Lyubeznik Functors in Local Cohomology

• Lyubeznik functors are additive covariant functors derived from local cohomology and exact sequences that unify and generalize invariants of Noetherian rings.
• They facilitate precise support comparisons under ring extensions by ensuring Zariski-closedness through control of minimal primes and leveraging flat and pure extensions.
• Connecting local cohomology with D-module theory, Lyubeznik functors enable computation of generalized Lyubeznik numbers and offer insights into singularity and invariants analysis.

A Lyubeznik functor is an additive covariant functor constructed systematically from local cohomology theories and the connecting morphisms in their long exact sequences. Originating in the paper of invariants of local rings via local cohomology, Lyubeznik functors provide a unified framework for generalizing Lyubeznik numbers, facilitate the comparison of supports under various types of ring extensions, and establish deep connections to D-module theory and singularity theory. The structure and properties of these functors are instrumental in understanding subtle algebraic and geometric features of Noetherian rings.

1. Formal Definition and Construction

Given a commutative Noetherian ring $R$ and the category $\Mod(R)$ of $R$-modules, Lyubeznik functors are defined via local cohomology and exact sequences. For a closed subset $Z \subseteq \Spec R$, the local cohomology functor $H^i_Z(-)$ is given by

$H^i_Z(M) = \varinjlim_{Z \subseteq V(I)} H^i_I(M),$

with $H^i_I(M)$ denoting the $i$-th local cohomology module with support in the ideal $I$. For each pair of closed subsets $Z_1 \subseteq Z_2$, there is a natural long exact sequence

$$\cdots \longrightarrow H^i_{Z_1}(M) \longrightarrow H^i_{Z_2}(M) \longrightarrow H^i_{Z_2 \setminus Z_1}(M) \longrightarrow H^{i+1}_{Z_1}(M) \longrightarrow \cdots.$$

A Lyubeznik functor $T \colon \Mod(R) \to \Mod(R)$ is any finite composite,

$T = T_t \circ T_{t-1} \circ \cdots \circ T_1,$

where each $T_j$ is either some $H^i_Z(-)$ or a kernel, image, or cokernel of one of the maps in the above exact sequence. These functors are additive and respect short exact sequences by construction.

2. Support and Zariski Closedness

For any $R$-module $M$, the support is

$\Supp_R(M) = \{ \mathfrak p \in \Spec R : M_{\mathfrak p} \neq 0 \}.$

This set is specialization-closed; it is Zariski-closed if and only if there exists an ideal $I$ such that $\Supp_R(M) = V(I)$. The following equivalence holds for Noetherian $R$: $\Supp_R(M) \text{ is Zariski-closed} \iff \min_R(M) < \infty,$ where $\min_R(M)$ are the minimal primes in the set of associated primes $\Ass_R(M)$. When closed,

$\Supp_R(M) = V\left( \bigcap_{\mathfrak p \in \min_R(M)} \mathfrak p \right).$

Thus, for Lyubeznik functors $T$, the Zariski-closedness of $\Supp_R T(R)$ equates to the finiteness of minimal primes in $T(R)$.

3. Behavior Under Ring Extensions

The behavior of Lyubeznik functors under extensions $R \to S$ of Noetherian rings is determined by the properties of the extension.

3.1. Flat and Faithfully Flat Extensions

If $R \to S$ is flat,

$T(S) \cong T(R) \otimes_R S,$

and Zariski-closedness of support ascends: if $\Supp_R(E)$ is closed, so is $\Supp_S(E \otimes_R S)$. A converse holds under a finiteness condition; if only finitely many $\mathfrak p \in \min_R(E)$ are not proper in $S$, closedness descends. In the faithfully flat case, there is equivalence: $\Supp_R T(R) \text{ is closed} \iff \Supp_S T(S) \text{ is closed}.$ The criterion relies on controlling minimal primes through flat base change.

3.2. Pure and Cyclically Pure Extensions

An extension $R \to S$ is pure if $M \to M \otimes_R S$ is injective for every $R$-module $M$. Cyclically pure means that the extension splits after tensoring with every cyclic module $R/I$. For pure $R \to S$, if $\Supp_S(H^i_I(M) \otimes_R S)$ is Zariski-closed, so is $\Supp_R(H^i_I(M))$. If $(R, \mathfrak m)$ is local and $R \to S$ is pure, then

$\Supp_S T(S) \text{ closed} \implies \Supp_R T(R) \text{ closed},$

for any Lyubeznik functor $T$. Cyclically pure extensions in analytically unramified local rings also yield purity, and the same descent of closedness applies.

