10
votes
1answer
169 views

Homological criteria for finite generation and finite presentation of modules?

(I'm new here; if I'm doing something wrong please help me out.) In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of ...
2
votes
2answers
194 views

local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...
2
votes
1answer
95 views

Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...
5
votes
2answers
219 views

Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...
3
votes
1answer
176 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 ...
2
votes
1answer
59 views

Kernel of the induced map of the wedge product

Let $A$ be a noetherian ring and let $M$ be a finitely generated $A$-module. Let $F$ be a free $A$-module and let $d: F \to M$ be a homomorphism which maps a basis of $F$ to a minimal set of ...
2
votes
0answers
91 views

syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...
1
vote
1answer
239 views

Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...
3
votes
0answers
154 views

Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...
0
votes
1answer
90 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
4
votes
1answer
205 views

Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms $f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$ for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put ...
4
votes
1answer
173 views

Algebra structure $Tor(A,A)$

This is a question i asked on math.stackexchange but i didn't get any answer. Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...
4
votes
1answer
113 views

Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?
5
votes
1answer
357 views

Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference. Let $A$ be a regular commutative noetherian ring (and ...
6
votes
3answers
349 views

zero homology of augmented Koszul complex implies the sequence is regular?

Let $A$ be a Noetherian ring, $M$ a finite $A$-module and $I=(y_1,\cdots,y_n)$ an ideal of $A$ such that $M \neq IM$. Denote by $H_i(y_1,\cdots,y_n;M)$ the homology at dimension $i$ of the augmented ...
13
votes
1answer
369 views

History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...
4
votes
0answers
151 views

What is known about this short exact sequence in Lie algebra cohomology?

In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact $ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
4
votes
1answer
282 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, ...
0
votes
1answer
169 views

cofree modules and dual

1, Why do people pay special attention to Q/Z in the definition of cofree modules instead of ordinary abelian groups? 2, Over a PID, is every injective module cofree? Just like the relationship ...
3
votes
1answer
130 views

is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module. Then, more generally, is every finitely ...
4
votes
1answer
329 views

example of Local cohomology

Let $S=k[x_1,...,x_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x_1,...,x_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0_m(M)=H^1_m(M)=0$ and $H^2_m(M)\neq 0$ be ...
3
votes
0answers
127 views

Is Hochschild cohomology finitely generated?

I am sorry if this is a naive question. Let $k$ be a field, and let $A$ be a finitely generated commutative $k$-algebra. Let $M$ be a finite $A$-module. Consider the Hochschild cohomologies of $A$ ...
6
votes
1answer
199 views

When are MCM ideals principal?

Suppose $(R,\mathfrak m, k)$ is a $d$-dimensional Cohen-Macaulay local ring with canonical module $\omega_R$ and $d>1$. Suppose $I\subset R$ is an ideal which is MCM (=maximal Cohen-Macaulay, ...
7
votes
0answers
349 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, ...
3
votes
1answer
207 views

finiteness of local cohomology

Well-known Theorem: Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated $R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a (M)$ is finitely generated for all $j < ...
1
vote
2answers
201 views

Decomposition of a quotient module

Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let $f:R^2 \rightarrow R^2$ denote the map given by the matrix $$M=\begin{pmatrix} v & y \\ x & z ...
2
votes
1answer
142 views

Is there an $A$ such that $B$ injective iff 1st Ext functor vanishes?

In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective iff ...
2
votes
1answer
269 views

An example of a tensor product consisting of only simple tensors?

Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\pi: A' \to A$ be a ...
2
votes
0answers
130 views

Coordinate free Koszul-Tate resolution

Tate's original construction of the Koszul-Tate resolution involved choosing cocycles representing the cohomology to be killed. Where is it written in a coordinate free treatment, perhaps via a ...
2
votes
1answer
129 views

Compatibility of connecting homomorphisms for Tor/Ext

This is a simple question about Tor and Ext functors. Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. ...
2
votes
0answers
141 views

About free resolutions of graded commutative algebras

Hi, I'm having troubles in adapting certain algebraic constructions to graded cases. We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the ...
1
vote
1answer
598 views

question on an exercise on homological algebra?

