′)

is defined as follows: Let Φ as above be given. For convenience we set E_{n+1} := F_{n}, F_{n+1} := E_{n} and we let F_{n+1} → F_{n} and E_{n} ← E_{n+1} be the isomorphism E_{n} →^{∼} F_{n} and its inverse respectively, whereas we let \xymatrix{E_{n} \ar[r]|{⊗} & E_{n+1}}

Compactification of the General Linear Group 585

and \xymatrix{F_{n+1} & F_{n} \ar[l]|{⊗}} both be the zero morphism. For what follows, the picture below may help to keep track of the indices:

Φ

|^{0} . . . . . . |^{n−j_{1}} … |^{i_{1}} … |^{i_{2}} . . . . . . |^{i_{r}} … |^{n}|^{n} … |^{j_{s}} . . . . . . |^{j_{2}} … |^{j_{1}} … |^{n−i_{1}} . . . . . . |^{0}
Φ′ φ_{1} φ_{r} ψ_{s} ψ_{1} Φ′

Let F_{0}E = F_{0}F := 0, F_{r+s+1}E := E, F_{r+s+1}F := F and

F_{p}E := \{ \begin{matrix}( \begin{matrix}image of ker(E_{n} → F_{j_{s−p+1}}) by \\ the morphism E_{S} ← E_{n} \end{matrix} ) , if 1 ≤ p ≤ s \\ ker( \xymatrix{E_{S} \ar[r]|{⊗} & E_{i_{p−s}+1}} ) , if s + 1 ≤ p ≤ r + s \end{matrix}
F_{q}F := \{ \begin{matrix}( \begin{matrix}image of ker(E_{i_{r−q+1}} ← F_{n}) by \\ the morphism F_{n} → F_{S} \end{matrix} ) , if 1 ≤ q ≤ r \\ ker( \xymatrix{F_{j_{q−r}+1} & F_{S} \ar[l]|{⊗}} ) , if r + 1 ≤ q ≤ r + s \end{matrix}

It is then clear from the definition of generalized isomorphisms that

F_{•}E := (0 = F_{0}E ⊆ ⋯ ⊆ F_{r+s+1}E = E_{S})

and  F_{•}F := (0 = F_{0}F ⊆ ⋯ ⊆ F_{r+s+1}F = F_{S})

are flags of type d and δ in E_{S} and F_{S} respectively. Let 1 ≤ p ≤ r. We set

E_{0}^{(p)} := ker( \xymatrix{E \ar[r]|{⊗} & E_{i_{p+1}+1}} )/ ker( [figure: E; ⊗; E_{i_{p}+1}] ) = F_{s+p+1}E/F_{s+p}E
M_{0}^{(p)} := ⨂_{i=0}^{i_{p}} M_{i} ,  μ_{0}^{(p)} := 0

and

E_{k}^{(p)} := ker(E_{i_{p}} ← E_{i_{p}+k}) ∩ ker( \xymatrix{E_{i_{p}+k} \ar[r]|{⊗} & E_{i_{p+1}+1}} ) (1 ≤ k ≤ m_{p})
M_{k}^{(p)} := M_{i_{p}+k} , μ_{k}^{(p)} := μ_{i_{p}+k} (1 ≤ k ≤ m_{p} − 1)

where m_{p} = i_{p+1} − i_{p}. Observe that the sheaves E_{k}^{(p)} thus defined are locally free of rank m_{p}. Indeed, this is clear for k = 0. For k ≥ 1 it suffices to show that E_{i_{p}+k} is generated by the two subsheaves ker(E_{i_{p}} ← E_{i_{p}+k}) and ker( \xymatrix{E_{i_{p}+k} \ar[r]|{⊗} & E_{i_{p+1}+1}} ). For this in turn, it suffices to show that the image

of ker( \xymatrix{E_{i_{p}+k} \ar[r]|{⊗} & E_{i_{p+1}+1}} ) by the morphism E_{i_{p}} ← E_{i_{p}+k} is im(E_{i_{p}} ← E_{i_{p}+k}).

But this is clear, since by the definition of generalized isomorphisms we have

ker( \xymatrix{E_{i_{p}+k} \ar[r]|{⊗} & E_{i_{p+1}+1}} ) ⊇ im(E_{i_{p}+k} ← E_{i_{p+1}+1})

and  im(E_{i_{p}} ← E_{i_{p}+k}) = im(E_{i_{p}} ← E_{i_{p+1}+1}) .

