Let X be a quasi-compact and quasi-separated algebraic space. Let T⊂|X| be a closed subset such that |X| – T is quasi-compact. The category D_{QCoh,T}(O_X) of complexes with quasi-coherent cohomology sheaves supported on T is generated by a single perfect object. See Lemma Tag 0AEC

This result for schemes is in the paper “Dimensions of triangulated categories” by Raphaël Rouquier

A regular local ring is a UFD. See Lemma Tag 0AG0.

Let (A,I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). For any torsion abelian sheaf F on X_{e´tale} we have H^q_{e´tale}(X, F) = H^q_{e´tale}(Z, F|Z). See Theorem Tag 09ZI.

Slogan: Affine analogue of the proper base change theorem (due to Gabber; can also be found in a paper by Huber)

Let k be a field. Let G be a separated group algebraic space locally of finite type over k. There does not exist a nonconstant morphism f : P^1_k → G over Spec(k). See Lemma Tag 0AEN.

Slogan: no (complete) rational curves on groups.

Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a (U ∪ V)-admissible blowup b : X′ → X such that X′ is a disjoint union of open subschemes X′ = X′1 ⨿ X′2 with b^{−1}(U) ⊂ X′1 and b^{−1}(V) ⊂ X′2. See Lemma Tag 080P.

Slogan: separate irreducible components by blowing up.

Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X′→X of Z in X is a final object of C. See Lemma Tag 0806.

Slogan: Universal property of blowing up

Let S be a scheme. Let {X_i → X}_{i ∈ I} be an fppf covering of algebraic spaces over S. Assume I is countable (we can allow larger index sets if we bound the size of the algebraic spaces or if we don’t worry about set theoretic issues). Then any descent datum for algebraic spaces relative to {X_i → X}_{i ∈ I} is effective. See Lemma Tag 0ADV.

Slogan: fppf descent data for algebraic spaces are effective.

Can somebody give me an explicit example of a Noetherian local domain A which does not have resolution of singularities? In particular, I would like an example where the completion A^* of A defines an isolated singularity, i.e., where Spec(A^*) – {m^*} is a regular scheme?

A related question (I think) is this: Does there exist a Noetherian local ring A such that there exists a proper morphism Y —> Spec(A^*) whose source Y is an algebraic space which is an isomorphism over the punctured spectrum such that Y is not isomorphic to the base change of a proper morphism X —> Spec(A) whose source is an algebraic space? [Edit 20 October 2014: This cannot happen; follow link below.]

I wanted to make an example for the second question by taking a Noetherian local ring A which does not have a resolution whose completion A^* is a domain and defines an isolated singularity. Then a resolution of singularities Y of A^* would presumably not be the base change of an X, because if so then X would presumably be a resolution for A. [Edit 20 October 2014: This cannot happen; follow link below.]

Anyway, I searched for examples of this sort on the web but failed to find a relevant example. Can you help? Thanks!

[Edit 23 July 2014: The discussion is continued in this blog post.]