Operational K-Theory

• Operational K-Theory is a bivariant extension of classical K-theory characterized by natural transformation operators that act on Grothendieck groups of coherent sheaves.
• It generalizes the K-theory of smooth varieties to singular and equivariant settings by incorporating descent, homotopy invariance, and duality principles.
• The framework employs localization and combinatorial models, such as piecewise exponential functions on toric and GKM varieties, for explicit computational applications.

Operational $K$-theory is the bivariant theory associated to the Grothendieck group of coherent sheaves, constructed via the formalism of Fulton–MacPherson, and provides contravariant and bivariant generalizations of algebraic $K$-theory. It is characterized by a collection of natural transformation operators acting covariantly with respect to proper maps and contravariantly with respect to flat morphisms and regular embeddings on the Grothendieck groups $K_0$ of coherent sheaves (or $T$-equivariant coherent sheaves). Operational $K$-theory recovers the usual $K$-theory on smooth varieties and provides a more robust framework for singular (especially equivariant) contexts, as well as enabling connections to localization, duality, and piecewise-exponential function rings on combinatorial models of varieties such as toric or Schubert varieties.

1. Bivariant Formalism and Definition

Let $f:X\to Y$ be a morphism of separated finite-type schemes over a fixed field $k$, possibly equipped with a torus $T$-action. The group $\operatorname{op} K_T^*(X\rightarrow Y)$ consists of collections of operators

$K$0

for every pullback square

$K$1

satisfying

• (A1) Proper push-forward compatibility:

$K$2

• (A2) Gysin (refined pullback) compatibility for flat maps and regular embeddings:

$K$3

The collection of these groups, with product, pushforward, and pullback (as above), forms a bivariant theory in the sense of Fulton–MacPherson.

For the identity morphism $K$4, one defines the (contravariant) operational $K$5-theory: $K$6 which is a commutative, associative $K$7-algebra with unit ($K$8 the representation ring).

On smooth schemes, operational $K$9-theory recovers the Grothendieck ring of vector bundles, i.e.,

$K_0$0

as $K_0$1-modules (Anderson et al., 2013).

2. Algebraic and Geometric Properties

Operational $K_0$2-theory is constructed to satisfy descent, homotopy invariance, and appropriate duality:

• Descent for Envelopes: For a (possibly equivariant) envelope $K_0$3, the sequence

$K_0$4

is exact (Anderson et al., 2013, Gonzales, 2014), paralleling Gillet's exact descent for Chow theory.

• $K_0$5-Homotopy Invariance: For any $K_0$6-equivariant affine bundle $K_0$7, pullback induces an isomorphism

$K_0$8

and in particular,

$K_0$9

(Anderson et al., 2013, Gonzales, 2014).

• Duality for T-linear Varieties: For a complete T-linear variety $T$0,

$T$1

by using Künneth-type decompositions (Anderson et al., 2013).

3. Explicit Descriptions: Toric, Spherical, and GKM Varieties

Operational $T$2-theory admits explicit combinatorial descriptions on highly structured varieties:

• Toric Varieties: For a toric variety $T$3 (possibly singular),

$T$4

where $T$5 is the ring of integral piecewise exponential functions on the fan $T$6; this generalizes the GKM/Klyachko description for smooth cases (Anderson et al., 2013, Gonzales, 2014).

• GKM Varieties and Chang–Skjelbred Property: For a $T$7-skeletal (GKM) variety, the restriction map

$T$8

identifies operational $T$9-theory as the subring of tuples $K$0 where

$K$1

for each $K$2-invariant curve of weight $K$3 joining $K$4 and $K$5 (Gonzales, 2014). The image is the intersection of analogous images for all codimension one subtori (Chang–Skjelbred).

• Spherical Varieties: For a complete spherical $K$6-variety,

$K$7

with image determined by explicit divisibility and congruence relations dictated by the weights of $K$8-invariant curves and higher-dimensional $K$9-stable loci (Anderson et al., 2019).