Since

⨂_{i=0}^{i_{p}} M_{i}^{∨} ⊗ (E_{S}/F_{s+p}E) = im(⨂_{i=0}^{i_{p}} M_{i}^{∨} ⊗ E_{S} → E_{i_{p}+1}) = ker(E_{i_{p}} ← E_{i_{p}+1}) ,

Documenta Mathematica 5 (2000) 553–594

we have a natural isomorphism E_{0}^{(p)} = F_{s+p+1}E/F_{s+p}E →^{∼} M_{0}^{(p)} ⊗ E_{1}^{(p)}. We

define E_{0}^{(p)} ← E_{1}^{(p)} to be the zero morphism. Thus we have a bf-morphism

[figure: E_{0}^{(p)}; ⊗; E_{1}^{(p)}; 0; (M_{0}^{(p)},μ_{0}^{(p)}=0)] of rank zero. For 1 ≤ k ≤ m_{p} − 1 let [figure: E_{k}^{(p)}; ⊗; E_{k+1}^{(p)}; k; (M_{k}^{(p)},μ_{k}^{(p)})] be

the bf-morphism induced by the bf-morphism [figure: E_{i_{p}+k}; ⊗; E_{i_{p}+k+1}; i_{p}+k; (M_{ip+k},μ_{ip+k})] . Observe

that ker( \xymatrix{E_{i_{p+1}} \ar[r]|{⊗} & E_{i_{p+1}+1}} ) = im(E_{i_{p+1}} ← E_{n}) = E_{n}/ ker(E_{i_{p+1}} ← E_{n}) and

that the morphism E_{n} → F_{S} maps ker(E_{i_{p}} ← E_{n}) injectively into F_{S}. Therefore we have a natural isomorphism E_{m_{p}}^{(p)} ≅ F_{r−p+1}F/F_{r−p}F by which we identify these two sheaves. It is not difficult to see that

φ_{p} := ( [figure: E_{0}^{(p)}; ⊗; E_{1}^{(p)}; 0; (M_{0}^{(p)},μ_{0}^{(p)}); …; E_{m_{p}−1}^{(p)}; ⊗; E_{m_{p}}^{(p)}; m_{p}; (M_{mp}^{(p)} ,μ_{mp}^{(p)} )] )

is a complete collineation in the sense of 8.1 from E_{m_{p}}^{(p)} = F_{r−p+1}F/F_{r−p}F to E_{0}^{(p)} = F_{s+p+1}E/F_{s+p}E. In a completely symmetric way the generalized isomorphism Φ induces also complete collineations

ψ_{q} = ( [figure: F_{n_{q}}^{(q)}; n_{q}; (L_{nq}^{(q)},λ_{nq}^{(q)}); F_{n_{q}−1}^{(q)}; ⊗; …; F_{1}^{(q)}; 0; (L_{0}^{(q)},λ_{0}^{(q)}); F_{0}^{(p)}; ⊗] )

from F_{n_{q}}^{(q)} = F_{s−q+1}F/F_{s−q}E to F_{0}^{(q)} = F_{r+q+1}F/F_{r+q}F for s ≥ q ≥ 1. It remains to construct the generalized isomorphism Φ′. We set

E′_{k} := ker( \xymatrix{E_{n−j_{1}+k} \ar[r]|{⊗} & E_{i_{1}+1}} )/im(E_{n−j_{1}+k} ← ker(E_{n} → F_{j_{1}}))
F′_{k} := ker( \xymatrix{F_{j_{1}+1} & F_{n−i_{1}+k} \ar[l]|{⊗}} )/im(ker(E_{i_{1}} ← F_{n}) → F_{n−i_{1}+k}))

for 0 ≤ k ≤ n′ := i_{1} + j_{1} − n and

M′_{k} := M_{n−j_{1}+k} , μ′_{k} := μ_{n−j_{1}+k}
L′_{k} := L_{n−i_{1}+k} , λ′_{k} := λ_{n−i_{1}+k}

for 0 ≤ k ≤ n′ − 1. It is then clear that the E′_{k} and F′_{k} are locally free of rank n′ = i_{1} +j_{1} −n. It follows from definition 5.2.2. that the μ_{i} and λ_{j} are nowhere vanishing for 0 ≤ i ≤ n − j_{1} − 1 and n − i_{1} − 1 ≥ j ≥ 0. Therefore we may identify E_{i} with E_{S} and F_{j} with F_{S} for 0 ≤ i ≤ n−j_{1} −1 and n−i_{1} −1 ≥ j ≥ 0 respectively. This implies in particular that we have E′_{0} = F_{s+1}E/F_{s}E and F′_{0} = F_{r+1}F/F_{r}F. Let

