Lemma of the day

Let X be a quasi-separated algebraic space. Let E be an object of DQCoh(OX). Let a ≤ b. The following are equivalent

  1. E has tor amplitude in [a,b], and
  2. for all F in QCoh(OX) we have Hi(E ⊗L F)=0 for i not in [a,b].

See Tag 08IL.

Proposition of the day

Let X be a scheme. Let a : X —> Spec(k1) and b : X —> Spec(k2) be morphisms from X to spectra of fields. Assume a,b are locally of finite type, and X is reduced, and connected. Then we have k′1 = k′2, where k′i ⊂ Γ(X,OX) is the integral closure of ki in Γ(X,OX). See Tag 04MK.