HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

Modules over algebraic cobordism

We prove that the $\infty $ -category of $\mathrm{MGL} $ -modules over any scheme is equivalent to the $\infty $ -category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$ -loop spaces, we deduce that very effective $\mathrm{MGL} $ -modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$ , $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$ .

Milnor fibers and links of local complete intersections

There are essentially no previously-known results which show how Milnor fibers, real links, and complex links "detect" the dimension of the singular locus of a local complete intersection. In this paper, we show how a good understanding of the derived category and the perverse t-structure quickly yields such results for local complete intersections with singularities of arbitrary dimension.

A note on thick subcategories of stable derived categories

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

A note on thick subcategories of stable derived categories

For an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

Which Schubert varieties are local complete intersections?

International audience We characterize by pattern avoidance the Schubert varieties for $\mathrm{GL}_n$ which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighbourhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials. Nous caractérisons par l’évitement des motifs les variétés de Schubert qui sont localement des intersections complètes. Pour les variétés de Schubert qui sont localement des intersections complètes, nous donnons des ensembles explicites des polynômes qui définissent leurs voisinages à l’identité. Bien que notre caractérisation n'utilise que les motifs ordinaire, nous avons besoin des notions plus générales des motifs dans notre démonstration. Les variétés de Schubert définies par des inclusions, introduites par Gasharov et Reiner, sont une sous-classe importante, et nous développons davantage leurs combinatoire. Une application est une nouvelle formule pour une spécialisation des polynômes de Schubert.