[figure: E′_{k}; ⊗; E′_{k+1}; k; (M′_{k},μ′_{k}); and; F′_{k+1}; k; (L′_{k},λ′_{k}); F′_{k}; ⊗]

be the bf-morphisms induced by the bf-morphisms

[figure: E_{n−j_{1}+k}; ⊗; E_{n−j_{1}+k+1}; n−j_{1}+k; (M_{n−j_{1}+k},μ_{n−j_{1}+k}); and; F_{n−i_{1}+k+1}; n−i_{1}+k; (L_{n−i_{1}+k},λ_{n−i_{1}+k}); F_{n−i_{1}+k}; ⊗]

respectively. We have

ker( \xymatrix{E_{i_{1}} \ar[r]|{⊗} & E_{i_{1}+1}} ) = im(E_{i_{1}} ← E_{n}) = E_{n}/ ker(E_{i_{1}} ← E_{n})

and therefore

E′_{n′} = E_{n}/(ker(E_{i_{1}} ← E_{n}) + ker(E_{n} → F_{j_{1}})) .

By the same argument:

F′_{n′} = F_{n}/(ker(E_{i_{1}} ← F_{n}) + ker(F_{n} → F_{j_{1}})) .

Thus the isomorphism E_{n} →^{∼} F_{n} induces an isomorphism E′_{n′} →^{∼} F′_{n′}. Again it is not difficult to check that Φ′ :=

( [figure: E′_{0}; ⊗; (M′_{0},μ′_{0}); E′_{1}; 0; (M′_{1},μ′_{1}); ⊗; E′_{2}; 1; …; E′_{n′−1}; (M′_{n′−1},μ′_{n′−1}); ⊗; E′_{n′}; n′−1; ∼; F′_{n′}; n′−1; F′_{n′−1}; ⊗; (L′_{n′−1},λ′_{n′−1}); …; F′_{2}; 1; F′_{1}; 0; (L′_{1},λ′_{1}); ⊗; F′_{0}; (L′_{0},λ′_{0}); ⊗] )

is a generalized isomorphism from E′_{0} = F_{s+1}E/F_{s}E to F′_{0} = F_{r+1}F/F_{r}F. This completes the construction of the mapping

Φ ↦ ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) .

We proceed by constructing the inverse of this mapping. Let data ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) be given. Let E_{i} := E_{S} and F_{j} := F_{S} for 0 ≤ i ≤ n − j_{1} and n − i_{1} ≥ j ≥ 0 respectively. Now let n − j_{1} + 1 ≤ i ≤ i_{1} and j_{1} ≥ j ≥ n − i_{1} + 1. Let Ẽ_{i} and F̃_{j} be defined by the cartesian diagrams

\xymatrix{Ẽ_{i} \ar[dd] \ar[r] & E′_{i+j_{1}−n} \ar[dd] &  & F̃_{j} \ar[dd] \ar[r] & F′_{j+i_{1}−n} \ar[dd] \\  &  & and \\ F_{s+1}E \ar@{->>}[r] & F_{s+1}E/F_{s}E &  & F_{r+1}F \ar@{->>}[r] & F_{r+1}F/F_{r}F}

respectively. For a moment let M := ⨂_{k=0}^{i+j_{1}−n−1} M′_{k}. We have a commutative diagram

\xymatrix{M^{∨} ⊗ F_{s+1}E \ar[d] \ar[r] & E′_{i+j_{1}−n} \ar[d] \\ F_{s+1}E \ar@{->>}[r] & F_{s+1}E/F_{s}E}  (∗)

where the left vertical arrow is induced by ⨂_{k=0}^{i+j_{1}−n−1} μ′_{k} : O_{S} → M and the upper horizontal arrow is the composition

M^{∨} ⊗ F_{s+1}E → M^{∨} ⊗ F_{s+1}E/F_{s}E = M^{∨} ⊗ E′_{0} → E′_{i+j_{1}−n}

The diagram (∗) induces a morphism M^{∨} ⊗ F_{s+1}E → Ẽ_{i}. Analogously, we have a morphism L^{∨} ⊗F_{r+1}F → F̃_{j}, where we have employed the abbreviation L := ⨂_{k=0}^{j+i_{1}−n−1} L′_{k}. Let E_{i} and F_{j} be defined by the cocartesian diagrams

