Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator

Abstract: We describe the obstruction to decomposing in degrees $\leq p$ the de Rham complex of a smooth variety over a perfect field $k$ of characteristic $p$ that lifts over $W_2(k)$, and show that there exist liftable smooth projective varieties of dimension $p+1$ whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. We also describe the action of the Sen operator on the de Rham complex in degrees $\leq p$ and give examples of varieties with a non-semisimple Sen operator. Our methods rely on the commutative algebra structure on de Rham and Hodge-Tate cohomology, and are inspired by the properties of Steenrod operations on cohomology of cosimplicial commutative algebras. The example of a non-degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing result on cohomology of groups of Lie type. We give applications to other situations such as describing extensions in the canonical filtration on de Rham, Hodge, and \'etale cohomology of an abelian variety equipped with a group action. We also show that the de Rham complex of a smooth variety over $k$ is formal as an $E_{\infty}$-algebra if and only if the variety lifts to $W_2(k)$ together with its Frobenius endomorphism.

• The paper demonstrates that the de Rham complex of smooth varieties does not decompose in higher degrees, countering earlier conjectures by Deligne and Illusie.
• The paper reveals that the Sen operator is non-semisimple, which impacts the degeneration of the Hodge-to-de Rham spectral sequence.
• The paper utilizes commutative algebra techniques and Steenrod operations to highlight implications for computational cohomology in positive characteristic.

Non-Decomposability of the de Rham Complex and Non-Semisimplicity of the Sen Operator

Introduction to the Paper

The paper "Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator" explores the mathematical intricacies surrounding the de Rham complex associated with smooth varieties over fields of positive characteristic $p$. Specifically, the paper addresses the conditions under which the de Rham complex fails to decompose in certain degrees and explores the non-semisimplicity of the Sen operator. This exploration reveals that certain smooth projective varieties exhibit a non-degenerate Hodge-to-de Rham spectral sequence, providing insights into their algebraic topology. The methods utilized rely heavily on the commutative algebra structure on de Rham and Hodge-Tate cohomology, alongside properties of Steenrod operations on the cohomology of cosimplicial commutative algebras.

Main Contributions

Non-decomposability in de Rham Complexes

The paper provides a clear answer to whether the de Rham complex of liftable varieties decomposes in degrees higher than $p$. The authors prove that this decomposition does not inherently occur, refuting a question posed by Deligne and Illusie. The existence of such non-decomposable complexes suggests that there are inherent algebraic structures preventing such decomposition, which in turn affect spectral sequences derived from these complexes.

Non-Semisimplicity of the Sen Operator

The paper provides examples of varieties where the Sen operator, associated with the de Rham complex, is non-semisimple. This characteristic is pivotal because it influences the spectral sequence's degeneration, which is vital for understanding the complex's topological and algebraic properties. The Sen operator, a derivation induced by the Frobenius lift, acts on the de Rham complex's cohomology sheaves, determining the algebraic structure of these complexes.

Implementation and Applications

Practical Implications

The non-decomposability in the de Rham complex and the non-semisimplicity of the Sen operator could have significant implications for algebraic geometers and those studying the topology of algebraic varieties in positive characteristic. The existence of non-degenerate spectral sequences directly impacts computational techniques in cohomology, potentially altering how these techniques are applied in complex geometric contexts.

Computational Techniques

In computational contexts, these results imply that automated systems or algorithms analyzing algebraic structures need to account for the possibility of non-decomposability and non-semi-simplicity. Algorithms that compute spectral sequences might need adjustments to detect and handle these non-standard behaviors accurately.

Conclusion

The paper contributes substantially to our understanding of algebraic topology in characteristic $p$. By exposing the limits of decomposability and the simplicity of involved operators, it sets a foundation for further exploration into the algebraic properties of smooth varieties. Such insights are crucial for algebraic geometers attempting to classify and analyze these varieties using computational methods. Moreover, the implications for algebraic manipulation software suggest that current tools might need re-evaluation to accommodate these newly understood structures. The research introduces a robust framework for approaching these complexities, enriching the field's discourse with precise mathematical proofs and examples.