nLab generalized lambda-structure

Overview

The usual notion of Lambda-ring is directly related to the Banach ring $(\mathbb{Z},|\cdot|_0)$ of integers equipped with their trivial norm in the following way: a Lambda-ring is a usual ring equipped with an action of the monoid $\mathbb{N}=\mathbb{Z}$-$\{0\}/\{\pm 1\}$. Remark that some important Lambda-rings, such as K-theory, are actually equipped with an additional $\mathbb{Z}/2$-grading, that may be combined with the Lambda-structure to get an action of the full monoid $\mathbb{Z}$-$\{0\}$. It is important to remark here that the $\Lambda$-structure on $K$-theory allows one to get back (as the spectrum of the Lambda-operations) the full $\mathbb{Z}$-grading on Betti cohomology.

From the perspective of global analytic geometry, one thinks of Lambda-structures as related to the Banach ring of integers with their trivial norm, so that one may seek for various generalizations, that will be called “generalized Lambda-structure”, associated to more general Banach or ind-Banach rings.

Definition

Let $(R,|\cdot|)$ be an integral Banach ring equipped with a multiplicative norm. We will denote $\Lambda(R,|\cdot|)$ the monoid given by

$$\Lambda(R,|\cdot|):=\mathrm{Frac}(R)\cap \{a\in R,\;|a|\leq 1\}.$$

Examples

There is not yet a precise notion of generalized Lambda-structure, but one may easily give various of its concrete incarnations.

1. The classical notion of ($\mathbb{Z}/2$-graded) Lambda-ring may be seen as a $\Lambda(\mathbb{Z},|\cdot|_0)=\mathbb{Z}$-$\{0\}$-structure. The (geometric/Weil) cohomology theories of arithmetic geometry (i.e. for schemes over $\mathbb{Z}$) are often equipped with an $\mathbb{N}$-grading, that one may interpret as a classical Lambda-ring structure.

2. The absolute cohomology theories in arithmetic geometry such as Beilinson-Deligne cohomology or motivic cohomology are equipped with a natural bi-graduation, related to the fact that they are defined by the (homotopical: think of a Leray spectral sequence) combination of geometric methods (Lambda-structures/Frobenii) and of differential methods (Hodge filtration of de Rham cohomology or $\mathbb{G}_m$-stabilization in motivic homotopy theory, that corresponds to the “Tate twist” grading). The corresponding Banach ring may be simply given by the Banach ring

   $$(R,|\cdot|):=(\mathbb{Z}[T],|\cdot|_0)$$

   of polynomials equipped with their trivial norm (analytic functions on the non-archimedean global analytic unit disc). The bigrading is given by the action of the monoid

   $$\Lambda(R,|\cdot|):= \mathbb{Q}(T)^\times\cap \mathbb{Z}[T]=\mathbb{Z}[T]-\{0\}.$$

   The (Tate twist) motivic and cohomological gradings are given respectively by the actions of the monoid $\mathbb{Z}-\{0\}$ and the monoid of powers of $T$.

3. The (yet to be properly defined) cohomology theories in global analytic geometry have a different type of bigrading (that is related to the idea of the algebra of polynomials over the field with one element, formulated precisely, e.g., in Durov’s setting of generalized rings, i.e., commutative algebraic monads). We will now extend the above definition of the monoid $\Lambda$ to the setting of ind-Banach ring, since this operation seems necessary to understand absolute cohomologies. The corresponding (ind-)Banach ring may be simply given by the ind-Banach ring

   $$R:=\mathbb{Z}\{T\}^\dagger$$

   of overconvergent power series on the unit disc with coefficients in the Banach ring $(\mathbb{Z},|\cdot|_\infty)$: the geometric classical Lambda-structure is given by the base Banach ring, and the differential/absolute graduation is given by the $T$-part of the monoid (we may need to make a completion here)

   $$\Lambda(R):= \mathrm{Frac}(\mathbb{R}\{T\}^\dagger)^\times\cap \{P\in \mathbb{Z}\{T\}^\dagger,|P|_{\infty,1}\leq 1\}=\mathbb{F}_{\{\pm 1\}}[T]-\{0\}.$$