\xymatrix{M^{∨} ⊗ F_{s+1}E \ar[dd] \ar@{^{(}->}[r] & M^{∨} ⊗ E_{S} \ar[dd] &  & L^{∨} ⊗ F_{r+1}F \ar[dd] \ar@{^{(}->}[r] & L^{∨} ⊗ F_{S} \ar[dd] \\  &  & and \\ Ẽ_{i} \ar[r] & E_{i} &  & F̃_{j} \ar[r] & F_{j}}

respectively. We define E_{n} = F_{n} by the cartesian diagram

\xymatrix{E_{n} = F_{n} \ar[d] \ar[rr] &  & F̃_{j_{1}} \ar@{->>}[d] \\ Ẽ_{i_{1}} \ar@{->>}[r] & E′_{i_{1}+j_{1}−n} \ar[r]^{≅} & F′_{i_{1}+j_{1}−n}}

Observe that the composed morphism E_{n} → Ẽ_{i_{1}} → F_{s+1}E maps the submodule ker(E_{n} → F̃_{j_{1}}) of E_{n} isomorphically onto the submodule F_{s}E of F_{s+1}E. Therefore we have canonical injections

F_{p}E ↪ F_{s}E →^{∼} ker(E_{n} → F̃_{j_{1}}) ↪ E_{n}

for 0 ≤ p ≤ s. Analogously, we have canonical injections

F_{q}F ↪ F_{r}F →^{∼} ker(F_{n} → Ẽ_{i_{1}}) ↪ F_{n}

for 0 ≤ q ≤ r. Now let 1 ≤ p ≤ r, i_{p} + 1 ≤ i ≤ i_{p+1} and s ≥ q ≥ 1, j_{q+1} ≥ j ≥ j_{q} + 1. We want to define E_{i} and F_{j} in this case. Let first Ẽ_{i} and F̃_{j} be defined by the cocartesian diagrams

\xymatrix{M^{∨} ⊗ F_{s+p+1}E/F_{s+p}E \ar[r] \ar@{^{(}->}[dd] & E_{i−i_{p}}^{(p)} \ar[dd] &  & L^{∨} ⊗ F_{r+q+1}F/F_{r+q}E \ar[r] \ar@{^{(}->}[dd] & F_{j−j_{q}}^{(q)} \ar[dd] \\  &  & and \\ M^{∨} ⊗ E_{S}/F_{s+p}E \ar[r] & Ẽ_{i} &  & L^{∨} ⊗ F_{S}/F_{r+q}F \ar[r] & F̃_{j}}

where we have set M := ⨂_{k=0}^{i−i_{p}−1} M_{k}^{(p)} and L := ⨂_{k=0}^{j−j_{q}−1} L_{k}^{(q)}. Let furthermore Ê_{i} and F̂_{j} be defined by the cocartesian diagrams

\xymatrix{F_{r−p+1}F/F_{r−p}F \ar[r] \ar@{^{(}->}[dd] & E_{i−i_{p}}^{(p)} \ar[dd] &  & F_{s−q+1}E/F_{s−q}E \ar[r] \ar@{^{(}->}[dd] & F_{j−j_{q}}^{(q)} \ar[dd] \\  &  & and \\ F_{n}/F_{r−p}F \ar[r] & Ê_{i} &  & E_{n}/F_{s−q}E \ar[r] & F̂_{j}}

Now we define E_{i} and F_{j} by the cocartesian diagrams

[figure: E_{i−i_{p}}^{(p)}; Ẽ_{i}; Ê_{i}; E_{i}; and; F_{j−j_{q}}^{(q)}; F̃_{j}; F̂_{j}; F_{j}]

respectively. For p = r and i = i_{r+1} = n this gives formally a new definition of E_{n}, but it is clear that we have a canonical isomorphism between the two E_{n}’s. A similar remark applies to F_{n}. We define the invertible sheaves M_{i} together with their respective sections μ_{i} as follows:

