Infinitesimal Motivic Complexes

• Infinitesimal Motivic Complexes are specialized complexes that model first-order square-zero deformations in weight-two motivic cohomology.
• They employ an adapted Bloch complex and dual regulators that link differential form data with André–Quillen homology, offering concrete deformation-theoretic insights.
• Their explicit additive dilogarithm construction and cyclic homology comparisons underline a method to detail and compute infinitesimal motivic invariants.

Searching arXiv for the specified paper and closely related work on infinitesimal motivic/cohomological complexes. Infinitesimal motivic complexes are complexes designed to capture the first-order, square-zero deformation part of motivic cohomology. In the weight-two case studied in "Infinitesimal Bloch regulator" (Unver, 2019), they arise from a Bloch-complex construction adapted to a square-zero thickening $\underline X \hookrightarrow X$ of a smooth $k$-scheme, with $k$ of characteristic $0$. The resulting theory isolates an “infinitesimal part” of ${\rm H}^2_M(X,\mathbb Q(2))$, constructs regulators from it to a Kähler-differential quotient and to first André–Quillen homology, and shows that after removing the differential-form contribution, the remaining infinitesimal motivic cohomology is identified with $H^1(X,D_1(\mathcal O_X))$ (Unver, 2019). In this sense, infinitesimal motivic complexes provide a first-order deformation-theoretic analogue of classical weight-two motivic complexes and regulators.

1. Geometric setting and basic idea

The geometric input is a scheme $X/k$ of finite type over a field $k$ of characteristic $0$, together with a closed immersion

$\underline X \hookrightarrow X$

defined by a square-zero ideal sheaf $k$0 satisfying $k$1. The closed subscheme $k$2 is assumed smooth over $k$3, and the conormal sheaf is assumed locally free; equivalently, the infinitesimal thickening is Zariski locally split (Unver, 2019). This is a first-order deformation problem rather than a theory of arbitrary nilpotent thickenings.

Within this setup, the central objective is to construct an infinitesimal analogue of Bloch’s regulator. The classical regulator

$k$4

is replaced by a pair of regulators defined on the infinitesimal part of weight-two motivic cohomology. The first lands in a quotient of Kähler differentials, and the second lands in a cohomology group of the first André–Quillen homology sheaf (Unver, 2019).

A key structural feature is that the infinitesimal part of weight-two motivic cohomology splits into two pieces. One piece is detected by differential forms, while the kernel of that first detection map is controlled exactly by André–Quillen homology. This suggests that, in the square-zero setting, infinitesimal motivic cohomology admits a deformation-theoretic description that is much more explicit than its classical counterpart.

2. Weight-one and weight-two infinitesimal motivic complexes

The weight-one case serves as the basic model. If $k$5 is regular, the weight-one motivic complex is quasi-isomorphic to $k$6. For the thickening $k$7, the infinitesimal weight-one complex is the cone of

$k$8

written as $k$9, and its hypercohomology defines

$k$0

(Unver, 2019).

The weight-two object is built from the Bloch complex. For a ring $k$1, one defines $k$2 as the $k$3-vector space generated by symbols $k$4, $k$5, modulo the five-term relation

$k$6

whenever the expression makes sense. The Bloch complex is

$k$7

with $k$8 in degree $k$9 (Unver, 2019).

If $0$0 is nilpotent and $0$1, then the cone of $0$2 is the infinitesimal complex $0$3, explicitly identified with

$0$4

where $0$5 denotes the kernel of reduction to $0$6. After sheafification this gives the infinitesimal motivic complex $0$7, and its hypercohomology

$0$8

is the infinitesimal analogue of $0$9-motivic cohomology (Unver, 2019).

The construction is therefore not a general replacement for motivic cohomology in all weights. It is a precise weight-two theory tailored to first-order thickenings, with the Bloch complex providing the basic symbolic input.

3. Regulators and the decomposition of the infinitesimal part

The first regulator is defined using the infinitesimal differential quotient

${\rm H}^2_M(X,\mathbb Q(2))$0

It is a map

${\rm H}^2_M(X,\mathbb Q(2))$1

coming from the map of complexes

${\rm H}^2_M(X,\mathbb Q(2))$2

(Unver, 2019).