3.3. Direct Summand Extensions

If $R \to S$ splits as $R$-modules, $S \cong R \oplus U$ for some $S$-ideal $U$, then $R$ is pure in $S$, and

$\Supp_R T(R) \text{ is closed} \iff \Supp_S(T(R) \otimes_R S) \text{ is closed}.$

There is always containment

$\Supp_S(T(R)\otimes_R S) \subseteq \Supp_S T(S),$

but equality requires additional conditions (e.g., flatness). This enables checking closedness in the larger ring $S$ for direct-summand extensions.

4. Lyubeznik Functors and D-Module Theory

The functor $G : R\text{-Mod} \to D$, where $S = R[[x]]$, sends $M$ to $G(M) = M \otimes_R (S_x/S)$, with $S_x/S$ constructed as a direct sum of $R$-modules indexed by powers of $x^{-1}$. $G(M)$ admits a left action by the ring of $R$-linear differential operators $D(S,R)$. The functor $G$ is exact and establishes an equivalence of categories between $R$-modules and $D(S,R)$-modules supported on $x$. The quasi-inverse is $\widetilde G(N) = \Ann_N(xS)$. This equivalence allows for precise control over lengths and exactness properties between the module categories, and enables reinterpretation of local cohomology modules as holonomic $D$-modules with finite length.

Specific functorial properties include:

• Exactness: $G$ is exact by flatness.
• Compatibility: $G(H^i_I(M)) \cong H^{i+1}_{(I,x)S}(M \otimes_R S)$.
• Length preservation: $\length_R(M) = \length_{D(S,R)}(G(M))$.
• Behavior on injectives: Gorenstein $R$ implies $G$ sends injectives to injectives and vice versa.

5. Applications and Illustrative Examples

Lyubeznik functors are central in the following settings:

Context	Feature	Implication
Polynomial extensions	Faithfully flat	Closedness of support ascends and descends
Galois invariants	$S^G$ finite-projective over $S$	Closedness in regular $S$ induces closedness in $R = S^G$
Smooth algebras over mixed char.	Example from Bhattacharyya	All supports of $H^i_I(R)$ are closed
Cyclic purity	Inclusion of normal into integral extension	Cyclic purity yields descent of closedness
Stanley–Reisner rings	Explicit Lyubeznik characteristic	$$\chi_\lambda(R_m) = \sum_{i=-1}^n (-2)^{i+1}|F_i(\Delta)|$$

Applications involve tracking minimal primes under extension using flat base change formulas, injectivity and splitting results under purity, and leveraging D-module length invariants for local cohomology.

6. Generalized Lyubeznik Numbers and Invariants

The lengths of iterated local cohomology modules in the D-module category define generalized Lyubeznik numbers,

$\lambda^{i_s,\dots,i_1}_{I_s,\dots,I_1}(R) := \length_{D(S,K)}\left( H^{i_s}_{J_s}\cdots H^{n-i_1}_{J_1}(S) \right),$

for $S = K[[x_1,\dots,x_n]] \to \hat{R}$, with ideals $J_j$ pulled back from $I_j$ in $R$. These invariants are independent of choices and capture finer algebraic information than the original Lyubeznik numbers, including connections to F-singularities in positive characteristic and characteristic cycle multiplicities in characteristic zero.

Specific features:

• For determinantal ideals, the values can differ sharply between characteristic zero and positive characteristic.
• For monomial ideals, lengths can be computed explicitly using graded components.
• Stanley–Reisner rings admit exact formulas for Lyubeznik characteristics via face counting.

A plausible implication is that the Lyubeznik functor unifies the paper of local cohomology supports, D-module invariants, and singularity properties, facilitating further generalizations and refinements of ring-theoretic invariants.

7. Connections and Refinements

The development of Lyubeznik functors establishes equivalence-of-categories results, well-definedness of invariants, and allows for iteration of local cohomology along arbitrary chains of ideals. In characteristic zero, Lyubeznik numbers provide lower bounds for D-module multiplicities, attaining equality for monomial ideals. In positive characteristic, the length-one property of certain cohomology modules detects F-regularity or F-rationality. The refinement of the original framework substantially broadens the scope and utility of Lyubeznik-type invariants.