Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,496 questions
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Abstract number ring of any characteristic
Let $(n_1,\ldots, n_i,\ldots)$ be an infinite tuple of nonnegative integers. Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals (...
user16974
2
votes
1
answer
325
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Gersten for homotopy invariant K-theory of non-singular varieties.
Here is the question:
if $X$ is a separated, finite type scheme over a perfect field (but not necassarily smooth) is the map $KH_n(X) \to \prod_{x \in X^{(0)}} KH_n(k(x))$ injective?
If $X$ is ...
- 1,347
3
votes
2
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344
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Pseudo-idempotent matrix generating a free module
Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
- 33.8k
1
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1
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178
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Surjective and injective criteria via Hilbert polynomials
Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}_X\rightarrow \mathcal{M}$...
- 13
5
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0
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454
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If $p=0$ and $df=0$, is $f$ a $p$th power?
This question is a follow-up to When does the relative differential $df=0$ imply that $f$ comes from the base?. There it was asked, for an $A$-algebra $B$, under what conditions does $df=0$ (in the ...
- 4,487
4
votes
1
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398
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A terminology question: formally finite ??
Is there a name for a local homomorphism $\varphi:A\longrightarrow B$ of local rings $A$ and $B$, whose completion $\hat{\varphi}:\hat{A}\longrightarrow\hat{B}$ is a finite homomorphism? (that is, $\...
- 3,101
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0
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538
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Picard Group of a singular surface with a non-rational singularity
I've been spending some time looking at the surface $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$ in $\mathbb{A}^3$ (over an algebraically closed field of characteristic different from 3). The surface has four ...
- 319
5
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2
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732
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What is the completion at a family of ideals?
Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition
$$
\widehat{A} := \underset{\leftarrow}\lim A/I^n.
$$
This operation is ...
- 3,704
0
votes
1
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149
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Partial dehomogenization and smoothness
Let $P_1l_1+P_2l_2$ be a homogeneous degree $d$ polynomial in $\mathbb{C}[X_0,X_1,X_2,X_3]$ which defines a smooth surface in $\mathbb{P}^3$. Here $l_i$ are linear polynomials and $l_1 \not=\lambda ...
- 1,040
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0
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242
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separability of commutative rings
Before discussing on the main Question I should recall two notions in the area of commutative rings.
By $Max(R)$, we mean the set of all maximal ideals of the commutative unitary ring $R$.
...
- 1,788
1
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1
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815
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Can we characterise affine open subschemes of ${\rm Spec}(A)$?
Let $A$ be any ring, commutative with identity, and let $I\subset A$ be an ideal $\neq A$. Let $U\subset{\rm Spec}(A)$ be the open subscheme obtained by "removing" the closed set $V(I)$ of all the ...
- 11
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1
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133
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Ideal membership (concerning polynomial invariants of orthogonal groups)
Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials
$$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$
I'm ...
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0
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245
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Sums of Strongly z-ideals
In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...
- 1,788
3
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603
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Norms in Galois extensions
Let $k$ be a field of characteristic 0, and $\overline k$ be a fixed algebraic closure of $k$.
Let $k\subset F\subset E$ be a tower of finite Galois extensions in $\overline k$,
where both $\mathrm{...
- 14.2k
7
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0
answers
329
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Computer Algebra solution for simplicial resolutions for André-Quillen cohomology
Hello,
I would like to experiment with André-Quillen (co)homology. Especially for singular rings.
A key problem is that the construction of a simplicial resolution of a ring seems to require a rather ...
- 71
14
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0
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899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
3
votes
1
answer
171
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If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion?
Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion.
If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec}...
- 7,338
3
votes
2
answers
611
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Computing Integral Closures
I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of &#...
- 1,386
1
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0
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273
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Depth of intersection
Let $I$ be an ideal in $S=K[x_1,\dots,x_n]$. Can we compute $\operatorname{depth}(I\cap K[x_1,\dots,x_r])$ with $r \leq n$? Is there any relation between depth $I$ and $\operatorname{depth}(I\cap K[...
- 287
5
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1
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220
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When are these rings regular?
Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bX-a)$. Is $S$ regular? If this is not the case are there some ...
- 3,704
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votes
0
answers
166
views
The intersection complex and the Cohen-Macaulay property
Let $\Delta:Y\rightarrow X$ a closed immersion of $k$-schemes of finite type and equidimensionnal.
We assume that $\Delta^{*}[-d]IC_{X}=IC_{Y}$, if $X$ is Cohen-Macaulay, does it imply that $Y$ is ...
- 3,472
3
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0
answers
495
views
An elementary proof of a criterion for $k$ sufficiently large that $M_k = \Gamma(\mathbb{P}^n, \widetilde{M}(k))$
It is well known that coherent sheaves on $\mathbb{P}^n$ are equivalent, as a category, to finitely generated graded modules over the polynomial ring, provided that in the latter category, morphisms ...
- 7,338
0
votes
1
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136
views
pd finite for finite module over local CM ring?
Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?
- 287
3
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0
answers
112
views
Nullspace of a matrix modulo an ideal
Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.
Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$.
Thus, $M$ has a non-...
- 31
4
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1
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266
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Exotic uniform algebras
The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...
- 43
2
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1
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193
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Terminology for system of equations and...
