All Questions
73 questions
42
votes
4
answers
8k
views
Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 30.5k
31
votes
0
answers
1k
views
On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
- 19.6k
23
votes
1
answer
966
views
Do DG-algebras have any sensible notion of integral closure?
Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
- 52.6k
21
votes
6
answers
3k
views
A ring such that all projectives are stably free but not all projectives are free?
This question is motivated by this recent question. Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and ...
- 30.5k
15
votes
3
answers
2k
views
Why do modules with small support have high Exts?
Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...
- 13.2k
15
votes
5
answers
4k
views
Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension
Let $(A,m_A)$ and $(B,m_B)$ be noetherian local rings and $f:A\rightarrow B$ a local homomorphism. Let $F = B/m_AB$ be the fiber ring and assume that
$$\mathrm{dim}(B) = \mathrm{dim}(A) + \mathrm{dim}...
- 1,609
15
votes
0
answers
721
views
If a polynomial ring is a finite flat module over some subring, is that subring itself a polynomial ring?
A question motivated by If a polynomial ring is a free module over some subring, is that subring itself a polynomial ring? and If a polynomial ring is finite free over a subring, is the subring ...
- 6,622
14
votes
0
answers
1k
views
Is there a slick proof of the fundamental theorem of dimension theory?
The fundamental theorem of dimension theory in commutative algebra states that given a module $M$ over a noetherian local ring $A$, we have $s(M)=\text{dim}(M)=d(M)$ (where $s(M)$ is the infimum of ...
- 1,514
13
votes
0
answers
474
views
Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
- 8,717
12
votes
3
answers
3k
views
Can we say anything about the Krull dimension of a localization?
I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...
- 30.3k
12
votes
2
answers
1k
views
Graded or stacky Serre duality
I am considering the following situation. $A$ is a finitely generated ring over a field $K$ with non-negative grading and $A_0=K$ of Krull dimension n+1, but I don't necessarily assume A is generated ...
- 121
10
votes
2
answers
1k
views
periodic cyclic homology and tilting in the sense of Scholze
Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost ...
- 7,952
10
votes
3
answers
1k
views
Can injective modules over R give non-injective sheaves over Spec R?
In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. ...
- 27.8k
10
votes
1
answer
929
views
Grothendieck trace on local cohomology?
Let R be an augmented regular local ring over a field $k$ with maximal ideal m. There is the Grothendieck residue symbol:
$$Res: H^n_m(\Omega^n) \to k$$
If $k=\mathbb{C}$ and $R$ is affine space, ...
- 1,331
9
votes
2
answers
1k
views
Projective resolution of modules over rings which are regular in codimension n
All rings are Noetherian and commutative, modules are finitely generated.
It is a theorem of Serre that over a regular ring $R$, every module has a finite projective resolution.
More generally, if $...
- 951
8
votes
1
answer
1k
views
Is the sheaf of smooth functions flat?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
- 338
8
votes
1
answer
979
views
Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
- 1,761
7
votes
2
answers
2k
views
Local property of split exact sequence
In the module category of a ring $A$, is a short exact sequence split if and only if the localization of this sequence is split for every prime ideal?
Thanks!
- 496
6
votes
1
answer
2k
views
Computing cotangent complex
I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?
In my precise situation, ...
- 7,300
6
votes
1
answer
395
views
Tor functor and invertible elements
Let $A$ be a commutative ring, $a \subset A$ be an ideal. For $A$-module $M$ let $S \subset A$ be the set of elements, which are invertible in $M$, so $M$ is actually a $S^{-1}A$-module. It is not ...
- 332
6
votes
1
answer
2k
views
Hochschild and cyclic homology of smooth varieties
Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ...
- 3,758
5
votes
2
answers
387
views
The flat support of a module
Let $R$ be a Noetherian commutative ring and $M$ a finitely generated $R$-module.
What is known about the following subset of $ \mathrm {Spec}(R)$:
$$\mathrm{supp}_{fl}(M)=\{P\in \mathrm {Spec}...
- 1,418
5
votes
1
answer
317
views
Localization of a ring and the Hom functor
Let $R=\mathbb{Z}[x,x^{-1}]$ be the ring of Laurent polynomials in $x$, $\mathfrak{p}=(1-x)$ be an ideal in $R$ and $R_\mathfrak{p}$ be the localization. I want to know what $\text{Hom}_R(R_\mathfrak{...
user340793
5
votes
1
answer
649
views
How to compute this $\mathrm{Ext}^1$?
Let $A$ be a regular local $\mathbb{C}$-algebra of dimension $2$, such as the localization of $\mathbb{C}[x,y]$ at $(x,y)$, and let $\nu=(\nu_1\geq\nu_2\geq\cdots\geq\nu_{\ell}\geq0)$, $\mu=(\mu_1\geq\...
