Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem

Abstract

We define a new version of A 1 -homology, called cellular A 1 -homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular A 1 -homology and use them to determine the A 1 -fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly A 1 -invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.

Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field

“…Corollary 7. A scheme V of the form described in Theorem 4 is cohomologically trivial in the sense of Definition 2.9 in [15]. The claim follows from Corollary 6 and Lemma 13.…”

“…Lemma 13. Any scheme that is cohomologically trivial in the sense of Definition 7 is cohomologically trivial in the sense of Definition 2.9 in [15]. This follows from Lemma 12, since any strict homotopy invariant sheaf F defines the cohomology theory E(−) = l H l (−, F ).…”

“…The scheme V of the form described in Theorem 4 is cohomologically trivial in the sense of Definition 2.9 in [15].…”

Quadratic enrichment of the logarithmic derivative of the zeta function

“…For this, we use Morel and Sawant's A 1 -cellular homology [MS20], currently defined for socalled cellular schemes, which are a generalization of the classical notion of varieties with an affine stratification. In [MS20, Remark 2.44], they note the existence of an extension of this theory to a pro-object for all spaces in the sense of A 1 -homotopy theory and in particular, for all smooth schemes over a field.…”