Let X be a quasi-compact and quasi-separated algebraic space such that for every quasi-coherent O_{X}-module F we have H^{1}(X, F) = 0. Then X is an affine scheme. See Tag 07V6.

# Monthly Archives: July 2013

# Example of the day

There exist a zero dimensional local ring with a nonzero flat ideal. See Tag 05FZ.

# Lemma of the day

Let X be a quasi-separated algebraic space. Let E be an object of D_{QCoh}(O_{X}). Let a ≤ b. The following are equivalent

- E has tor amplitude in [a,b], and
- for all F in QCoh(O
_{X}) we have H^{i}(E ⊗^{L}F)=0 for i not in [a,b].

See Tag 08IL.

# Lemma of the day

# Lemma of the day

Let A be a ring. Let I ⊂ J ⊂ A be ideals. If M is J-adically complete and I is finitely generated, then M is I-adically complete. See Tag 090T.

# Proposition of the day

Let X be a scheme. Let a : X —> Spec(k_{1}) and b : X —> Spec(k_{2}) 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,O_{X}) is the integral closure of k_{i} in Γ(X,O_{X}). See Tag 04MK.