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52 votes
7 answers
5k views

What does a projective resolution mean geometrically?

For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
Justin DeVries's user avatar
16 votes
1 answer
754 views

When is every "solid" perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy. ...
Paul Balmer's user avatar
12 votes
1 answer
1k views

Fullness of pullback functor in algebraic geometry

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can ...
David Jordan's user avatar
  • 6,131
11 votes
1 answer
513 views

When does derived tensor product commute with arbitrary products?

Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
uno's user avatar
  • 412
11 votes
1 answer
1k views

Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \...
Theo Johnson-Freyd's user avatar
10 votes
0 answers
312 views

Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?

Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user avatar
10 votes
0 answers
241 views

Has anyone seen this construction of dg algebras?

Let $A$ be an associative algebra, $M$ a right $A$-module. Suppose we are given an $A$-module homomorphism $M \to A$. Then we can make $M$ itself into an associative algebra via the multiplication $$ ...
Dan Petersen's user avatar
  • 40.2k
8 votes
1 answer
413 views

Chain complexes split in the derived category over rings of global dimension 1

Let $R$ be a ring of global dimension $1$. Then I have seen the claim (in a paper, and in this MO post When do chain complexes decompose as a direct sum?) that any chain complex over $R$ is equivalent ...
user142700's user avatar
8 votes
0 answers
337 views

flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion. If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat. If $A$ is not noetherian, ...
prochet's user avatar
  • 3,472
7 votes
1 answer
533 views

Lifting isomorphisms between derived categories

(Remark: I first asked this question at math.stackexchange. As it received no answer, I'm posting it here). Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. ...
anonymous's user avatar
6 votes
1 answer
175 views

Relative Ext of Avramov-Martsinkovsky as a derived Hom

Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their ...
Daniel Pomerleano's user avatar
6 votes
0 answers
399 views

Unbounded derived Nakayama lemma

Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field. Let $X$ be a complex of $R$-modules with finitely generated ...
Maxime Ramzi's user avatar
  • 15.8k
6 votes
0 answers
607 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
Mahdi Majidi-Zolbanin's user avatar
5 votes
1 answer
379 views

Jacobson radical of a derived $I$-complete ring

Let $A$ be a commutative ring and $I \subseteq A$ a finitely generated ideal (I am not assuming that $A$ is Noetherian). Assume that $A$ is derived $I$-complete, meaning, let's say, that $\mathrm{...
Pavel Čoupek's user avatar
5 votes
1 answer
248 views

On the bounded derived category of sheaves with coherent cohomology

Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
FPV's user avatar
  • 541
5 votes
0 answers
586 views

When is the cotangent complex perfect?

Let $X\rightarrow S$ be a proper flat morphism of schemes. When is the cotangent complex $L_{X/S}$ perfect ? It is well known, that for local complete intersections the cotangent complex is perfect, ...
Can Yaylali's user avatar
4 votes
2 answers
285 views

Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?

Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
uno's user avatar
  • 412
4 votes
1 answer
267 views

A particular morphism being zero in the singularity category

Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
strat's user avatar
  • 361
4 votes
1 answer
558 views

derived tensor product and finite projective dimension

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules. Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ ...
strat's user avatar
  • 361
4 votes
1 answer
227 views

Literature Request: The derived category is Krull-Schmidt

I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question Literature request: $K^b(\text{...
Sebastian Pozo's user avatar
4 votes
1 answer
503 views

Mapping cone and derived tensor product

This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
slinshady's user avatar
  • 309
4 votes
0 answers
113 views

Determining whether a morphism is the induced morphism?

Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...
Avi Steiner's user avatar
  • 3,079
3 votes
1 answer
378 views

Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
bananastack's user avatar
  • 1,280
3 votes
1 answer
173 views

Characterizing the image of $D(A_f) \rightarrow D(A)$

Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A_f$-module. Now suppose that $C \in D(A)$ is a complex ...
Lisa S.'s user avatar
  • 2,663
3 votes
1 answer
92 views

Pure complex isomorphic to complex of free modules in derived category

This is a basic question, but I don't know the answer. Suppose $M$ is an $R$-module, considered as a complex concentrated in degree $0$. Let $F^*$ be an complex consisting of free modules. Recall that ...
user2520938's user avatar
  • 2,788
3 votes
0 answers
120 views

Derived tensor by perfect complex preserves exact triangle in singularity category?

Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
Snake Eyes's user avatar
3 votes
0 answers
106 views

Multiplication map by a ring element on an object vs. all its suspensions in singularity category

Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
uno's user avatar
  • 412
3 votes
0 answers
161 views

A Nakayama type of claim for countably generated modules on complex affine varieties

Let $U\subset \mathbb{A}^n_{\mathbb{C}}$ be any Zariski open affine subvarity. Let $M$ be an $\mathcal{O}(U)$-module. Suppose $M$ satisfies $M\overset{L}{\otimes}\mathbb{C}_{\mathfrak{M}}\cong 0$ for ...
Amanda Taylor's user avatar
2 votes
1 answer
361 views

If tensor product has finite length, do higher Tors also have finite length?

Let R be a local noetherian ring and let M, N be two finitely generated modules. Is it true that if $M \otimes_R N$ has finite length, then $Tor_i^R(M,N)$ also has finite length for all i? I know ...
Yosemite Sam's user avatar
  • 1,889
2 votes
1 answer
131 views

derived completion and flat base change

Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings. We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete. For a ...
OOOOOO's user avatar
  • 349
2 votes
1 answer
98 views

A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree

Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
Snake Eyes's user avatar
1 vote
1 answer
169 views

perfect modules over polynomial algebra

This may be obvious. My question is short: $R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...
John-Ed's user avatar
  • 117
1 vote
1 answer
149 views

Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
feder's user avatar
  • 73
1 vote
1 answer
124 views

Tensoring with complex of finite flat dimension in derived category

Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...
tessellation's user avatar
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}...
S.D.'s user avatar
  • 494
1 vote
0 answers
359 views

Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
Mahdi Majidi-Zolbanin's user avatar