All Questions
28 questions
1
vote
1
answer
610
views
The Krull dimension of the tensor product of rings
The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
- 35
2
votes
1
answer
184
views
Lazard module structure of rings with formal elliptic curve
Recently in algebraic topology I was working with a certain graded ring $R$ equipped with an elliptic curve $C$. Now completion at the identity gives a 1-dimensional formal group $G$. This induces a ...
- 23
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
2
votes
0
answers
169
views
The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
3
votes
0
answers
116
views
Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
3
votes
0
answers
249
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
4
votes
1
answer
466
views
Top local cohomology - recommendations
I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand ...
user340793
2
votes
0
answers
172
views
Intersection theory on normal crossing algebraic surfaces
Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...
- 494
1
vote
0
answers
250
views
Local cohomology: Polynomial ring vs Power series ring
I study algebraic topology and am currently examining the applications of local (co)homology in algebraic topology. We have the canonical inclusion of rings $\mathbb{Z}[x_1,\cdots,x_n]\subset \mathbb{...
user340793
1
vote
0
answers
173
views
The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
13
votes
2
answers
2k
views
Why doesn't local cohomology seem to be used as much in algebraic geometry?
In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations.
In algebraic geometry, we have local cohomology,...
- 711
3
votes
0
answers
225
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Complexity: Groebner bases method vs homotopy continuation method
Today, I came to know about homotopy continuation method to solve system of multivariate polynomials. This method finds its roots from the field of Numerical algebraic geometry.
I already know that ...
- 131
1
vote
0
answers
213
views
Defining path on the prime spectrum
If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...
5
votes
1
answer
384
views
Euler characteristic and rational Poincaré series
$\DeclareMathOperator\len{len}\DeclareMathOperator\Tor{Tor}$Let $(A,\mathfrak{m})$ be a regular local ring, and $x \in \mathfrak{m}^2$ be a non-zero prime element. So $R:=A/(x)$ is a non-regular Cohen-...
- 125
8
votes
0
answers
313
views
Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
13
votes
1
answer
2k
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Proj construction in derived algebraic geometry
The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...
- 133
1
vote
1
answer
218
views
stable sheaves in characteristic $0$
Let $K$ be a non-algebraically closed field of characteristic $0$ and $X_K$ a smooth, projective, geometrically connected curve defined over $K$. If $F$ is a stable locally free sheaf on $X_K$, is it ...
- 2,323
2
votes
1
answer
278
views
Intersection of two curves is not Cohen Macaulay
Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$.
(a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
34
votes
2
answers
933
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If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?
I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
- 1,462
18
votes
1
answer
770
views
Koszul complex for non-Koszul algebras
Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal $...
- 4,133
7
votes
1
answer
543
views
What are some examples of total derived functors that can't be computed from a functorial replacement?
(Or more generally, what are some examples of Kan extensions which are not pointwise?)
By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
- 63.9k
23
votes
6
answers
2k
views
Pathological Examples of Dimension
I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...
3
votes
2
answers
653
views
Does the first singular cohomology of an ACM surface vanish?
Hi everybody, I am interested in the following:
Let $I\subset S=\mathbb{C}[x_0,\ldots ,x_n]$ be a graded ideal such that $\operatorname{depth}(S/I)\geq 3$, and let $X^h$ denote the analytic space ...
15
votes
3
answers
3k
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State of the art for Gersten's conjecture for K-theory?
Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \...
- 1,347
9
votes
0
answers
513
views
E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product
Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 Ben-...
- 3,758
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
5
votes
1
answer
631
views
Showing an Ext^2 element is zero
If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1_X(G,E)$, we need to show that this sequence splits. To produce a splitting ...
- 145
7
votes
2
answers
637
views
An algebraic proof of Mumford's smoothness criterion for surfaces?
(Disclaimer: I'm a beginner in this area, so welcome corrections.)
Let $(X,x)$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $x$ an isolated ...
- 5,846