Frobenius Cokernel in Algebraic Geometry

• Frobenius cokernel is the cokernel of a Frobenius-induced map, capturing key exact sequences and duality in algebraic geometry.
• It links positivity of vector bundles to geometrical features in projective spaces, complete intersections, and Fano varieties.
• Its analogues extend to Witt vectors, invariant theory, and random-matrix models, underpinning universal cokernel distributions.

In algebraic geometry, arithmetic, and representation theory, the Frobenius cokernel denotes the cokernel of a Frobenius-induced map, but the concrete object depends on the ambient category. On a smooth projective variety in characteristic $p>0$, it is the vector bundle $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$; on Witt vectors, it is the quotient $W(R)/\operatorname{im}(F)$; in invariant theory, it appears as $R/R^p$ for the Frobenius map on a graded ring; and in recent random-matrix models, cokernels of polynomial endomorphisms over $\mathbb Z_p$ are treated as direct analogues of “Frobenius cokernels” (Carvajal-Rojas et al., 2021, Davis et al., 2014, Raedschelders et al., 2017, Cheong et al., 2023).

1. Basic definitions and exact sequences

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$, and let

$$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$

be the absolute Frobenius morphism. Since $X$ is reduced, there is a canonical injective map

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$0

The Frobenius cokernel is defined by

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$1

so that one has a short exact sequence of locally free sheaves

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$2

Using Grothendieck duality for the finite morphism $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$3, one identifies

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$4

so the dual of the Frobenius cokernel is the kernel of the Frobenius trace map (Mallory, 3 Oct 2025).

For the $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$5-th iterate $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$6, with $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$7, one writes

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$8

and obtains the exact sequences

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$9

and

$W(R)/\operatorname{im}(F)$0

Thus the cokernel of the Frobenius endomorphism is exactly $W(R)/\operatorname{im}(F)$1, while $W(R)/\operatorname{im}(F)$2 is the Frobenius trace kernel. These sequences are locally split, and $W(R)/\operatorname{im}(F)$3 is locally free of rank $W(R)/\operatorname{im}(F)$4 when $W(R)/\operatorname{im}(F)$5 (Carvajal-Rojas et al., 2021).

The same formal pattern recurs in higher degree. If

$W(R)/\operatorname{im}(F)$6

then the Cartier isomorphism yields short exact sequences

$W(R)/\operatorname{im}(F)$7

and $W(R)/\operatorname{im}(F)$8 is the kernel of the higher Cartier operator $W(R)/\operatorname{im}(F)$9 (Mallory, 3 Oct 2025).

2. Positivity, negativity, and the geometry they detect

A central theme is that positivity of the Frobenius trace kernel is equivalent to negativity of the Frobenius cokernel. If $R/R^p$0 is nef, then $R/R^p$1 is nef; if $R/R^p$2 is ample, then $R/R^p$3 is ample, so $R/R^p$4 is Fano. If $R/R^p$5 is globally generated and $R/R^p$6 is $R/R^p$7-split, then $R/R^p$8 is ample as well (Carvajal-Rojas et al., 2021).

These implications are not merely numerical. An ample Frobenius trace kernel rules out nontrivial smooth fibrations: if $R/R^p$9 is a fibration with smooth general fiber and $\mathbb Z_p$0 is ample, then the general fibers are zero-dimensional, and all fibers are zero-dimensional if $\mathbb Z_p$1 is flat. Blow-ups in codimension $\mathbb Z_p$2 likewise destroy ampleness of the dual Frobenius trace bundle, because the restriction of $\mathbb Z_p$3 to the exceptional divisor has a nontrivial trivial summand (Carvajal-Rojas et al., 2021).

The opposite positivity problem concerns the Frobenius cokernel itself rather than its dual. For a smooth projective variety over an $\mathbb Z_p$4-finite field of characteristic $\mathbb Z_p$5,

$\mathbb Z_p$6

This equivalence is proved by analyzing the Katz–Sun filtration on $\mathbb Z_p$7, whose graded pieces are truncated symmetric powers $\mathbb Z_p$8. It gives a direct bridge between classical positivity of the cotangent bundle and Frobenius-based positivity (Mallory, 3 Oct 2025).