   This is the monoid of polynomials whose terms are all equal to zero except exactly one, that is equal to $\{\pm 1\}$. It contains and extends the monoid $\Lambda(\mathbb{Z},|\cdot|_\infty)=\{\pm 1\}=(\mathbb{F}_{\{\pm 1\}})^\times$ in degree zero. This will be the natural grading monoid (generalized Lambda-structure) for absolute motives, i.e., motivic cohomology theories over $(\mathbb{Z},|\cdot|_\infty)$. Remark that the recent work of Peter Scholze on local Schtukas in mixed characteristic also uses in an essential way objects such as the unit disc over the given base Banach ring.

4. The notion of $\Lambda(\mathbb{Z},|\cdot|_\infty)=\{\pm 1\}$-structure is simply given by the notion of $\mathbb{Z}/2$-grading. Many cohomological invariants, such as K-theory, negative cyclic homology and the Chern character are equipped with a natural $\mathbb{Z}/2$-grading. It is quite probable that one can’t hope to get something more that a $\mathbb{Z}/2$-grading on a “really natural” cohomology theory in global analytic geometry.

5. In the theory of $(\Phi,\Gamma)$-modules, the monoid $\Lambda(\mathbb{Z}_p,|\cdot|_p)=\mathbb{Z}_p$-$\{0\}$ plays a central role. It looks like a not so hard but important task to clarify the relation of this theory with the classical notion of Lambda-ring.

6. It is an interesting question to try to understand the relation of classical Hodge theory (over $\mathbb{R}$ or $\mathbb{C}$) with the notion of Lambda-structure on the corresponding Banach ring. This may show interesting limits to the idea of generalizing Lambda-structures to other Banach rings. The case of $\mathbb{R}$ may (or may not) be treated using $\mathbb{Z}/2$-equivariant methods. An important point, in this perspective, is that the naive archimedean generalization of the notion of $(\Phi,\Gamma)$-module does not work, because $S^1$ does not act directly on the open complex unit disc $D^\circ(1,1)$. One only has an infinitesimal action (connection $\nabla$), whose combination with the infinitesimal generator $\Phi$ of $\mathbb{R}_+^*$ may be seen as an archimedean analog of the $p$-adic differential equations used in Berger’s thesis to prove the monodromy theorem of $p$-adic Hodge theory. An important drawback of this infinitesimal approach (in the $p$-adic setting) is that the functor from $p$-adic Hodge structures (i.e., $(\Phi,\Gamma)$-modules) to $p$-adic Frobenius-equivariant differential equations is not fully faithful: making the action of $U(1)=\mathbb{Z}_p^*$ infinitesimal kills an important part of the information (essentially, the Hodge filtration on de Rham cohomology). A possible solution to this problem may be to work with a multiplicative theory over $[\mathbb{A}^1/\mathbb{G}_m]$ instead of an additive one over $[D^\circ(1,1)/\Lambda]$ or $[D^\circ(1,1)/(\Phi,\nabla)]$, or to use a combination of the additive and multiplicative theory.

7. The monoid that should come in play into the theory of spectral interpretation for zeroes and poles of global arithmetic and automorphic L-functions may be given by the monoid $\Lambda(\mathbb{A})$, where $\mathbb{A}$ is the ind-Banach ring of adèles.

Geometric interpretation (to be checked very carefully: may be problematic)

One may take inspiration from the theory of $(\Phi,\Gamma)$-modules ($p$-adic Hodge structures) to define a natural notion of $\Lambda$-module in global analytic geometry. This gives a version of the notion of a “Hodge structure” that works over an integral base, which makes it quite well adapted to the global analytic situation.

