All Questions
Tagged with homological-algebra ag.algebraic-geometry
154 questions with no upvoted or accepted answers
2
votes
0
answers
101
views
Find a Morita equivalent finite cell DG category
I am trying to understand the following statement:
Suppose that $\mathcal{E}$ is a pre-triangulated proper DG category with a full exceptional collection. Then $\mathcal{E}$ is Morita equivalent to a ...
- 21
2
votes
0
answers
371
views
How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
- 711
2
votes
0
answers
173
views
de Rham cohomology of a specific ring
I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of:
$$
\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
- 1,077
2
votes
0
answers
174
views
Interpretation of some maps involving cohomology groups
I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
- 895
2
votes
0
answers
486
views
An alternative proof of Künneth spectral sequence, independent of Künneth formula for homology
I am currently reading Künneth spectral sequence, which is given below.
Let $R$ be a ring and A$=\big\{A_n,d_n:A_n\longrightarrow A_{n-1}\big|d_{n-1}\circ d_n=0\big\}_{n\in \Bbb Z}$ be a chain ...
- 632
2
votes
0
answers
136
views
A infinity structure on Yoneda Ext group
I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
- 213
2
votes
0
answers
118
views
Adjoining data about singularities to "correct" the category of pure motives?
There are a few well known constructions of potential categories of pure motives for smooth projective varieties over a field. My understanding is that modulo the standard conjectures these should be ...
- 1,635
2
votes
0
answers
177
views
Geometric meaning of Koszul modules
Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\...
- 439
2
votes
0
answers
833
views
Applications of Jordan-Holder theorem in an abelian category
The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length.
This theorem holds ...
- 129
2
votes
0
answers
213
views
Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
- 7,952
2
votes
0
answers
314
views
Derived category of coherent sheaves with a codimension $\geq$ 1 support
Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories:
$D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the ...
- 988
2
votes
0
answers
325
views
A question on direct limits of rings, and descent of ideals
Motivated by an étale cohomology calculation I am going to do, here is a question that should have positive and not too hard answer.
Let $A$ be a ring, $A_0$ a ring such that $A_0$ is equipped with ...
user92332
2
votes
0
answers
20
views
Cohomology of $\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a$
Let $X$ be a smooth projective variety over a field.
Is $$H^p(X_{Zar},\mathbf{G}_a\otimes_{\mathbf{Z}}\mathbf{G}_a)$$
at all related to $H^p(X_{Zar},\mathbf{G}_a) = H^p(X,\mathcal{O}_X)$ via tensor ...
user92332
2
votes
0
answers
100
views
Definition of an attractor of a stack under an action of $G_{m}$
For an algebraic space $Z$ with the action of multiplicative group scheme $G_{m}$ one can define the attractor space $Z^{+}$ as the functor which sends a scheme $S$ to the set $Map(S \times A^{1},Z)^{...
- 21
2
votes
0
answers
178
views
Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,...
- 7,789
2
votes
0
answers
868
views
depth of ideal in polynomial ring
Let $R$ be a Noetherian local ring. Then the "depth" of an ideal $I$ measures the maximal length of regular sequence inside $I$. And $depth (R/I)$ measures the maximal length of regular sequence in ...
- 183
2
votes
0
answers
232
views
Example of non-locally finite stability condition
I am trying to work out example 5.6 in Bridgeland's paper "Stability conditions on triangulated categories".
http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf
A standard ...
- 71
2
votes
0
answers
866
views
How to prove that any perfect complex on an affine scheme is strictly perfect?
Let $(X,\mathcal{A})$ be a ringed space. A complex $\mathcal{S}^{\bullet}$ of $\mathcal{A}$-modules is $\textit{perfect}$ if for any point $x\in X$, there exists an open neighborhood $U$ of $x$ and a ...
- 9,019
2
votes
0
answers
1k
views
What is a Beilinson spectral sequence?
I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...
