5
votes
1answer
220 views

Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be $$ \text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!, $$ where $ P_{M} $ denotes the Hilbert ...
4
votes
0answers
224 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 ...
1
vote
2answers
203 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 ...
4
votes
2answers
359 views

Homological characterization of smooth maps

Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings. As far as I understand, in various generalizations of this situation, such a map is called smooth if $B$ ...
0
votes
2answers
154 views

Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...
30
votes
3answers
3k views

What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...
2
votes
2answers
339 views

Homological dimensions of module

$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. ...