- 23.3k
5
votes
1
answer
408
views
On universally closed morphisms of reduced schemes
In this question I'd like to examine some properties of universally closed morphisms.
The question is self-contained. It can also be seen as a follow-up to this question.
Let $R$ be a discrete ...
user315884
5
votes
1
answer
1k
views
(geometric/intuitive) interpretation of ext
In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, but I don't have a feeling about the meaning of ext.
Is there a informal/intuitive ...
- 345
5
votes
0
answers
288
views
Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
- 51
5
votes
0
answers
232
views
Coherence of the monoid algebra of a non-finitely generated monoid
Let $P$ be an integral, sharp, finitely generated commutative monoid (say even torsion-free and saturated if you like), and consider the "rational cone" $P_\mathbb{Q}\subseteq P^{gp}\otimes_\mathbb{Z}...
- 1,030
5
votes
0
answers
331
views
Extensions of maps between graded modules
Let $R$ be a connected graded ring (like $R=\mathbb R[x_1,\dots,x_d]$ with the usual grading) and let $N \subset R^{\oplus n}$ be a graded submodule, i.e. $$N= \bigoplus_{i \in \mathbb N} (N \cap R^{\...
- 25.5k
4
votes
1
answer
694
views
Proper mapping theorem
Let $Z\to X$ be a closed immersion of schemes. Assume $\mathcal{O}_Z$ and $\mathcal{O}_X$ both are coherent sheaves of $\mathcal{O}_Z$, resp. $\mathcal{O}_X$-modules.
In particular, the coherent ...
user120812
4
votes
1
answer
244
views
Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$
Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...
- 790
4
votes
1
answer
357
views
Surjectivity of natural map of rings
$\DeclareMathOperator\Hom{Hom}$Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ then is the following natural map
$$\Hom_A(P,A)\otimes_A B\to \Hom_A(P,B)$$ surjective?
...
- 585
4
votes
1
answer
327
views
Detecting closed immersions on fibers
Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes.
Assume $X$ and $S$ are $R$-flat and universally closed.
If the special fiber of $X\to S$ is a closed immersion, is $X\...
user315884
4
votes
0
answers
218
views
derived symmetric powers of an ideal
Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$.
We have a canonical map $Lsym^k(m)\to m^k$ from the ...
- 4,189
4
votes
0
answers
74
views
self-cogenerator rings
Let $\mathbb{U}$ be a non-empty set (class) of objects of a
category $C$. An object $B$ in $C$ is said to be cogenerated by
$\mathbb{U}$ or $\mathbb{U}$-cogenerated if, for every pair of
distinct ...
- 41
3
votes
1
answer
509
views
Quotients of Gorenstein rings
Let $R$ be a reduced Noetherian ring. Assume $R$ is quasi-excellent and Cohen-Macaulay.
Is $R$ the quotient of a Gorenstein ring?
If the answer is yes, then $R$ has a dualizing complex. The question ...
user322153
3
votes
1
answer
260
views
K-projectivity for rings of finite homological dimension
Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$...
- 959
3
votes
1
answer
247
views
Support of cohomology of a dualizing complex
Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the ...
- 33
3
votes
1
answer
420
views
Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness ...
- 183
3
votes
1
answer
239
views
commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
- 1,209
3
votes
1
answer
496
views
Regular ring is smooth when the field is perfect
Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
3
votes
1
answer
243
views
Few commutative algebra questions
A few commutative algebra questions for which I have no reference
For $P$ = "catenarian", "coherent", " Jacobson":
1- is an arbitrary product of rings satisfying $P$, a ring satisfying $P$?
2- if a ...
user95222
3
votes
1
answer
422
views
Tor dimension in polynomial rings over Artin rings
I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
- 165
3
votes
0
answers
271
views
Explanation for devissage argument
Let $K$ be a local field of characteristic $0$ with the ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $k$ be the residue field of $K$ with $\text{card}(k)=q$. Let $\mathcal{O}_\mathcal{E}...
- 101
3
votes
0
answers
204
views
Yoneda extension and splittings
Let $X$ be a non-singular algebraic variety and $F$ be a coherent sheaf defined over $X$. Suppose that we have a locally free resolution
$$0 \to L_n \xrightarrow{f_n} L_{n-1} \to ... \to L_0 \to F \to ...
- 2,126
3
votes
0
answers
240
views
Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes
Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
- 642
3
votes
0
answers
175
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
- 26.5k
3
votes
0
answers
450
views
Ext groups of affine scheme
Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...
- 491
2
votes
2
answers
961
views
Characterization of projective modules in terms of Ext groups
This is from Hartshrone exercise 6.6 part (a).
Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following
$M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{...
- 395
2
votes
2
answers
1k
views
when tensor complex resolves S/I+J?
Assume that $I\subset k[x_1,\ldots,x_n]$ and $J\subset k[y_1,\ldots,y_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F_\cdot$ and $G_\cdot$, are ...
- 43