- 21
2
votes
0
answers
287
views
Flat and injective quasi-coherent sheaves
Let $X$ be a quasi-compact semi-separated scheme and
$$\varepsilon: 0\to A \to B \to C \to 0$$
be a short exact sequence of quasi-coherent sheaves. $\varepsilon$ is called a (categorical) pure exact ...
- 141
2
votes
0
answers
254
views
semi-orthogonal decompositions and embeddings
This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition
$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$
Suppose ...
- 3,779
2
votes
0
answers
88
views
Relationship of height zero hypercovers to co-cartesian condition on cosimplicial modules
Suppose given a cosimplicial ring $R^\bullet$ and a cosimplicial module $M^\bullet$ (i.e. a cosimplicial Abelian group such that $M^n$ is an $R^n$-(left/right/bi)module). I have seen it said that $M^\...
- 10.4k
2
votes
0
answers
275
views
Tilting objects and Koszul algebras
Let $X$ be a smooth variety over a field, and $T\in D^b(Coh(X))$ be a tilting object, i.e. (1) $Ext^i(T,T)=0$ for all $ i\neq 0$; (2)Tilting algebra $A:=End(T)$ has finite global dimension; (3)$add$-$...
- 21
1
vote
0
answers
96
views
Chain complexes indexed over measurable subsets of $\mathbb{R}$: Towards a measurable notion of Euler Characteristic
I have for a while tried to generalize the notion of a chain complex in a way to obtain a "continuous" or at least "measurable" notion of Euler Characteristic.
I have come up with ...
- 1,805
1
vote
0
answers
175
views
Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup
This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...
- 1,805
1
vote
0
answers
132
views
A question concerning cancellation of ideals
I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am ...
- 11
1
vote
0
answers
111
views
Kunneth formula for hypercohomology
Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
- 494
1
vote
0
answers
52
views
Is the Schofield semi invariant defined at $V/IV$?
Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
- 839
1
vote
0
answers
115
views
Prove that $B$ is a directing module
Let $A\cong\mathbb{K}Q/I$ be a finite dimensional, associative, basic $\mathbb{K}$ algebra, where $\mathbb{K}$ is algebraically closed and $Q$ is a finite Gabriel quiver on $n$ vertices and $I\...
- 839
1
vote
0
answers
93
views
Spectral sequences associated to cohomologies of simplicial type and derived-functor type: Proof of convergence
Assume I have two cohomology theories $\mathrm{\tilde{H}^{*}}$ and $\mathrm{H^{*}}$, the latter being defined over a Grothendieck site $X$ as the derived functor of some left-exact covariant functor $\...
- 1,805
1
vote
0
answers
157
views
Admissible subcategories of derived categories
Let $\mathcal A$ be an abelian category; $\mathcal B \subseteq \mathcal A$ a weak Serre subcategory. Does the inclusion $\mathbf D_{\mathcal B}(\mathcal A) \subseteq \mathbf D(\mathcal A)$ admit a ...
- 988
1
vote
0
answers
199
views
Künneth formula for local cohomology with support
In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
- 473
1
vote
0
answers
41
views
Homology of a complex constructed by semi-local simplices
Let $\Delta^{\bullet}$ be the cosimplicial scheme that at each level is given by the algebraic simplex over the field $k$ which contains $\mathbb{C}$. Let $\hat{\Delta}^{\bullet}$ be semi-localization ...
- 5,901
1
vote
0
answers
72
views
Bound on Hochschild dimension of a dg-algebra
Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$?
More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...
- 213
1
vote
0
answers
166
views
To see that the fundamental class of a local complete intersection is independent of choice of regular sequence
In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...
- 1,217
1
vote
0
answers
154
views
What does "standard Koszul morphism" mean?
I'm reading a paper 'D'Andrea, Carlos(RA-UBA), Dickenstein, Alicia(RA-UBA)Explicit formulas for the multivariate resultant. (English summary)
Effective methods in algebraic geometry (Bath, 2000).
