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2 votes
0 answers
53 views

When localization commutes with arbitrary intersection of ideals

For a commutative ring with identity we know that in general localization does not commute with arbitrary intersection of ideals. I am looking for a paper that considers equivalent condition(s) for ...
Ya MA e. r's user avatar
1 vote
1 answer
96 views

On "minimal presentation" of local rings essentially of finite type over a field

Let k be a field of characteristic 0. Let (R,m) be a local ring essentially of finite type over k (https://stacks.math.columbia.edu/tag/07DR). Then, R is the homomorphic image of ...
strat's user avatar
  • 361
5 votes
1 answer
542 views

When is it possible to localize a scheme along a closed subscheme?

If we have ZX a closed irreducible subscheme of an integral scheme X (which you can take to have various further niceness properties if you want), one can take its generic point ηZ ...
xir's user avatar
  • 2,054
2 votes
0 answers
136 views

Some relative GW calculations

I have a question about the ψ class in the following paper of Graber and Vakil: https://arxiv.org/abs/math/0309227 For k,d2, and a partition d=d1++dk of d into positive ...
Mohammad Farajzadeh-Tehrani's user avatar
1 vote
0 answers
125 views

Different ways to construct the isogeny category of abelian varieties

Let k be a field and let AV/k be the category of abelian varieties over k. I'm interested in different definitions of the isogeny category of AV/k. Of course, the ...
Lukas Heger's user avatar
6 votes
0 answers
370 views

Geometric meaning of localization at (1+I)?

Let I be an ideal of a commutative ring. Consider the submonoid 1+I\subset A. What is the geometric interpretation of localization at this submonoid? How does it relate to the ...
Arrow's user avatar
  • 10.5k
7 votes
0 answers
258 views

Ample divisors on T-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample? In more detail: I have a (normal, but usually rather singular) T-variety X, where T is an ...
Geordie Williamson's user avatar
6 votes
0 answers
180 views

Abelian localisation for K theory?

Let X be a scheme acted upon by \mathbf{G}_m and K(X)=K_0(\text{Perf}X) the Thomason-Trobaugh K theory. Is there a localisation theorem in this context? By this I mean something like $$\text{id}...
Pulcinella's user avatar
  • 5,711
1 vote
1 answer
161 views

Geometric meaning of colocalization of modules?

Let A be a commutative ring and S\subset A a subset. A localization of A at S is defined as a ring morphsim A\to A[S^{-1}] which is initial with respect to inverting S. Similarly, a ...
Arrow's user avatar
  • 10.5k
8 votes
1 answer
372 views

Commutative ring R with no nontrivial idempotents, with a localization R_r with infinitely many idempotents

I am looking for a commutative ring R with 1 such that R has no idempotents and there exists r\in R such that the localization ring R_r has infinitely many idempotents.
Anahita's user avatar
  • 101
7 votes
0 answers
376 views

Grothendieck Riemann Roch is abelian localisation on loop spaces

Abelian localisation says approximately that for a proper equivaraint map f:X\to Y between schemes with a \mathbf{G}_m action, the pushforward on cohomology f_*\omega can be computed by the ...
Pulcinella's user avatar
  • 5,711
3 votes
1 answer
251 views

Localization on varieties with toric singularities

Is there a written version of Atiyah-Bott localization formula for varieties/manifolds with toric singularities? More precisely, suppose F is a fixed locus of a torus action $\mathbb{T}={\mathbb{C}^*...
Mohammad Farajzadeh-Tehrani's user avatar
0 votes
1 answer
314 views

Localization and containment in commutative ring

Let R be a commutative ring with identity and x, y be fixed elements of R such that for each maximal ideal m of R we have \langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle ...
Asad Albani's user avatar
2 votes
1 answer
154 views

Special submodules over almost Dedekind domains

An integral domain R is an almost Dedekind domain if for each maximal ideal m of R, the ring R_m is a Dedekind domain, where R_m is the localization of R at m. Question: Let M ...
user140640's user avatar
3 votes
0 answers
343 views

Localization of the pushforward in equivariant cohomology

I am reading Nekrasov's paper and in page 2 he considers the G \times T^2 equivariant cohomology of the (compactified) moduli space \tilde{M_k} of U(N) instantons on \mathbb{C}^2. Here G ...
Marion's user avatar
  • 587
0 votes
1 answer
260 views

Analytic spread of localization of an ideal

Let J be an ideal in a Noetherian local ring (R,m). It is well known that for any prime ideal p\in Spec(R), l(J_p)\leq l(J), where l(J) is the analytic spread of J. Q) Are there ...
Cusp's user avatar
  • 1,713
1 vote
0 answers
172 views

Local cohomology commuting with fiber

Let A be a nice commutative ring (say, A=k[t_1,\ldots , t_n], ring of polynomials over an algebraically closed field k). Let M be an A[x]-module, which is finitely generated as an A-...
Sasha's user avatar
  • 5,562
4 votes
1 answer
580 views

Voevodsky's proof in any characteristic (for motivic and Chow)

Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at: http://imrn.oxfordjournals.org/content/2002/7/351.full....
Quetzalcoatl's user avatar
53 votes
2 answers
8k views

Is primary decomposition still important?

On p.50 of Atiyah and Macdonald's Introduction to Commutative Algebra, in the introduction to the chapter on primary decomposition, it says In the modern treatment, with its emphasis on ...
David Corwin's user avatar
  • 15.4k
1 vote
1 answer
245 views

Are these connecting homomorphisms commutative?

Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative? In other words, is the following statement true? If it is true, then, how can one prove it? ...
Hiro's user avatar
  • 945