Suppose R has finite global dimension n, N is a f.g. module, F is a free module and Ext^n(N,F) is not equal to zero, then Ext^n(N, R) is not trival either. Note, HERE R is not Noetherian necessarily. ...
2
votes
3answers
362 views

Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite. Let us assume we are given a group $G$ and a ...
5
votes
2answers
981 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
3
votes
1answer
367 views

Vanishing of Tor

Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x_1,...,x_t$ be an $M$-regular sequence and $I = (x_1,...,x_t)$. Is it true that ...
6
votes
3answers
409 views

A lost lemma about periodicity in a grid of long exact sequences?

This is a question about finding references and hopefully a larger context for a lemma in homological algebra I proved recently. The motivation is to understand properties of characteristic classes of ...
4
votes
1answer
549 views

(geometric/intuitive) interpretation of ext

Hi folkz, 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 ...
2
votes
1answer
152 views

Proving indecomposability of special modules

Hi, I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf On page 6 there are the definitions (in a diagrammatical way) of some $A_n$ modules, whereupon ...
0
votes
1answer
175 views

Representation dimension of a special algebra

Hi, I'm reading the following paper: http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf I've come across a piece of information, which I don't understand, and wanted to ask, if I ...
4
votes
1answer
139 views

Turning a resolution of an algebra into a resolution of its field of fractions

Is there a way to turn a free resolution of a $k$-algebra $A$ into a resolution of the field of fractions $Q(A)$? Specifically, I'm interested in the ring of polynomials in two variables: $A = ...
2
votes
2answers
454 views

vanishing of local cohomology $H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$

Edited: I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ We know that if $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, then $$\operatorname{Supp} ...
0
votes
2answers
169 views

Projective dimension of cohomology over regular rings

Suppose $R$ is a regular ring and $F^{\bullet}: 0\to F^0 \to F^1 \to \dots \to F^d \to 0$ is a complex of finite rank free $R$-modules. Is is true that $\mathrm{projdim}H^i(F^{\bullet}) \leq d$ for ...
3
votes
0answers
131 views

Minimal Koszul-Tate resolutions

In what generality of commutative associative algebras does there exist a minimal Koszul-Tate resolution? Or what is the most general condition known?
11
votes
3answers
666 views

Serre's theorem about regularity and homological dimension

One of the nicest results I know of is (Auslander-Buchsbaum-)Serre's theorem asserting that a (commutative!) local ring is regular iff it has finite global dimensional. I'd like to ask a somewhat ...
3
votes
1answer
158 views

Do the classes of cotorsion and strongly cotorsion modules coincide?

Cotorsion modules A module M is called cotorsion if for all flat modules X, $Ext_R^1(X,M)=0$ . Strongly cotorsion modules M is called strongly cotorsion if for all modules X of finite flat ...
12
votes
2answers
1k views

Why are injective modules more complicated than projective modules?

For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. For example, for finitely generated modules over a noetherian ...
10
votes
1answer
617 views

When tensor reflects exact sequences?

Is there a characterization of modules $N$ for which the functor $N\otimes-$ reflects exact sequences?
4
votes
2answers
498 views

Why depth, dimension, etc?

Recently I came across a paper by Stephen McAdam "GRADE SCHEMES AND GRADE FUNCTIONS" (Transactions of the American Mathematical Society, Vol. 288, No. 2 (Apr., 1985), pp. 563-590). So far I have just ...
14
votes
0answers
347 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 ...
12
votes
3answers
641 views

Injective dimension of graded-injective modules.

In "Existence theorems..." Van den Bergh proposes the following "pleasant excercise in homological algebra": Let $A$ be a connected graded noetherian $k$-algebra (that is, $\mathbb N$-graded with ...