I am looking for the standard term for a system that consists of things of the form
$p_i(x_1,\ldots ,x_n)=0$ and of the form $q_j(x_1,\ldots,x_n)\neq 0$ with the $p_i$ and $q_j$ polynomials. I have ...
1
vote
1
answer
307
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A problem on Moebius transformations
We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (...
- 73
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279
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Uncountable Lüroth problem
Question. Let $F(X)$ be the field obtained by adding an uncountable collection of indeterminates (mutually transcendental elements) to a prime field $F$. Is there an example of a subfield $E$ of $F(X)$...
- 17.7k
2
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1
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323
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Presentation of finite modules with null annihilator
Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: $R^n\stackrel{\varphi}{\...
- 3,101
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72
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Decomposition of polynomials with three variables
We use $\bigtriangleup _i$ to denote either multiplication or addition.
Suppose we have a polynomial $P(x,y,z)$ over some algebraic closed field such that:
There are $Q(x), W_1(x,y),W_2(x,z)$ sucht ...
- 11
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0
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355
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Is the invariant part of the canonical module by finite group action nonzero?
This is further question to this.
Let $q \in V=(f=0) \subset \mathbb{C}^{n+1}$ be an isolated rational singularity of dimension $n$. Suppose that $ G:=\mathbb{Z}/m \mathbb{Z}$ acts on $V$ freely ...
- 909
4
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What is the abstract relationship between an indecomposable representation and a sum of irreducible representations with the same character?
$\mathbb{Z}$ is the simplest example of a group with indecomposable representations which are not irreducible. Over $\mathbb{C}$, the isomorphism classes of $n$-dimensional representations of $\...
- 118k
6
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0
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106
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Irreducibility testing and factoring
It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
- 96.4k
1
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531
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Krull's intersection theorem for commutative local not necessarily noetherian rings
Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module ...
4
votes
1
answer
382
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Is the subspace of DVR's of the Zariski-Riemann space still quasi-compact?
If $S$ is the Zariski-Riemann space of a noetherian subring $k$ of a field $K$, Zariski-Samuel prove that $S$ is quasi-compact. If $S'$ is the subspace of valuations that are discrete (i.e. that ...
- 1,347
6
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1
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301
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Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 2,179
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1
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325
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Is a particular element of a particular ring a nonzerodivisor?
Let $A$ be the ring $\Bbbk[\alpha_0, \alpha_1, \alpha_2, x_0, x_1, x_2]$ (where $\Bbbk$ is an infinite field, algebraically closed if it matters). Let $g \in \Bbbk[\alpha_0, \alpha_1, \alpha_2]$ be a ...
- 7,338
0
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1
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315
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Generalized Picard group (reflexive fractional ideals, principal ideals)
Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, $y^...
- 1,391
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93
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The structure of symmetric powers of finite-dimensional local rings
Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the ...
- 3,637
0
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0
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346
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Length of $\mathfrak{m}$-torsion module
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module such that $\mathfrak{m}^tM=0$ for some non-negative integer $t$. Then the length of $M$ is finite.
Is that right?...
- 259
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139
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ideal generated by highest weight vectors
Let $S$ be a polynomial ring which carries the action of a semi-simple linear algebraic group $G$ (I'm interested in a product of $GL$'s). Take $S$ and $G$ to be over an algebraically closed field.
...
- 31
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234
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Set of Curves Passing through a smooth point of a Variety is Zariski-Dense
In a paper by F. Pop he claims the following fact-
Let $K$ be a field. The set (by which I believe he means the union) of all smooth $K$-curves passing through a smooth $K$-rational point of an ...
- 11
1
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0
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35
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Projectivity of a faithfully balanced self-orthogonal bimodule
Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right $S$...
- 1,207
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2
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205
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Structure of Homomorphisms of commutative C^* algebra
Being new to $C$* algebra, I'm trying to understand basic properties of -homomorphisms of such algebras. Let $P$ be a set of commuting projections on Hilbert space ${\cal H}$.
Let ${\cal P}$ be the $...
- 45
3
votes
1
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268
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Universal catenarity and Laurent algebras
A Noetherian (commutative) ring $A$ is called universally catenary if every $A$-algebra of finite type is catenary. If one wants to know whether $A$ is universally catenary, then this definition ...
- 6,700
3
votes
0
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197
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which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?
Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, $...
- 2,232
2
votes
1
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326
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Flatness on the fiber
Hi.
Let $f:A\rightarrow B$ be a local morphism of locally noetherian (reduced) rings with $B$ $A$-flat. Let $M$ be an $B$-module of finite type.
Question: Which conditions ensure the following:
$N\...
- 435
4
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0
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259
views
Software for Computations with Complexes
What software would you recommend for working with chain complexes? In particular I'd like to be able to compute cohomology of a finite complex of free modules over polynomial ring. Is it possible in ...
- 607
6
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0
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881
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
2
votes
1
answer
118
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Dual of a semilinear morphism
Let $R$ be a commutative ring and let $M$ and $N$ be $R$-modules. Let $\sigma:R\rightarrow R$ be a ring automorphism.
Let $f: M\rightarrow N$ be a $\sigma$-semilinear map, i.e. a map of abelian ...
- 11.1k