This suggests two complementary regimes. When $\mathbb Z_p$9 is ample, the variety is forced toward Fano-type geometry; when $X$0 is ample, the cotangent bundle is ample, so the geometry is controlled by strong negativity of the tangent bundle. The literature treats these as distinct but formally parallel manifestations of Frobenius-controlled positivity (Carvajal-Rojas et al., 2021, Mallory, 3 Oct 2025).

3. Projective space, quadrics, and classification results

Projective space is the basic model. On $X$1,

$X$2

Hence $X$3 is ample, and the Frobenius cokernel is strongly negative. This behavior motivated the question whether ampleness of the Frobenius trace kernel characterizes projective space (Carvajal-Rojas et al., 2021).

In low dimension, the answer is affirmative. If $X$4 is a smooth projective curve and $X$5 is ample, then $X$6. If $X$7 is a smooth projective surface, then $X$8 is ample if and only if $X$9. For smooth projective threefolds, ampleness of $k$0 forces $k$1 to be a Fano threefold of Picard rank $k$2 (Carvajal-Rojas et al., 2021).

The threefold classification was sharpened later. If $k$3 is a smooth Fano threefold over an algebraically closed field of characteristic $k$4, then

$k$5

The key obstruction is geometric: if $k$6 contains a smooth positive-dimensional subvariety $k$7 with normal bundle $k$8 such that $k$9 is effective, then $p>0$0 is not ample. In particular, if $p>0$1 contains a smooth rational curve $p>0$2 with $p>0$3, then $p>0$4 is not ample. This excludes Fano threefolds containing lines or conics with respect to $p>0$5 (Mallory, 3 Oct 2025).

The naive projective-space characterization fails in higher dimension. Quadrics provide the basic counterexample: for $p>0$6, $p>0$7 is ample if and only if $p>0$8, and $p>0$9 is ample for all $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$0 (Carvajal-Rojas et al., 2021). More broadly, for $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$1 and quadric hypersurfaces, the full set of kernels of the higher Cartier operators does not characterize projective space (Mallory, 3 Oct 2025). This is the main correction to the optimistic expectation that Frobenius positivity should single out $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$2.

Complete intersections furnish another obstruction class. If $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$3 is a smooth complete intersection of degrees $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$4 with $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$5 and $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$6 or $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$7, then $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$8 is not ample. The argument uses the existence of lines in $$F:X\to X,\qquad F^\sharp:\mathcal O_X\to\mathcal O_X,\ f\mapsto f^p$$9 together with the inequality $X$0 for such lines (Mallory, 3 Oct 2025).

4. Witt vector Frobenius and its cokernel

For a commutative ring $X$1 and a prime $X$2, the $X$3-typical Witt vector Frobenius

$X$4

and its finite-level truncations

$X$5

provide another classical setting for the Frobenius cokernel. Here

$X$6

and vanishing of the cokernel is equivalent to surjectivity of the Frobenius map (Davis et al., 2014).

The structure of the kernel is governed by a descending sequence of ideals

$X$7

An element $X$8 occurs as the first component of an element in $X$9 if and only if $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$00 (Davis et al., 2014).

The finite-level cokernel vanishes precisely under a ring-theoretic $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$01-root condition. Surjectivity of

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$02

is equivalent to surjectivity of the $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$03-th power map on each quotient $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$04. At infinite level, surjectivity is stricter: $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$05 Equivalently, full surjectivity is characterized by finite-level surjectivity together with Teichmüller density, or by surjectivity of the $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$06-th power map on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$07 together with the same completeness condition (Davis et al., 2014).

Examples show that finite and infinite behavior diverge sharply. For $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$08, finite-level surjectivity fails, so the Frobenius cokernel is already nonzero at finite levels. For the ring of integers $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$09 in an algebraic closure of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$10, finite-level Frobenius is surjective for all $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$11, but spherical completeness fails, so the infinite-level cokernel is nonzero. For a spherically complete valuation ring of the kind constructed by Poonen, the infinite Witt Frobenius is surjective (Davis et al., 2014).

In the mixed-characteristic direction, the finite-level vanishing of the Frobenius cokernel is described as the integral perfectoid condition of Scholze, repackaged in Witt-vector language. The same condition is stable under certain integral extensions and underlies the almost-purity statement proved there (Davis et al., 2014).

