Lefschetz-Riemann-Roch for Singular Schemes

• Lefschetz–Riemann–Roch for singular projective schemes is a generalization that extends classical identities to schemes with singularities and diagonalizable group actions.
• It employs equivariant G-theory, relative K-theory, and fixed-point methods to address nontrivial weight decompositions and singular structures.
• The approach strategically uses λ₋₁ correction factors to bypass higher K-theory, enabling applications in arithmetic intersections, moduli space analysis, and motivic integration.

The Lefschetz–Riemann–Roch theorem for singular projective schemes provides a fundamental generalization of the classical Lefschetz–Riemann–Roch identities to the case of possibly singular schemes equipped with an action by a diagonalizable group scheme. The theorem, as established by Fu–Tang, extends the framework of Baum–Fulton–Quart from singular varieties to more general singular schemes, incorporating equivariant G-theory, relative K-theory, and the fixed-point theory in the context of group actions (Fu et al., 14 Dec 2025).

1. Statement of the Lefschetz–Riemann–Roch Theorem

Let $D$ be a Noetherian regular integral ring, $N \cong \mathbb{Z}/n$ a cyclic group of order $n$ invertible on $D$, and $\mu_n = \operatorname{Spec} D[N]$ the diagonalizable group scheme over $D$. The representation ring is $R(\mu_n) = K_0(D)[N] \cong K_0(D)[T]/(1-T^n)$. For a flat $R(\mu_n)$-algebra $\mathcal{R}$ where $1-T^k$ ($k=1,\ldots,n-1$) become invertible, consider separated $D$-schemes $X$ and $Y$ of finite type with $\mu_n$-actions, each admitting a $\mu_n$-projective regular envelope.

For a $\mu_n$-equivariant proper morphism $f:X \to Y$, there is a commutative “Lefschetz–Riemann–Roch” push-forward square:

$$\begin{array}{ccc} K'_0(X,\mu_n) & \xrightarrow{L.} & K'_0(X_{\mu_n},\mu_n) \otimes_{R(\mu_n)} \mathcal{R} \ \downarrow f_* && \downarrow f_{\mu_n,*} \ K'_0(Y,\mu_n) & \xrightarrow{L.} & K'_0(Y_{\mu_n},\mu_n) \otimes_{R(\mu_n)} \mathcal{R} \end{array}$$

(Theorem 4.21). In the case where $X$ is regular, an explicit formula is given:

$$L.[\mathcal{O}_X] = \lambda_{-1}^{-1}(N_{X/X_{\mu_n}}) \cap [\mathcal{O}_{X_{\mu_n}}]$$

in $K'_0(X_{\mu_n},\mu_n) \otimes \mathcal{R}$ (cf. Definition 2.35).

Given a $\mu_n$-projective regular envelope $j:X \to Z$ and any $\mu_n$-equivariant coherent sheaf $\mathcal{F}$ on $X$, the fixed-point formula (Corollary 4.22) asserts:

$$L.\left( \sum_{i \geq 0} (-1)^i R^i f_* \mathcal{F} \right) = f_{\mu_n,*} \left( \lambda_{-1}^{-1}(N_{Z/Z_{\mu_n}}) \cap \sum_{j \geq 0} (-1)^j \operatorname{Tor}_j^{\mathcal{O}_Z}(j_*\mathcal{F}, \mathcal{O}_{Z_{\mu_n}}) \right)$$

in $K'_0(Y_{\mu_n},\mu_n) \otimes \mathcal{R}$.

2. Equivariant Set-Up and Key Definitions

• $\mu_n$-scheme: A separated $D$-scheme of finite type with a $\mu_n$-action.
• $\mu_n$-projective regular envelope: A $\mu_n$-equivariant closed immersion into a $\mu_n$-projective (resp. regular) scheme (Sect 1.2).
• $K'_0(X, \mu_n)$: Grothendieck group of $\mu_n$-equivariant coherent sheaves on $X$, an $R(\mu_n)$-module (Defn 1.6).
• Relative equivariant K-group $K_X(Y, \mu_n)$: Bounded complexes of $\mu_n$-equivariant locally free sheaves on $Y$ acyclic off $X$ (Defn 2.21).
• Homology map $h$: $K_X(Y,\mu_n)\rightarrow K'_0(X,\mu_n)$, an isomorphism when $Y$ is $\mu_n$-projective regular (Prop 2.210).
• Thom–Gysin push-forward $j_*$: $K_X(Y, \mu_n)\to K_X(Z, \mu_n)$ for a closed immersion into a $\mu_n$-projective regular $Z$.
• Fixed-point subscheme $X_{\mu_n}$: Closed subscheme where $\mu_n$ acts trivially (Prop 2.31).

Weight decomposition on fibers above $X_{\mu_n}$ is crucial: for equivariant bundles $F$, restriction to $Y_{\mu_n}$ decomposes into weight-parts; the nonzero-weight part $F^{(\times)}$ enters explicitly in correction terms (Prop 2.34).