J. ...
- 663
1
vote
0
answers
32
views
A regular sequence in a quotient by a "half lattice" defined by a toric manifold
I am interested in some properties of polynomial algebras associated with smooth compact toric varieties. Recall that a toric manifold can be obtained as a quotient $$P^{-1}(p) / \mathbb{K}$$ by the ...
- 1,227
1
vote
0
answers
140
views
On finite dimensional commutative algebras and regular sequences
Let $\mathbb{C}[u]:= \mathbb{C}[u_1,...,u_n]$ the algebra of polynomial functions on $\mathbb{C}^n$. I consider two ideals $I$, and $J$, where $I$ is the ideal generated by the polynomials vanishing ...
- 1,227
1
vote
0
answers
74
views
On some finiteness properties of cohomological algebras of complex tori
Denote by $A := \mathbb{C}[u,u^{-1}]$, $u = (u_1,...,u_n)$, the algebra of polynomial functions on the complex torus $(\mathbb{C} \setminus \{ 0 \})^n$, which we consider as a $\mathbb{C}[u]$-module.
...
- 1,227
1
vote
0
answers
83
views
Direct limit of complexes from cocycle property up to homotopy
Suppose that we have a directed set $(I, \leq)$, and a set of maps
\begin{equation}
f_{i,j} : C^*(A_i) \to C^*(A_j), \quad i \leq j,
\end{equation}
of singular cochain complexes of topological spaces $...
- 1,227
1
vote
0
answers
113
views
Reference request. The adjunction $\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$
We have the adjunction
$$\hom_{CDGA}(\Omega^\bullet(A),B^\bullet)\cong\hom_{CA}(A,B^0)$$
where $CDGA$ is the category of commutative diffferential graded algebras and $CA$ is the category of ...
- 1,615
1
vote
0
answers
103
views
Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences
I hope this question is not too basic.
Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.
I am ...
1
vote
0
answers
135
views
Relation of primary decomposition of two ideals
I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...
1
vote
0
answers
606
views
Is local-to-global spectral sequence functorial?
Consider a lower term of local-to-global spectral sequence
$0 \to H^1(X,\mathcal{Hom}(\mathcal{F},\mathcal{G})) \to Ext^1(\mathcal{F},\mathcal{G}) \to H^0(X,\mathcal{Ext}^1(\mathcal{F},\mathcal{G})) \...
- 81
1
vote
0
answers
235
views
Récollement of stable $t$-structures
Given a recollement
$$
\mathbf{D}^0 \underset{\underset{i_R}\leftarrow}{\overset{\overset{i_L}\leftarrow}\to} \mathbf{D} \underset{\underset{q_R}\leftarrow}{\overset{\overset{q_L}\leftarrow}\to} \...
- 13.6k
1
vote
0
answers
165
views
Ext and cup products and subvarieties
I am trying to understand Remark 11.3 in Huybrechts's amazing book on derived categories (FM transforms in AG).
He starts with smooth projective varieties $j\colon Y \subset X$ and aims to describe ...
- 1,280
1
vote
0
answers
125
views
Realization of a formal duality isomorphism via integration
This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of $X$...
- 1,331
0
votes
0
answers
168
views
Theorems related to Chevalley's theorem
Recently I have read Chevalley's theorem of a complete local ring which basically says that if $(R,\mathfrak{m})$ is a complete local ring and if $\{b_n\}$ be a sequence of ideals such that $b_n \...
0
votes
0
answers
132
views
Example of a periodic free resolution over a hypersurface
I'm reading "HOMOLOGICAL ALGEBRA ON A COMPLETE INTERSECTION,
WITH AN APPLICATION TO GROUP REPRESENTATIONS" by David Eisenbud
I'm wondering what would be a nice example illustrating Theorem 6....
- 839
0
votes
0
answers
86
views
On the dimension of the cohomology of toric manifolds
Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...
- 1,227