5. Graded rings, invariant theory, and homogeneous varieties

In invariant theory, the Frobenius cokernel appears module-theoretically. Let

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$12

be the homogeneous coordinate ring of the Grassmannian $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$13, with $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$14 and $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$15. The Frobenius on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$16 is

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$17

with image $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$18. Viewing $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$19 as an $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$20-module, the Frobenius cokernel is

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$21

The decomposition of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$22 as a graded $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$23-module therefore controls the structure of the cokernel (Raedschelders et al., 2017).

The indecomposable Frobenius summands are expressed in terms of modules of covariants $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$24 and certain indecomposable Cohen–Macaulay modules $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$25. Forgetting degrees, the Frobenius summands of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$26 lie among

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$27

and all of these appear when $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$28. This is the paper’s explicit instance of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$29-uniformity: for sufficiently large $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$30, the set of indecomposable summands does not depend on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$31, although multiplicities and grading shifts still vary (Raedschelders et al., 2017).

Geometrically, the same analysis yields a decomposition of the Frobenius pushforward $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$32 into indecomposable homogeneous bundles. Up to multiplicity, the summands are twists of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$33, the bundles $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$34, and twists of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$35. This Frobenius pushforward is a tilting bundle only for $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$36 and $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$37; otherwise it is not tilting (Raedschelders et al., 2017).

The module-theoretic consequence for the cokernel is that the Frobenius cokernel is far from projective in general. The nonfree indecomposable summands $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$38 contribute to the nonfree part of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$39 over $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$40, and hence to the nonfree part of the Frobenius cokernel. The same summands are then used to construct a noncommutative resolution $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$41 of finite global dimension for a suitable direct sum $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$42 of Frobenius summands (Raedschelders et al., 2017).

6. Random and cohomological analogues

Recent random-matrix work treats polynomial cokernels over $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$43 as direct analogues of Frobenius cokernels. For

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$44

with $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$45 fixed and $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$46 random, and for a non-constant monic polynomial $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$47, the central object is

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$48

More naturally, one equips it with the structure of an $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$49-module for

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$50

via letting the image of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$51 act by $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$52. The key linearization is

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$53

which replaces a polynomial in $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$54 by an $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$55-linear map. The paper proves exact finite-$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$56 formulas for the distribution of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$57 when $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$58 is Haar-random, establishes universality for a broader concentrated-residue class, and derives a Cohen–Lenstra style limit law in the non-concentrated regime (Cheong et al., 2023).

This random-matrix perspective is explicitly presented as directly analogous to “Frobenius cokernels.” Conceptually, one studies an endomorphism $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$59 of a $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$60-adic module and the modules

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$61

while the random model replaces $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$62 by $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$63. The resulting distribution depends on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$64 and on the factorization of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$65, and in the squarefree case becomes a direct generalization of the Cohen–Lenstra measure to modules over $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$66 (Cheong et al., 2023).

A complementary universality principle holds for random integral matrices. For an $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$67 random integral matrix $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$68 with independent entries from a very broad class of distributions, the limiting probability that $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$69 is

$$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$70

This is a Cohen–Lenstra-type law, and it agrees with the distribution defined by the Haar model on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$71. The paper interprets this as support for the principle that random endomorphisms modelling Frobenius produce canonical cokernel distributions independent of microscopic details of the entry distribution (Nguyen et al., 2018).

Representation-theoretic work on Frobenius kernels gives a different, but related, analogue. For the $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$72-th Frobenius kernels $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$73, the cohomology ring $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$74 is Cohen–Macaulay, and its maximal ideal spectrum is homeomorphic to $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$75. That work is about kernels of Frobenius rather than cokernels, but it is formulated so that explicit knowledge of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$76 constrains possible kernels and cokernels of Frobenius-induced maps on cohomology (Ngo, 2012).

Across these settings, the Frobenius cokernel is best viewed not as a single invariant but as a family of formally parallel constructions. In geometry it is a vector bundle attached to $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$77; in Witt theory it measures the failure of Frobenius surjectivity on $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$78; in invariant theory it records the nonfree part of $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$79 over $$\mathcal B_X:=\operatorname{coker}(\mathcal O_X\to F_*\mathcal O_X)$$80; and in random-matrix models it becomes a probabilistic proxy for cokernels of polynomial functions of Frobenius-like endomorphisms (Carvajal-Rojas et al., 2021, Davis et al., 2014, Raedschelders et al., 2017, Cheong et al., 2023).