M_{i} := O_{S} , μ_{i} := 1 (0 ≤ i ≤ n − j_{1} − 1)
M_{i} := M′_{i+j_{1}−n} , μ_{i} := μ′_{i+j_{1}−n} (n − j_{1} ≤ i ≤ i_{1} − 1)
M_{i} := M_{i−i_{p}}^{(p)} , μ_{i} := μ_{i−i_{p}}^{(p)} (1 ≤ p ≤ r , i_{p} < i < i_{p+1})

M_{i_{1}} := M_{0}^{(1)} ⊗ ⨂_{k=0}^{i_{1}+j_{1}−n−1}(M′_{k})^{∨} , μ_{i_{1}} := 0

M_{i_{p}} := M_{0}^{(p)} ⊗ ⨂_{k=0}^{i_{p}−i_{p−1}−1}(M_{k}^{(p−1)})^{∨} , μ_{i_{p}} := 0 (2 ≤ p ≤ r)

Let the L_{j} and λ_{j} be defined symmetrically (i.e. by replacing in the above definition the letter M with L, μ with λ, i with j, j with i and r with s). It remains to define the bf-morphisms

[figure: E_{i}; ⊗; E_{i+1}; i; (M_{i},μ_{i}); and; F_{j+1}; j; (L_{j},λ_{j}); F_{j}; ⊗]

for n − j_{1} ≤ i ≤ n − 1 and n − 1 ≥ j ≥ n − i_{1}. Again we restrict ourselves to the left hand side, since the right hand side is obtained by the symmetric construction. For n − j_{1} ≤ i ≤ i_{1} − 1 (respectively for 1 ≤ p ≤ r , i_{p} ≤

i ≤ i_{p+1} − 1) the bf-morphism [figure: E_{i}; ⊗; E_{i+1}] is induced in an obvious way

by the bf-morphism [figure: E′_{i+j_{1}−n}; ⊗; E′_{i+j_{1}−n+1}] (respectively by the bf-morphism

[figure: E_{i−i_{p}}^{(p)}; ⊗; E_{i−i_{p}+1}^{(p)}] ). For the definition of the bf-morphism [figure: E_{i_{1}}; ⊗; E_{i_{1}+1}]

consider the two canonical exact sequences

\xymatrix{0 \ar[r] & Ẽ_{i_{1}} \ar[r] & E_{i_{1}} \ar[r] & M^{∨} ⊗ E_{S}/F_{s+1}E \ar[r] & 0 \\ 0 \ar[r] & _{Ẽ_{i_{1}+1}} \ar[r] & E_{i_{1}+1} \ar[r] & Ê_{i_{1}+1}/E_{1}^{(1)} \ar[r] & 0}

where M := ⨂_{k=0}^{i_{1}−1} M_{k}. Observe that we have canonical isomorphisms

\xymatrix{Ẽ_{i_{1}} \ar[r]^{a}_{≅} & F_{n}/F_{r}F \ar[r]^{b}_{≅} & Ê_{i_{1}+1}/E_{1}^{(1)} \\ _{Ẽ_{i_{1}+1}} \ar[rr]^{c}_{≅} &  & M_{i_{1}}^{∨} ⊗ M^{∨} ⊗ E_{S}/F_{s+1}E}

The isomorphism a follows from the observation we made after the definition of E_{n} = F_{n}, namely that the composed morphism F_{n} → F̃_{j_{1}} → F_{r+1}F maps ker(E_{n} → Ẽ_{i_{1}}) isomorphically to F_{r}F. The isomorphism b comes from the fact that for i = i_{1} +1 the left vertical arrow in the defining diagram for Ê_{i} vanishes, and the isomorphism c follows since for i = i_{1} + 1 the left vertical arrow in the defining diagram for Ẽ_{i} is an isomorphism. Thus we have morphisms

\xymatrix{E_{i_{1}} \ar@{->>}[r] & M^{∨} ⊗ E_{S}/F_{s+1} \ar[r]^{c^{−1}} & M_{i_{1}} ⊗ Ẽ_{i_{1}+1} \ar@{^{(}->}[r] & M_{i_{1}} ⊗ E_{i_{1}+1} \\ E_{i_{1}} & _{Ẽ_{i_{1}}} \ar@{^{(}->}[l] & Ê_{i_{1}+1}/E_{1}^{(1)} \ar[l]_{a^{−1}b^{−1}} & E_{i_{1}+1} \ar@{->>}[l]}

which make up the bf-morphism [figure: E_{i_{1}}; ⊗; E_{i_{1}+1}] . For 2 ≤ p ≤ r the bf-

morphism [figure: E_{i_{p}}; ⊗; E_{i_{p}+1}] is constructed similarly from the exact sequences