4. Bivariant Grothendieck Transformations and Riemann–Roch

Operational $K$0-theory admits natural bivariant Chern character and Riemann–Roch transformations:

• Grothendieck Transformation: There are natural bivariant transformations:

$K$1

where the first map is completion and tensoring with $K$2, and the second is the bivariant Chern character, an isomorphism (Anderson et al., 2019).

• Bivariant Riemann–Roch: For equivariant lci morphisms $K$3, the specialized Riemann–Roch formula holds:

$K$4

in the operational equivariant Chow theory.

• Adams Operations and Localization: Bivariant Adams operations $K$5 act on operational $K$6-theory, and there exist equivariant localization isomorphisms after inverting the multiplicative set generated by $K$7 for all characters $K$8 (Anderson et al., 2019).

5. Computational Techniques and Applications

Operational $K$9-theory provides a powerful computational toolkit for singular and equivariant algebraic geometry:

• Localization and Fixed Point Theorems: A Borel–Atiyah–Segal type localization theorem for operational $f:X\to Y$0-theory holds for arbitrary $f:X\to Y$1-schemes, enabling reduction to fixed-point loci computations and combinatorial models (Gonzales, 2014, Anderson et al., 2019).
• Envelope Descent and Singular Varieties: Gillet–Kimura's cohomological descent (envelope descent) furnishes exact sequences relating the operational $f:X\to Y$2-theory of singular schemes to smooth models, enabling inductive calculations (Anderson et al., 2013, Gonzales, 2014).
• Explicit Cases and Failures of Surjectivity: For certain projective toric threefolds, $f:X\to Y$3 does not surject onto $f:X\to Y$4, with operational $f:X\to Y$5-theory capturing more refined phenomena due to singularities (Anderson et al., 2019).
• Extensions: Operational $f:X\to Y$6-theory has been generalized to derived schemes, with bivariant theories defined via perfect complexes and Grothendieck transformations linking algebraic and operational $f:X\to Y$7-theory [(Anderson et al., 2019), Appendix B].

6. Relation to Other Theories and Distinctions

Operational $f:X\to Y$8-theory is a bivariant extension of algebraic $f:X\to Y$9-theory, designed to be parallel to operational Chow theory yet retaining subtle $k$0-theoretic features:

Property	Operational $k$1-theory	Operational Chow theory
Structure	Bivariant with descent, duality	Bivariant, similar pattern
Behavior on smooth varieties	Recovers $k$2	Recovers Chow cohomology
Combinatorial description	Piecewise exponentials (toric/GKM)	Piecewise polynomials
Singularities	Detects more $k$3-torsion, negative $k$4-theory	Simpler rationally

Operational $k$5-theory often involves richer $k$6-torsion and accommodates subtleties arising from negative $k$7-theory and higher homotopy invariants, especially in singular contexts (Anderson et al., 2013).

7. Future Directions and Open Problems

Several generalizations and open questions remain:

• Extensions to more general classes of algebraic stacks and derived algebraic geometry have been explored, with Grothendieck transformations from bivariant $k$8-theory of relatively perfect complexes to operational $k$9-theory (Anderson et al., 2019).
• Classification of operational $T$0-theory rings for increasingly general classes of singular varieties remains open.
• Determining the precise relationship and difference between operational and classical $T$1-theory (e.g., surjectivity and injectivity of natural maps) in various equivariant and singular settings is an ongoing area of research.
• The piecewise-exponential presentation for broader classes of varieties, particularly spherical, group embeddings, and Schubert varieties continues to be developed for geometric and combinatorial applications (Gonzales, 2014, Anderson et al., 2019).

Operational $T$2-theory serves as a bivariant, functorial, and combinatorially explicit framework encompassing and extending much of algebraic $T$3-theory, equivariant geometry, and localization, with an active research program investigating its limits and applications.