3. The Lefschetz–Riemann–Roch Identity and Functoriality

The central identity takes diagrammatic and class-formula forms, expressing how the Lefschetz functor $L.$ interfaces with push-forward in equivariant G-theory and fixed-point data (Theorem 4.21, Corollary 4.22). The correction factor $\lambda_{-1}^{-1}(N_{Z/Z_{\mu_n}})$ and cap product reflect contributions from nontrivial weights and normal bundle geometry at the fixed locus.

The methods avoid higher K-theory by grounding the construction in equivariant G-theory and direct use of $\lambda_{-1}$-classes, inverting only the “bad” weights $1-T^k$ for $k\neq 0$ (Prop 2.36). Functoriality is guaranteed by deformation to the normal cone, Koszul complexes, and weight decompositions, ensuring independence of choices of regular envelopes and compatibility for general projective morphisms.

4. Proof Methods and Technical Ingredients

• Resolution by regular envelopes: Any $X$ as above can be immersed into a $\mu_n$-projective regular $Z$ (Lemmas 2.11, 2.15–2.16).
• Reduction to relative K-theory: The main statements on G-theory are deduced from properties of relative $K_X(Y,\mu_n)$ (Prop 2.210).
• Construction of $L.$ and corrections: The functor $L.$ is defined via pull-back between relative K-groups, extension to $\mathcal{R}$, and insertion of $\lambda_{-1}^{-1}$ factors to correct for weights (Defs 3.1, 2.37).
• Functoriality by deformation to the normal cone: Verified explicitly for closed immersions (Thm 3.1, Sects 3.1–3), and by standard traces for projective morphisms (Props 4.13–4.16).
• Independence of envelopes: External products and projective space arguments (Prop 4.11, Thm 4.17).

These techniques allow bypassing the machinery of Todd classes and higher chern characters, and are tailored to the singular and equivariant setting.

5. Role of the Diagonalizable Group Scheme Action

The presence of a $\mu_n$-action endows $X$ and $Y$ with a finite weight decomposition on fibers over the fixed-point locus. The nontrivial weights in the normal bundle produce the $\lambda_{-1}^{-1}$ correction, which is only invertible after passing to a localization $\mathcal{R}$. The fixed-point subschemes $X_{\mu_n}$ and $Y_{\mu_n}$, and their normal bundles, govern the contribution of each component under push-forward (Props 2.31–2.36).

For $n=1$ (trivial group), the theorem recovers the non-equivariant Baum–Fulton–Quart Lefschetz–Riemann–Roch theorem by taking $\mu_n$ trivial. All products, pull-backs, push-forwards, and formation of fixed-point subschemes are compatible in this setting (Props 2.32–2.34).

6. Special Cases and Examples

Several key cases illustrate the scope of the theorem:

Special Case	Description	Reduction
Trivial group ($\mu_n$ acts trivially)	$X_{\mu_n} = X$, $N_{X/X_{\mu_n}} = 0$, $\lambda_{-1}^{-1} = 1$, $L.$ is identity	Ordinary Grothendieck–Riemann–Roch
$X$ smooth	$X$ can serve as its own regular envelope. Formula (⋆) recovers Donovan’s theorem	[Do]
$Y$ regular, no singularities	One may pick $Z = Y$, so the Tor-formula holds directly on $Y$ (Remark 4.23)	—
$n=1$ (no group)	$X_{\mu_n} = X$, abelian case, recovers Baum–Fulton–Quart [BFQ]	[BFQ]

For example, with $X$ a $\mu_n$-invariant hypersurface in a projective space with global action, Koszul complex computations in Section 3.1 yield the explicit fixed-point and correction terms.

7. Comparison to Classical Results and Applications

In the classical smooth case (Donovan, Baum–Fulton–Quart), equivariant algebraic K-theory $K_0(X,\mu_n)$ relies on locally free resolutions and Todd/Chern character constructions. The approach of Fu–Tang instead centers on $K_0'$ (equivariant G-theory), using $\lambda_{-1}$-classes and only inverting “bad” weights, extending the statement to arbitrary singular $X$ and $Y$ admitting regular envelopes.

Applications include:

• Explicit Lefschetz fixed-point formulas for coherent sheaves (Corollary 4.22).
• Thomason’s fixed-point formula after removing projectivity via Chern character into higher K-theory or $\ell$-adic cohomology ([Th, 3.5]).
• Arithmetic intersections in Arakelov geometry (Köhler–Roessler [KR]), moduli spaces with torus actions, and trace computations in motivic integration.

This advances the toolkit for handling singularities and group actions in modern intersection theory and geometry, unifying prior approaches and extending their field of applicability in both algebraic and arithmetic geometry contexts (Fu et al., 14 Dec 2025).