\xymatrix{0 \ar[r] & Ê_{i_{p}} \ar[r] & E_{i_{p}} \ar[r] & Ẽ_{i_{p}}/E_{i_{p}−i_{p−1}}^{(p−1)} \ar[r] & 0 \\ 0 \ar[r] & _{Ẽ_{i_{p}+1}} \ar[r] & E_{i_{p}+1} \ar[r] & Ê_{i_{p}+1}/E_{1}^{(p)} \ar[r] & 0}

and the canonical isomorphisms

\xymatrix{Ê_{i_{p}} \ar[r]^{≅} & F_{n}/F_{r−p+1}F \ar[r]^{≅} & Ê_{i_{p}+1}/E_{1}^{(p)} \\ _{Ẽ_{i_{p}+1}} \ar[r]^{≅} & M_{i_{p}}^{∨} ⊗ M^{∨} ⊗ E/F_{s+p}E \ar[r]^{≅} & M_{i_{p}}^{∨} ⊗ Ẽ_{i_{p}}/E_{i_{p}−i_{p−1}}^{(p−1)}}

where M := ⨂_{k=0}^{i_{p}−1} M_{k}. This completes the construction of

Φ = ( [figure: E_{S}; ⊗; (M_{0},μ_{0}); E_{1}; 0; (M_{1},μ_{1}); ⊗; E_{2}; 1; …; E_{n−1}; (M_{n−1},μ_{n−1}); ⊗; E_{n}; n−1; ∼; F_{n}; n−1; F_{n−1}; ⊗; (L_{n−1},λ_{n−1}); …; F_{2}; 1; F_{1}; 0; (L_{1},λ_{1}); ⊗; F_{S}; (L_{0},λ_{0}); ⊗] )

It is not difficult to see that Φ is a generalized isomorphism from E_{S} to F_{S} and that the mapping ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) ↦ Φ is inverse to the mapping Φ ↦ ((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) constructed before. We leave the details to the reader.

In the situation of theorem 9.3 we denote by Gl(E) the group scheme over T, whose S-valued points are the automorphisms of E_{S}. There is a natural left operation of Gl(E) ×_{T} Gl(F) on KGl(E, F), which is given on S-valued points by

(f, g) ( [figure: E_{S}; ⊗; u^{♭}; E_{1}; u^{♯}; …; F_{1}; v^{♯}; F_{S}; ⊗; v^{♭}] ) := ( [figure: E_{S}; ⊗; u^{♭}∘f^{−1}; E_{1}; f∘u^{♯}; …; F_{1}; g∘v^{♯}; F_{S}; ⊗; v^{♭}∘g^{−1}] )

Corollary 9.4. The orbits of the Gl(E)×_{T} Gl(F)-operation on KGl(E, F) are the locally closed subvarieties

O_{I,J}(E, F) := \overline{O}_{I,J}(E, F) \ (⋃_{i∉I} Z_{i}(E, F) ∪ ⋃_{j∉J} Y_{j}(E, F)) ,

where I, J ⊆ [0, n − 1] with min I + min J ≥ n, and where Z_{i}(E, F) := \overline{O}_{\{i\},∅}(E, F) and Y_{j}(E, F) := \overline{O}_{∅,\{j\}}(E, F).

Proof. The S-valued points of O_{I,J}(E, F) are the generalized isomorphisms

Φ = ( [figure: E_{S}; ⊗; (M_{0},μ_{0}); E_{1}; 0; (M_{1},μ_{1}); ⊗; E_{2}; 1; …; E_{n−1}; (M_{n−1},μ_{n−1}); ⊗; E_{n}; n−1; ∼; F_{n}; n−1; F_{n−1}; ⊗; (L_{n−1},λ_{n−1}); …; F_{2}; 1; F_{1}; 0; (L_{1},λ_{1}); ⊗; F_{S}; (L_{0},λ_{0}); ⊗] )

where μ_{i} = λ_{j} = 0 for i ∈ I and j ∈ J and where μ_{i}, λ_{j} are nowhere vanishing for i ∉ I, j ∉ J. It is clear that O_{I,J}(E, F) is invariant under the operation of Gl(E) ×_{T} Gl(F). From the proof of theorem 9.3 it follows that we have the following isomorphism

