{'fraction_non_alphanumeric': 0.05110697289468239, 'fraction_numerical': 0.015932133250569004, 'mean_word_length': 4.1153439153439155, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nilpotent cone and bivariant theory

We exhibit a new proof, relying on bivariant theory, that the nilpotent cone is rationally smooth. Our approach enables us to prove a slightly more general statement.


Introduction
In [2] Borho and MacPherson proved that the nilpotent cone is a rational homology manifold. The proof relies on the celebrated Decomposition Theorem by Beilinson, Bernstein, Deligne and Gabber [1] and on the Springer's theory of Weyl group representations (see [2] and the references therein).
The aim of this paper is to present a new proof, in our opinion conceptually very simple, based on the bivariant theory founded by Fulton and MacPherson in [4]. Actually, our approach enables us to prove a slightly more general statement (see Remark 2.4 below). By bivariant theory we intend the topological bivariant homology theory with coefficients in a Noetherian commutative ring with identity A [4,pp. 32,83 and p. 86,Corollary 7.3.4].
That the nilpotent cone is a rational homology manifold can be seen as an easy consequence of a characterization of homology manifolds we recently proved in [3,Theorem 6.1]: given a resolution of singularities π : N → N of a quasi-projective variety N , then N is a homology manifold if and only if there exists a bivariant class of degree one for π. A bivariant class of degree one for π is an element η ∈ H 0 ( N π → N ) such that the induced Gysin homomorphism η 0 : H 0 ( N ) → H 0 (N ) sends 1 N to 1 N .

The main result
Theorem 2.1. Let π ′ : g → g be a projective morphism between complex quasiprojective nonsingular varieties of the same dimension. Assume that π ′ is generically finite, of degree δ. Let N ⊂ g be a closed irreducible subvariety. Consider the induced fibre square diagram: If N is irreducible and nonsingular and π is birational, then N is an A-homology manifold for every Noetherian commutative ring with identity A for which δ is a unit.
Proof. Since π ′ : g → g is a projective morphism between complex quasi-projective nonsingular varieties of the same dimension, it is a local complete intersection morphism of relative codimension 0 [4, p. 130]. Let Notations, (ii)]. At this point, our claim follows by [3,Theorem 6.1]. For the Reader's convenience, let us briefly summarize the argument.
Since δ −1 ·θ is a bivariant class of degree one for π, it follows that Now, set ν = dim N = dim N and let [ N ] ∈ H 2ν ( N ) be the fundamental class of N . We have: , where D denotes Verdier dual. Therefore, [ N ] corresponds to a morphism (2) A , whose induced map in hypercohomology is nothing but the duality morphism If we assume that N is nonsingular (actually it suffices that N is an A-homology manifold, the morphisms (2) and (3) are isomorphisms. The first one induces an isomorphism which in turn, via decomposition (1), induces two projections Making explicit the isomorphism induced in cohomology and homology by (1), one may prove [3,Corollary 5.1] that D N is the direct sum of P 1 and P 2 , where are the maps induced in hypercohomology by the projections (4). It follows that P 1 is an isomorphism, because so is D N , and this holds true when restricting to every open subset U of N . For instance (see also [3,Corollary 5.1]), if U = π −1 (U ), the vanishing of the morphism H • (K U [ν]) → H 2ν−• (U ) derives from projection formula [4, p. 26, G4, (ii)]: where λ * is the morphism induced in hypercohomology by , which is equivalent to say that N is an A-homology manifold.
Remark 2.2. Observe that, as a scheme, N could also be nonreduced, but what matters is that, for the usual topology, it is a nonsingular variety [4, p. 32, 3.1.1] Corollary 2.3. The nilpotent cone is a rational homology manifold.
Proof. Let π : N → N be the Springer resolution of the nilpotent cone N . It extends to a generically finite projective morphism π ′ : g → g, known as the Grothendieck simultaneous resolution, between complex quasi-projective nonsingular varieties of the same dimension [2, p. 49]. Therefore, Theorem 2.1 applies.
Remark 2.4. If the Grothendieck simultaneous resolution π ′ : g → g has degree δ, by Theorem 2.1 we deduce that the nilpotent cone N is an A-homology manifold for every Noetherian commutative ring with identity A for which δ is a unit. For instance, for the variety N of nilpotent matrices in GL(n, C), we have δ = n!. Therefore, in this case, N is also a Z h -homology manifold for every integer h relatively prime with n! in Z.

Statements and Declarations.
Funding. The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Data availability. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Competing Interests. The authors have no relevant financial or non-financial interests to disclose.