The second regulator is defined on the kernel of the first: ${\rm H}^2_M(X,\mathbb Q(2))$3 where ${\rm H}^2_M(X,\mathbb Q(2))$4 is the Zariski sheaf associated to first André–Quillen homology (Unver, 2019). For a commutative ${\rm H}^2_M(X,\mathbb Q(2))$5-algebra ${\rm H}^2_M(X,\mathbb Q(2))$6,

${\rm H}^2_M(X,\mathbb Q(2))$7

and for ${\rm H}^2_M(X,\mathbb Q(2))$8 and ${\rm H}^2_M(X,\mathbb Q(2))$9 one writes $H^1(X,D_1(\mathcal O_X))$0. If $H^1(X,D_1(\mathcal O_X))$1 with $H^1(X,D_1(\mathcal O_X))$2 smooth over $H^1(X,D_1(\mathcal O_X))$3, then

$H^1(X,D_1(\mathcal O_X))$4

so the target can be described concretely by the cotangent sequence (Unver, 2019).

The main theorem identifies the kernel of $H^1(X,D_1(\mathcal O_X))$5 with the hypercohomology of a subcomplex $H^1(X,D_1(\mathcal O_X))$6, obtained by replacing $H^1(X,D_1(\mathcal O_X))$7 with $H^1(X,D_1(\mathcal O_X))$8: $H^1(X,D_1(\mathcal O_X))$9 Moreover,

$X/k$0

is an isomorphism (Unver, 2019).

This is the central structural result. The infinitesimal part of weight-two motivic cohomology is governed first by differential forms and then, on the residual kernel, by André–Quillen homology. A plausible implication is that the deformation-theoretic content of $X/k$1-type motivic classes becomes accessible through sheaf-theoretic linear invariants once one passes to square-zero thickenings.

4. Additive dilogarithm and local-to-global construction

The main local tool is a generalization of the additive dilogarithm. For a local $X/k$2-algebra $X/k$3 with square-zero ideal $X/k$4, with $X/k$5 smooth over $X/k$6, and a chosen $X/k$7-algebra splitting $X/k$8 of the projection $X/k$9, the paper constructs

$k$0

(Unver, 2019).

On generators $k$1, using a lift $k$2 of $k$3 to a smooth presentation $k$4, the construction sends $k$5 to an element represented by

$k$6

in

$k$7

where $k$8 is the image of $k$9 under the chosen splitting (Unver, 2019). A second construction interprets the same map more conceptually by integrating the infinitesimal form

$0$0

against a suitable antiderivative. The five-term functional equation for the additive dilogarithm is proved, so the formula descends from symbols to the Bloch group $0$1 (Unver, 2019).

The globalization step compares different local splittings. On overlaps, local branches are related by homotopy maps

$0$2

constructed from an explicit formula measuring the difference between the two branches. These homotopies ensure that local cocycles patch into a global class in $0$3, which yields a globally well-defined and functorial $0$4 (Unver, 2019).

A common misconception is to treat the infinitesimal regulator as merely a formal linearization of the classical dilogarithm. The construction is more rigid: the five-term relation, the explicit symbolic formula, and the homotopy comparison of splittings are all essential to descend from local symbolic data to a global cohomology class.

5. Goodwillie, cyclic homology, and the isomorphism theorem

The identification of $0$5 as an isomorphism depends on comparison with cyclic homology via Goodwillie’s theorem. For a nilpotent extension, Goodwillie identifies the infinitesimal part of algebraic $0$6-theory with cyclic homology, and in the setting of a smooth algebra $0$7 with square-zero ideal $0$8,

$0$9

(Unver, 2019).

Under the additional assumption that $\underline X \hookrightarrow X$0 is free as an $\underline X \hookrightarrow X$1-module, the composite

$\underline X \hookrightarrow X$2

is an isomorphism (Unver, 2019). The key point is that the regulator on the infinitesimal part of cyclic homology is identified with multiplication by $\underline X \hookrightarrow X$3 on a symmetric cube model, hence invertible over $\underline X \hookrightarrow X$4, and the induced map on the motivic side is therefore an isomorphism as well (Unver, 2019).