O_{I,J}(E, F) ≅ P_{1}^{o} ×_{Fl} … ×_{Fl} P_{r}^{o} ×_{Fl} Q_{s}^{o} ×_{Fl} … ×_{Fl} Q_{1}^{o} ×_{Fl} K′^{o} ,

where

P_{p}^{o} := PGl(V_{r−p+1}/V_{r−p}, U_{s+p+1}/U_{s+p}) (1 ≤ p ≤ r)
Q_{q}^{o} := PGl(U_{s−q+1}/U_{s−q}, V_{r+q+1}/V_{r+q}) (1 ≤ q ≤ s)
K′^{o} := Isom(U_{s+1}/U_{s}, V_{r+1}/V_{r}) .

There is a left Gl(E) ×_{T} Gl(F)-operation on the right-hand side of this isomorphism, given on S-valued points by

(f, g)((F_{•}E, F_{•}F), φ_{1}, …, φ_{r}, ψ_{s}, …, ψ_{1}, Φ′) :=

((f(F_{•}E), g(F_{•}F)), f^{−1}φ_{1}g, …, f^{−1}φ_{r}g, gψ_{s}f^{−1}, …, gψ_{1}f^{−1}, gΦ′f^{−1}),

where φ_{p} is an isomorphism (up to multiplication by an invertible section of O_{S}) from F_{r−p+1}F/F_{r−p}F to F_{s+p+1}E/F_{s+p}E for 1 ≤ p ≤ r, ψ_{q} an isomorphism (up to multiplication by an invertible section of O_{S}) from F_{s−q+1}E/F_{s−q}E to F_{r+q+1}F/F_{r+q}F for s ≥ q ≥ 1 and Φ′ is an isomorphism from F_{s+1}E/F_{s}E to F_{r+1}F/F_{r}F. It is easy to see that this operation is transitiv and that the isomorphism

O_{I,J}(E, F) ≅ P_{1}^{o} ×_{Fl} … ×_{Fl} P_{r}^{o} ×_{Fl} Q_{s}^{o} ×_{Fl} … ×_{Fl} Q_{1}^{o} ×_{Fl} K′^{o} ,

is Gl(E) ×_{T} Gl(F)-equivariant.

10. A morphism of KGl_{n} onto the Grassmannian compactification of the general linear group

Let V be an n-dimensional vector space over some field. As mentioned in the introduction, there is another natural compactification of the general linear group Gl(V): The Grassmannian Gr_{n}(V ⊕ V) of n-dimensional subspaces of a V ⊕ V -dimensional vector space. The embedding Gl(V) ↪ Gr_{n}(V ⊕ V) is given by associating to an automorphism V →^{∼} V its graph in V ⊕ V. We will see in this section that there exists a natural morphism from KGl(V) to Gr_{n}(V ⊕ V). Our motivation here is to obtain a better understanding of the relation between the Gieseker-type degeneration of moduli spaces of vector bundles and the torsion-free sheaves approach as developed in [NS] and [S2]. As in the previous section, we work over an arbitrary base scheme T. Let E, F be two locally free O_{T} -modules of rank n. Denote by Gr_{n}(E ⊕ F) the Grassmanian variety over T which parametrizes subbundles of rank n of E ⊕F. Let S be a T -scheme and let

Φ = ( [figure: E_{S}; ⊗; (M_{0},μ_{0}); E_{1}; 0; (M_{1},μ_{1}); ⊗; E_{2}; 1; …; E_{n−1}; (M_{n−1},μ_{n−1}); ⊗; E_{n}; n−1; ∼; F_{n}; n−1; F_{n−1}; ⊗; (L_{n−1},λ_{n−1}); …; F_{2}; 1; F_{1}; 0; (L_{1},λ_{1}); ⊗; F_{S}; (L_{0},λ_{0}); ⊗] )

be a generalized isomorphism from E_{S} to F_{S}. By 5.2.2, the morphism E_{n} → E_{S} ⊕ F_{S} induced by the two composed morphisms

E_{n} → E_{n−1} → ⋯ → E_{1} → E_{S}

E_{n} →^{∼} F_{n} → F_{n−1} → ⋯ → F_{1} → F_{S}

is a subbundle of E_{S} ⊕ F_{S}. Let

KGl(E, F) → Gr_{n}(E ⊕ F)

be the morphism, which on S-valued points is given by Φ ↦ (E_{n} → E_{S} ⊕ F_{S}). Observe that the following diagram commutes

\xymatrix{ & Isom(E, F) \ar@{^{(}->}[dl] \ar@{^{(}->}[dr] \\ KGl(E, F) \ar[rr] &  & Gr_{n}(E ⊕ F)}

and that furthermore all the arrows in this diagram are equivariant with respect to the natural action of Gl(E) ×_{T} Gl(F) on the three schemes. In the next proposition we compute the fibres of the morphism KGl(E, F) → Gr_{n}(E ⊕ F).

