Noncommutative standard conjectures of type C+ and D for smooth proper dg categories in positive characteristic

Grothendieck's standard conjectures of type C+ and D ask, for a smooth projective variety, that the even Künneth projector on crystalline cohomology be induced by an algebraic cycle and that homological and numerical equivalence of algebraic cycles coincide; both are open since the late 1960s. Tabuada (2017) transplanted both conjectures from varieties to smooth proper dg categories — the noncommutative spaces of noncommutative algebraic geometry — replacing crystalline cohomology by topological periodic cyclic homology (identified with crystalline cohomology after inverting p, by a theorem of Scholze) and Chow motives by noncommutative Chow motives. The resulting conjectures $\mathrm{C}^{+}_{\mathrm{nc}}(\mathcal{A})$ and $\mathrm{D}_{\mathrm{nc}}(\mathcal{A})$ are known for finite-dimensional algebras of finite global dimension, for geometric categories exactly when the classical conjectures hold, conditionally under homological projective duality, for orbifold categories $[X/G]$ with $p\nmid|G|$ (reducing to the classical conjectures on fixed-point loci), and (for $\mathrm{D}_{\mathrm{nc}}$, in characteristic zero) for matrix factorization categories. The problem is whether $\mathrm{C}^{+}_{\mathrm{nc}}(\mathcal{A})$ and $\mathrm{D}_{\mathrm{nc}}(\mathcal{A})$ hold for every smooth proper dg category $\mathcal{A}$ over a perfect field of positive characteristic, or whether an explicit counterexample exists. Since the geometric case is equivalent to the classical conjectures, the problem is a strict generalization of Grothendieck's original question and remains open.

Background

Grothendieck's standard conjectures, formulated in 1968–1969, concern the cohomology of smooth projective varieties and remain among the central open problems of algebraic geometry. Let $k$ be a perfect field of characteristic $p>0$ (a field in which every element admits a $p$-th root, so that $k$ has no purely inseparable algebraic extensions), let $W(k)$ be its ring of $p$-typical Witt vectors — the canonical complete discrete valuation ring with residue field $k$ — and let $K=W(k)[1/p]$ be its fraction field. For a smooth projective $k$-scheme $X$, crystalline cohomology $H^{\ast}_{\mathrm{crys}}(X/W(k))$, developed by Berthelot as the characteristic-$p$ analogue of de Rham cohomology, becomes after tensoring with $K$ a Weil cohomology theory: it satisfies Poincaré duality and Künneth formulas and carries cycle classes for algebraic cycles (formal $\mathbb{Q}$-linear combinations of subvarieties). The Künneth projectors $\pi_X^i$ are the endomorphisms of $H^{\ast}_{\mathrm{crys}}(X)\otimes_{W(k)}K$ that act as the identity on the $i$-th cohomological graded piece and as zero on the others; $\pi_X^{+}:=\sum_i\pi_X^{2i}$ is the even Künneth projector. An endomorphism of cohomology is called algebraic when it is induced by an algebraic cycle via the cycle class map. Grothendieck's conjecture $\mathrm{C}^{+}(X)$ asserts that $\pi_X^{+}$ is algebraic, and conjecture $\mathrm{D}(X)$ asserts that homological equivalence of cycles (vanishing of the crystalline cycle class) coincides with numerical equivalence (vanishing of all intersection numbers). Kleiman's survey recalls the precise statements and their pivotal role in the theory of motives, where they would make the category of numerical motives abelian and semisimple. Both conjectures hold when $\dim(X)\le 2$, and $\mathrm{C}^{+}$ holds for abelian varieties and whenever $k$ is finite (Katz–Messing); in general they are wide open.

Noncommutative algebraic geometry replaces a variety by its category of perfect complexes. Concretely, one works with differential graded (dg) categories over $k$: categories enriched in complexes of $k$-vector spaces. A dg category $\mathcal{A}$ is smooth when its diagonal bimodule is a perfect $\mathcal{A}^{\mathrm{op}}\otimes\mathcal{A}$-module, and proper when its Hom-complexes have finite-dimensional total cohomology; following Kontsevich, smooth proper dg categories are regarded as noncommutative smooth proper varieties. Examples include $\mathrm{perf}_{\mathrm{dg}}(X)$, a dg enhancement of the category of perfect complexes on a smooth proper scheme or algebraic stack $X$; finite-dimensional $k$-algebras of finite global dimension; and dg categories of matrix factorizations, attached to isolated hypersurface singularities. Two dg categories related by a Morita equivalence (an equivalence after passing to modules) share all invariants considered here.