This comparison is conceptually significant because it links symbolic motivic data, represented by Bloch-group generators and five-term relations, to the linearized deformation theory encoded by André–Quillen homology. This suggests that infinitesimal motivic complexes occupy an intermediate position between motivic cohomology and homological invariants such as cyclic homology.

The result should not be conflated with a full identification of motivic cohomology and André–Quillen homology. The isomorphism concerns the kernel of the first regulator in the specific square-zero, weight-two setting.

6. Relation to Deligne–Vologodsky and to broader motivic-complex formalisms

When $\underline X \hookrightarrow X$5 is smooth over the dual numbers $\underline X \hookrightarrow X$6, the theory is reinterpreted in terms of the infinitesimal Deligne–Vologodsky crystalline complex $\underline X \hookrightarrow X$7. In this setting,

$\underline X \hookrightarrow X$8

so the same two target pieces reappear in crystalline form (Unver, 2019). Vologodsky’s theorem gives an abstract isomorphism between the infinitesimal part of $\underline X \hookrightarrow X$9-theory and the hypercohomology of $k$00, although the explicit regulator constructed in (Unver, 2019) is not fully identified there with Vologodsky’s Chern-character map.

Within the broader theory of motivic complexes, infinitesimal motivic complexes occupy a specialized rather than universal role. "A commutative $k$01-spectrum representing motivic cohomology over Dedekind domains" (Spitzweck, 2012) constructs a motivic Eilenberg–MacLane spectrum representing Levine’s motivic cohomology via Bloch–Levine cycle complexes, étale sheaves, de Rham–Witt objects, and an arithmetic square. That work supplies a global spectrum-level formalism for motivic cohomology over Dedekind domains, including localization and six functors (Spitzweck, 2012). By contrast, the infinitesimal Bloch-regulator construction is a first-order deformation-theoretic refinement in weight two, specialized to square-zero thickenings (Unver, 2019).

Likewise, "Motivic complexes and special values of zeta functions" (Milne et al., 2013) enhances étale motivic complexes with a $k$02-integral structure using a dg fibred product involving the Raynaud ring and crystalline or rigid realizations. That paper is not organized around infinitesimal thickenings, though it incorporates de Rham–Witt and crystalline data (Milne et al., 2013). This contrast helps delimit the term “infinitesimal motivic complexes”: in (Unver, 2019) it refers specifically to complexes modeling first-order nilpotent deformation data, not to every motivic formalism with crystalline or $k$03-adic content.

7. Significance, scope, and limitations

The principal contribution of the weight-two theory is the construction of an explicit, functorial infinitesimal Bloch regulator for square-zero thickenings of smooth schemes, together with a concrete additive dilogarithm formula and an isomorphism theorem identifying the residual infinitesimal part with André–Quillen homology (Unver, 2019). The resulting picture is that the infinitesimal part of $k$04 is governed by two layers: $k$05 and then on the kernel,

$k$06

This gives a workable model for infinitesimal motivic complexes in weight two: a Bloch complex adapted to first-order thickenings, with hypercohomology detecting the infinitesimal part of $k$07-motivic cohomology and with explicit interfaces to cyclic homology, Goodwillie’s theorem, and the crystalline Deligne–Vologodsky complex (Unver, 2019).

The scope of the theory is nevertheless narrow in several respects. It is formulated for characteristic $k$08, for square-zero thickenings with locally free conormal sheaf, and in practice for the first-order deformation situation in which the thickening is locally split (Unver, 2019). It is also a weight-two construction. A plausible implication is that higher-weight infinitesimal motivic complexes would require additional symbolic and homotopical input beyond the Bloch-group framework used here.

In that form, infinitesimal motivic complexes should be understood as a deformation-theoretic refinement of motivic cohomology rather than a replacement for general motivic-complex formalisms. Their distinctive feature is the explicit passage from symbolic motivic relations to linear infinitesimal invariants.