Proposition 10.1. Let S′ be a T -scheme and let H ↪ E_{S′} ⊕ F_{S′} be an S′- valued point of Gr_{n}(E, F) such that im(H → E_{S′}) and im(H → F_{S′}) are subbundles of E_{S′} and F_{S′} respectively. Then the fibre product

KGl(E, F) ×_{Gr_{n}(E⊕F)} S′

is isomorphic to

\overline{PGl}(ker(H → E_{S′}), coker(H → E_{S′})) ×_{S′} \overline{PGl}(ker(H → F_{S′}), coker(H → F_{S′})),

where by convention \overline{PGl}(N , N) := S′ for the zero-sheaf N = 0 on S′.

Proof. Let S be an S′-scheme. An S-valued point of the fibre product KGl(E, F) ×_{Gr_{n}(E⊕F)} S′ is given by a generalized isomorphism

Φ = ( [figure: E_{S}; ⊗; (M_{0},μ_{0}); E_{1}; 0; (M_{1},μ_{1}); ⊗; E_{2}; 1; …; E_{n−1}; (M_{n−1},μ_{n−1}); ⊗; E_{n}; n−1; ∼; F_{n}; n−1; F_{n−1}; ⊗; (L_{n−1},λ_{n−1}); …; F_{2}; 1; F_{1}; 0; (L_{1},λ_{1}); ⊗; F_{S}; (L_{0},λ_{0}); ⊗] )

from E_{S} to F_{S} such that the induced morphism E_{n} ↪ E_{S} ⊕F_{S} identifies E_{n} with the subbundle H_{S}. Let i_{1} and j_{1} be the ranks of im(H → E_{S′}) and im(H → F_{S′}) respectively. Observe that i_{1} + j_{1} ≥ n. We restrict ourselves to the case, where i_{1} and j_{1} are both strictly smaller than n. (The cases where one or both of i_{1}, j_{1} are equal to n are proved analogously). Then the sections μ_{0}, …, μ_{i_{1}−1} and λ_{j_{1}−1}, …, λ_{0} are invertible and μ_{i_{1}} = λ_{j_{1}} = 0. From the proof of theorem 9.3 it follows that such a Φ may be given by a tupel ((F_{•}E, F_{•}F), φ, ψ, Φ′) where

F_{•}E = (0 = F_{0}E ⊆ F_{1}E ⊆ F_{2}E ⊆ F_{3}E = E_{S})
F_{•}F = (0 = F_{0}F ⊆ F_{1}F ⊆ F_{2}F ⊆ F_{3}F = F_{S})

are the filtrations given by

F_{1}E := im(ker(H_{S} → F_{S}) → E_{S})
F_{2}E := im(H_{S} → E_{S})
F_{1}F := im(ker(H_{S} → E_{S}) → F_{S})
F_{2}F := im(H_{S} → F_{S}) ,

φ is a complete collineation from F_{1}F/F_{0}F ≅ ker(H_{S} → E_{S}) to F_{3}E/F_{2}E ≅ coker(H_{S} → E_{S}), ψ is a complete collineation from F_{1}E/F_{0}E ≅ ker(H_{S} → F_{S}) to F_{3}F/F_{2}F ≅ coker(H_{S} → F_{S}), and Φ′ is the isomorphism

F_{2}E/F_{1}E →^{∼} H_{S}/(ker(H_{S} → E_{S}) + ker(H_{S} → F_{S})) →^{∼} F_{2}F/F_{1}F .

We see in particular that the tupel ((F_{•}E, F_{•}F), φ, ψ, Φ′) is already determined by the subbundle H_{S} ↪ E_{S} ⊕ F_{S} (i.e. by the morphism S → S′) and the pair (φ, ψ).

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Ivan Kausz

NWF I - Mathematik Universität Regensburg 93040 Regensburg Germany ivan.kausz@mathematik.uni-regensburg.de