Chapter of the day

There is a new chapter entitled “Pushouts of Algebraic Spaces”. Here is a link to the introduction. It contains a section on something that is often called formal glueing of quasi-coherent modules. The name presumably comes from the fact that this can be used and is often used to glue coherent sheaves on Noetherian schemes given on an open and on the formal completion along the closed. However, the mathematics is more like what one does when studying elementary distinguished squares in the Nishnevich topology — in fact, I wonder what topology we get if we replace elementary distinguished squares by the notion in Situation Tag 0AEV? Hmm…

Lemma of the day

Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds

  1. A → A^* is a regular ring map,
  2. A is a Noetherian G-ring, or
  3. (A,I) is the henselization (More on Algebra, Lemma 15.7.10) of a pair (B,J) where B is a Noetherian G-ring.

Given f_1, …, f_m ∈ A[x_1, ..., x_n] and a_1, …, a_n ∈ A^* such that f_j(a_1, …, a_n) = 0 for j = 1, …, m, for every N ≥ 1 there exist b_1, …, b_n ∈ A such that a_i − b_i ∈ I^N and such that f_j(b_1, …, b_n) = 0 for j = 1, …, m. See Lemma Tag 0AH5.

Slogan: Approximation for henselian pairs.

Lemma of the day

Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules.

  1. Hom_{D(A)}(M, N) = Hom_{D(I^∞-torsion)}(M, N),
  2. Ext^1_{D(A)}(M, N) = Ext^1_{D(I^∞-torsion)}(M, N),
  3. Ext^2_{D(I^∞-torsion)}(M, N) → Ext^2_{D(A)}(M, N) is not surjective in general,
  4. (0A6N) is not an equivalence in general.

See Lemma Tag 0592.

Discussion: Let A be a ring and let I be an ideal. The derived category of complexes of A-modules with I-power torsion cohomology modules is not the same as the derived category of the category of I-power torsion modules in general, even if I is finitely generated. However, if the ring is Noetherian then it is true, see Lemma Tag 0955.

Fixed all issues

Hello again. This just a quick post to let you know that I worked through your recent comments and updated the Stacks project website with the latest version. So get back there and find more mistakes. Thank you all!

Lemma of the day

Let F be a predeformation category which has a versal formal object. Then

  1. F has a minimal versal formal object,
  2. minimal versal objects are unique up to isomorphism, and
  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

See Lemma Tag 06T5.

What is fun about this lemma is that it produces a minimal versal object (as defined in Definition Tag 06T4) from a versal one without assuming Schlessinger’s axioms. If Schlessinger’s axioms are satisfied and one is in the classical case (see Definition Tag 06GC), then a minimal versal formal object is a versal formal object defined over a ring with minimal tangent space. This is discussed in Section Tag 06IL.

Lemma of the day

Let h : X –> Y, g : Y –> B be morphisms of algebraic spaces with composition f : X –> B. Let b ∈ |B| and let Spec(k) → B be a morphism in the equivalence class of b. Assume

  1. X → B is a proper morphism,
  2. Y → B is separated and locally of finite type,
  3. one of the following is true:
    1. the image of |X_k| → |Y_k| is finite,
    2. the image of |f|^{−1}({b}) in |Y| is finite and B is decent.

Then there is an open subspace B′ ⊂ B containing b such that X_{B′} → Y_{B′} factors through a closed subspace Z ⊂ Y_{B′} finite over B′. See Lemma Tag 0AEJ.

Slogan: Collapsing a fibre of a proper family forces nearby ones to collapse too.

Lemma of the day

Let (A,m) be a Noetherian local ring. Let I ⊂ J ⊂ A be proper ideals. Assume

  1. A/J has finite tor dimension over A/I, and
  2. J is generated by a regular sequence.

Then I is generated by a regular sequence and J/I is generated by a regular sequence. See Lemma Tag 09PW.

Here is the graph of this lemma
09PW