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Dat Minh Ha
Dat Minh Ha
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Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like how if $\Lambda$ is a Noetherian ring then constructible sheaves of $\Lambda$-modules will span a thick subcategory of the abelian category of sheaves of $\Lambda$-modules ? Will such a generalisation of constructible sheaves involve defining stratifications of sites/sheaf topoi, and if so, how would we go about defining such notions of stratifications ?

In particular, I am interested in "direct" definitions of constructible étale, lisse-étale, and fppf sheaves over a given scheme/algebraic space/Artin stack $X$. Certain restrictions that I am willing to work under are that $X$ is (locally) of finite type over some base scheme $S$ that is affine, regular, Noetherian, of dimension $\leq 1$, and of characteristic $p \geq 0$, and that the ring of coefficient $\Lambda$ is a Gorenstein local ring of dimension $0$ and characteristic $\ell \not = p$.

Is it possible to define (possibly derived) categories constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like how if $\Lambda$ is a Noetherian ring then constructible sheaves of $\Lambda$-modules will span a thick subcategory of the abelian category of sheaves of $\Lambda$-modules ? Will such a generalisation of constructible sheaves involve defining stratifications of sites/sheaf topoi, and if so, how would we go about defining such notions of stratifications ?

In particular, I am interested in "direct" definitions of constructible étale, lisse-étale, and fppf sheaves over a given scheme/algebraic space/Artin stack $X$. Certain restrictions that I am willing to work under are that $X$ is (locally) of finite type over some base scheme $S$ that is affine, regular, Noetherian, of dimension $\leq 1$, and of characteristic $p \geq 0$, and that the ring of coefficient $\Lambda$ is a Gorenstein local ring of dimension $0$ and characteristic $\ell \not = p$.

Is it possible to define (possibly derived) categories of constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like how if $\Lambda$ is a Noetherian ring then constructible sheaves of $\Lambda$-modules will span a thick subcategory of the abelian category of sheaves of $\Lambda$-modules ? Will such a generalisation of constructible sheaves involve defining stratifications of sites/sheaf topoi, and if so, how would we go about defining such notions of stratifications ?

In particular, I am interested in "direct" definitions of constructible étale, lisse-étale, and fppf sheaves over a given scheme/algebraic space/Artin stack $X$. Certain restrictions that I am willing to work under are that $X$ is (locally) of finite type over some base scheme $S$ that is affine, regular, Noetherian, of dimension $\leq 1$, and of characteristic $p \geq 0$, and that the ring of coefficient $\Lambda$ is a Gorenstein local ring of dimension $0$ and characteristic $\ell \not = p$.

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Dat Minh Ha
Dat Minh Ha
  • 1.5k
  • 1
  • 8
  • 21

Stratified sites/topoi and constructible sheaves

Is it possible to define (possibly derived) categories constructible sheaves over sites more general than those of open subsets of topological spaces while still retaining essential features, like how if $\Lambda$ is a Noetherian ring then constructible sheaves of $\Lambda$-modules will span a thick subcategory of the abelian category of sheaves of $\Lambda$-modules ? Will such a generalisation of constructible sheaves involve defining stratifications of sites/sheaf topoi, and if so, how would we go about defining such notions of stratifications ?

In particular, I am interested in "direct" definitions of constructible étale, lisse-étale, and fppf sheaves over a given scheme/algebraic space/Artin stack $X$. Certain restrictions that I am willing to work under are that $X$ is (locally) of finite type over some base scheme $S$ that is affine, regular, Noetherian, of dimension $\leq 1$, and of characteristic $p \geq 0$, and that the ring of coefficient $\Lambda$ is a Gorenstein local ring of dimension $0$ and characteristic $\ell \not = p$.