The cohomology theory playing the role of crystalline cohomology on the noncommutative side is topological periodic cyclic homology $TP$: a 2-periodic, cyclotomic refinement of Hochschild homology built from topological Hochschild homology, the natural homology theory of ring spectra and dg categories. For a smooth proper dg category $\mathcal{A}$, the groups $TP_{\pm}(\mathcal{A})_{1/p}$ — a $\mathbb{Z}/2$-graded $K$-vector space obtained after inverting $p$ — play the role of $\oplus_iH_{\mathrm{crys}}^i\otimes K$, graded by parity. A theorem of Scholze, recorded with proof in [1], identifies $TP_{\pm}(\mathrm{perf}_{\mathrm{dg}}(X))_{1/p}$ with $(\bigoplus_{i\,\mathrm{even}}H_{\mathrm{crys}}^i(X)\otimes K,\bigoplus_{i\,\mathrm{odd}}H_{\mathrm{crys}}^i(X)\otimes K)$ for every smooth proper $k$-scheme $X$; this comparison is what makes the following definitions faithful noncommutative analogues.

On the motivic side, every Morita-invariant additive invariant of dg categories (algebraic $K$-theory, $TP$, Hochschild homology, ...) factors through the universal additive invariant $U(-)_{\mathbb{Q}}$ with values in the $\mathbb{Q}$-linear category $\mathrm{NChow}(k)$ of noncommutative Chow motives, the noncommutative analogue of the category of Chow motives (the category built from smooth projective varieties and algebraic correspondences modulo rational equivalence). Quotienting $\mathrm{NChow}(k)_{\mathbb{Q}}$ by the kernel of the $TP$-realization, respectively by the kernel of the Euler pairing, yields the categories $\mathrm{NHom}(k)_{\mathbb{Q}}$ of noncommutative homological motives and $\mathrm{NNum}(k)_{\mathbb{Q}}$ of noncommutative numerical motives; the Euler pairing on the Grothendieck group $K_0(\mathcal{A})$ is $\chi([a],[b])=\sum_i(-1)^i\dim_k\mathrm{Ext}^i(a,b)$. Tabuada [1] then defines the noncommutative standard conjecture $\mathrm{C}^{+}_{\mathrm{nc}}(\mathcal{A})$ — algebraicity of the even Künneth projector $\pi_{+}^{\mathcal{A}}=(\mathrm{id},0)$ of $TP_{\pm}(\mathcal{A})_{1/p}$, meaning it lifts through $TP_{\pm}(-)_{1/p}$ to an endomorphism of $U(\mathcal{A})_{\mathbb{Q}}$ — and $\mathrm{D}_{\mathrm{nc}}(\mathcal{A})$ — the equality $K_0(\mathcal{A})_{\mathbb{Q}}/_{\sim\mathrm{hom}}=K_0(\mathcal{A})_{\mathbb{Q}}/_{\sim\mathrm{num}}$, both sides being $\mathrm{Hom}(U(k)_{\mathbb{Q}},U(\mathcal{A})_{\mathbb{Q}})$ computed in $\mathrm{NHom}(k)_{\mathbb{Q}}$, respectively $\mathrm{NNum}(k)_{\mathbb{Q}}$.