Let $R$ be a Banach ring, and $\overset{\circ}{D}(1,1)$ be the open unit disc on $R$. We denote (be careful, this differs from the previously used notation, because it is a different kind of object)

$$\Lambda\subset \overset{\circ}{D}(1,1)\times \overset{\circ}{D}(1,1)$$

the (non-strict) analytic subgroupoid of the groupoid of pairs acting on $\overset{\circ}{D}(1,1)$ given by pairs of the form $(1+x,(1+x)^a)$ where $a\in D(0,1)\cap \GL_1\subset \mathbb{A}^1$, and

$$(1+x)^a:=\sum_{k\geq 0}\binom{a}{k}x^k.$$

If $R=(\mathbb{Q}_p,|\cdot|_p)$, then we have actually that $\mathbb{Z}_p-\{0\}\subset D(0,1)(\mathbb{Q}_p)$.

A $\Lambda$-module over $R$ is a module over the analytic stack $B\Lambda$ that one may denote as a quotient stack $[\overset{\circ}{D}(1,1)/\Lambda]$. We then have, if $R$ contains the rational numbers, a natural logarithm map

$$log:[\overset{\circ}{D}(1,1)/\Lambda]\to [\mathbb{A}^1/\GL_1]$$

that allows us to give a relation between the classical Hodge filtration of a (say) proper or logarithmically proper analytic space over $R$ to its $R$-Hodge structure, that should be a module over $[\overset{\circ}{D}(1,1)/\Lambda]$.

If we suppose given a (say) strict analytic space over $R$, and one wants to define the associated $R$-Hodge structure, one may simply try to adapt Simpson’s construction of the deformation to the normal bundle, to get what one wants. Actually, one needs a loop space analog of this construction, that is due to Vezzosi for a derived scheme. Recall that in this derived scheme case, we have

$$LX=\Hom_{dSt}(B\mathbb{Z},X)\cong \Hom_{dSt}(B\mathbb{G}_a,X).$$

We may use the action by multiplication of $\mathbb{A}^1$ on $B\mathbb{G}_a$ to define a family of actions of $B\mathbb{G}_a$ on $LX$, parametrized by $\mathbb{A}^1$. This gives a $\mathbb{G}_m$-equivariant family

$$Hod(LX)\to \mathbb{A}^1$$

whose fiber at $0$ is $LX$ with the trivial action of $B\mathbb{G}_a$ and whose fiber at $1$ is $LX$ equipped with the usual action of $B\mathbb{G}_a$.

If we want to define a construction that is related to the loop space Hodge filtration through the logarithm map

$$log:[\overset{\circ}{D}(1,1)/\Lambda]\to [\mathbb{A}^1/\mathbb{G}_m],$$

we need to replace $B\mathbb{G}_a$ by $BD^\circ:=B\overset{\circ}{D}(1,1)$, and the multiplicative action of $\mathbb{A}^1$ on $B\mathbb{G}_a$ by the (partial) action of $D(0,1)$ on $BD^\circ$ through the power map

$$(1+x,a)\mapsto (1+x)^a.$$

We thus replace the derived loop space by the space

$$L^D X:=Hom_{dSt}(B\overset{\circ}{D}(1,1),X),$$

together with its (partial) action of $B\overset{\circ}{D}(1,1)$ given by the (partial) multiplication of $\overset{\circ}{D}(1,1)\subset \mathbb{G}_m$. There is actually a family of such actions parametrized by $D(0,1)$ through the (partial) map

$$D(0,1)\times B\overset{\circ}{D}(1,1)\times L^D X\to L^D X$$

given by $(a,d,\gamma)\mapsto \gamma\circ m_{d^a}$. This family of actions gives a space

$$Hod(L^D X)\to D(0,1)$$

whose fiber at $0$ is the trivial action of $B\overset{\circ}{D}(1,1)$ on $L^D X$ and whose fiber at $1$ is the standard action.

Related notion

generalized higher loop spaces?