Tabuada [1] proves: the equivalences $\mathrm{C}^{+}(X)\Leftrightarrow\mathrm{C}^{+}_{\mathrm{nc}}(\mathrm{perf}_{\mathrm{dg}}(X))$ and $\mathrm{D}(X)\Leftrightarrow\mathrm{D}_{\mathrm{nc}}(\mathrm{perf}_{\mathrm{dg}}(X))$, so the noncommutative conjectures simultaneously generalize and transfer the classical ones; both conjectures for every finite-dimensional $k$-algebra of finite global dimension (via the semisimplicity of the Artin–Wedderburn quotient); stability under Morita equivalence, tensor products, and direct sums and summands; a transfer theorem under homological projective duality (a framework producing paired component categories from Lefschetz decompositions), conditional on the conjectures for the components; and new geometric cases, such as linear sections of determinantal varieties and low-dimensional orbifolds $[X/G]$ with $p\nmid|G|$. Subsequent work of Brown–Walker and Kim–Kim established $\mathrm{D}_{\mathrm{nc}}$ for dg categories of matrix factorizations of isolated hypersurface singularities and their equivariant versions — but in characteristic zero — while Tabuada's companion papers developed the noncommutative Weil-conjecture and zeta-function picture. The conjectures themselves for a general smooth proper dg category in positive characteristic, and even the matrix-factorization case in positive characteristic, remain open.

Problem Statement

Let $k$ be a perfect field of characteristic $p>0$, $W(k)$ its ring of $p$-typical Witt vectors, and $K=W(k)[1/p]$. Recall that a dg category $\mathcal{A}$ over $k$ is smooth and proper when the diagonal $\mathcal{A}$–$\mathcal{A}^{\mathrm{op}}$-bimodule is perfect and the Hom-complexes have finite-dimensional total cohomology; up to Morita equivalence, these are the noncommutative smooth proper varieties, and they include $\mathrm{perf}_{\mathrm{dg}}(X)$ for smooth proper $k$-schemes and stacks, finite-dimensional $k$-algebras of finite global dimension, and dg categories of matrix factorizations. For such $\mathcal{A}$, let $TP_{\pm}(\mathcal{A})_{1/p}$ be the $\mathbb{Z}/2$-graded $K$-vector space obtained from topological periodic cyclic homology after inverting $p$, let $\pi_{+}^{\mathcal{A}}=(\mathrm{id},0)$ be its even Künneth projector, and let $U(\mathcal{A})_{\mathbb{Q}}$ be the noncommutative Chow motive of $\mathcal{A}$.

Conjecture $\mathrm{C}^{+}_{\mathrm{nc}}(\mathcal{A})$: the projector $\pi_{+}^{\mathcal{A}}$ is algebraic, i.e. there exists an endomorphism $\underline{\pi}_{+}^{\mathcal{A}}$ of $U(\mathcal{A})_{\mathbb{Q}}$ such that $TP_{\pm}(\underline{\pi}_{+}^{\mathcal{A}})_{1/p}=\pi_{+}^{\mathcal{A}}$.

Conjecture $\mathrm{D}_{\mathrm{nc}}(\mathcal{A})$: the canonical map $\mathrm{Hom}_{\mathrm{NHom}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(\mathcal{A})_{\mathbb{Q}})\to\mathrm{Hom}_{\mathrm{NNum}(k)_{\mathbb{Q}}}(U(k)_{\mathbb{Q}},U(\mathcal{A})_{\mathbb{Q}})$ is an isomorphism; equivalently $K_0(\mathcal{A})_{\mathbb{Q}}/_{\sim\mathrm{hom}}=K_0(\mathcal{A})_{\mathbb{Q}}/_{\sim\mathrm{num}}$, where $\sim\mathrm{num}$ is equivalence modulo the (left = right) kernel of the Euler pairing $\chi([a],[b])=\sum_i(-1)^i\dim_k\mathrm{Ext}^i(a,b)$ on $K_0$.

The problem: prove, or refute by an explicit counterexample, that $\mathrm{C}^{+}_{\mathrm{nc}}(\mathcal{A})$ and $\mathrm{D}_{\mathrm{nc}}(\mathcal{A})$ hold for every smooth proper dg category $\mathcal{A}$ over every perfect field of positive characteristic. The two conjectures may be addressed separately. The question concerns precisely the remaining generality, including arbitrary smooth proper dg algebras, matrix factorization and fractional Calabi–Yau categories (smooth proper categories generalizing derived categories of Calabi–Yau varieties, in which a power of the Serre duality autoequivalence equals a power of the suspension functor) in positive characteristic, and quotient-stack categories with $p\mid|G|$.

References

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