Introduction to resolutions [Lecture notes]

30 Pages·2011·0.24 MB·English

by  Irena Swanson

Introduction to resolutions
Irena Swanson, MSRI, Lecture 1
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz 1
d d
A complex: · · · → M −i→+1 M −→i M → · · · , where the M are groups
i+1 i i−1 i
(or left modules), the d are group (or left module) homomorphisms, and for
i
all i, d ◦ d = 0. Abbreviate as C , M or (M , d ), et cetera.
i i+1 • • • •
(M , d ) is bounded below if M = 0 for all sufficiently small (negative) i;
• • i
it is bounded above if M = 0 for all sufficiently large (positive) i;
i
it is bounded if M = 0 for all sufficiently large |i|.
i
(M , d ) is exact at the ith place if ker(d ) = im(d ).
• • i i+1
A complex is exact if it is exact at all places. (Also called exact sequence.)
A complex is free (resp. flat, projective, injective) if all the M are free
i
(resp. flat, projective, injective).
i p
An exact complex 0 → M0 −→ M −→ M00 → 0 is called a short exact
sequence.
http://www.reed.edu/~iswanson/MSRI11SwansonIntroResolutions.pdf
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
d d
A complex: · · · → M −i→+1 M −→i M → · · · , where the M are groups
i+1 i i−1 i
(or left modules), the d are group (or left module) homomorphisms, and for
i
all i, d ◦ d = 0. Abbreviate as C , M or (M , d ), et cetera.
i i+1 • • • •
(M , d ) is bounded below if M = 0 for all sufficiently small (negative) i;
• • i
it is bounded above if M = 0 for all sufficiently large (positive) i;
i
it is bounded if M = 0 for all sufficiently large |i|.
i
(M , d ) is exact at the ith place if ker(d ) = im(d ).
• • i i+1
A complex is exact if it is exact at all places. (Also called exact sequence.)
A complex is free (resp. flat, projective, injective) if all the M are free
i
(resp. flat, projective, injective).
i p
An exact complex 0 → M0 −→ M −→ M00 → 0 is called a short exact
sequence.
http://www.reed.edu/~iswanson/MSRI11SwansonIntroResolutions.pdf
A cocomplex is a complex that is numbered in the opposite order: C• :
di−1 di
· · · → Mi−1 −→ Mi −→ Mi+1 → · · ·.
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
What do we do with complexes?
(1) Given a module, build some corresponding complexes −→ resolutions of
modules (free, projective, flat, injective, minimal).
(2) Applying functors to (exact) complexes −→ Hom, tensor products, tensor
products of two complexes.
(3) Koszul complex: as tensor product of complexes (two ways), and there
is a definition via exterior powers.
(4) Kernels and images in a complex −→ homology.
(5) When are resolutions finite?
(6) When is a complex exact?
(7) What invariants (ranks?) appear in exact complexes?
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
d d d
Suppose that · · · → F −i→+1 F −→i F → · · · −→ F −→1 F → M → 0
i+1 i i−1 1 0
is exact and all the F are free (respectively projective, flat) modules over R,
j
and M is an R-module. Then we call both the exact complex above, as well
d d d
as the complex · · · → F −i→+1 F −→i F → · · · → F −→1 F → 0, a free
i+1 i i−1 1 0
(resp. projective, flat) resolution of M.
Every R-module has a free resolution, and hence a projective resolution. If
for F = 0 for some/all large n, we say that M has a finite free/projective
n
resolution. The smallest n such that F = F = · · · is called the length
n+1 n+2
of this resolution. The free/projective dimension of M is the smallest
such n (it could be ∞). This is denoted as pd(M).
Minimal resolutions
Suppose that 0 → M → I0 → I1 → I2 → I3 → · · · is exact, where each I is an
j
injective R-module. This cocomplex, as well as 0 → I0 → I1 → I2 → I3 → · · ·
are called an injective resolution of M.
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
Ways to make complexes from a given one
d d
C = · · · → C −→n C −n→−1 C → · · ·
• n n−1 n−2
(1) Hom from M: Hom (M, C ), with Hom(M, d ) = d ◦ :
R • n n
Hom(M,d ) Hom(M,d )
n n−1
· · · → Hom(M, C ) −−−→ Hom (M, C ) −−−−−−→ Hom (M, C ) → · · ·
n R n−1 R n−2
(2) Hom to M: Hom (C , M) is a cocomplex, with Hom(d , M) = ◦ d :
R • n n
Hom(d ,M) Hom(d ,M)
n n+1
· · · → Hom(C , M) −−−−−−→ Hom (C , M) −−−−−−→ Hom (C , M) → · · ·
n−1 R n R n+1
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
(3) Tensor product C ⊗ M:
• R
d ⊗id d ⊗id
C ⊗ M : · · · → C ⊗ M −−n−→ C ⊗ M −n−−−1 → C ⊗ M → · · · .
• R n R n−1 R n−2 R
(We say that M is flat if C ⊗ M is exact for every exact complex C .)
• •
e e
(4) Tensor product of complexes: Let K = · · · → K −→n K −n→−1 K →
• n n−1 n−2
· · · be another complex. The tensor product of complexes of C and K
• •
yields a kind of a bicomplex, as follows:
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m+1 n−1 m+1 n−2 m+1
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m n−1 m n−2 m
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m−1 n−1 m−1 n−2 m−1
↓ ↓ ↓
(cid:80)
(Total complex along 45◦ angle) G = C ⊗K , g : G → G defined
n i n−i n n n−1
i
on C ⊗ K as d ⊗ id + (−1)iid ⊗ e , where the first summand is
i n−i i K C n−i
n−i i
in C ⊗ K and the second in C ⊗ K .
i−1 n−i i n−i−1
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
(Here we show that the total complex of the tensor product of complexes is a
complex)
Recall (C , d ) ⊗ (K , e ) is:
• • • •
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m+1 n−1 m+1 n−2 m+1
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m n−1 m n−2 m
↓ ↓ ↓
C ⊗ K → C ⊗ K → C ⊗ K →
n m−1 n−1 m−1 n−2 m−1
↓ ↓ ↓
(cid:80)
Total complex: G = C ⊗ K , g : G → G restricted to C ⊗ K
n i n−i n n n−1 i n−i
i
is d ⊗ id + (−1)iid ⊗ e .
i K C n−i
n−i i
This new construction is still a complex:
i
g ◦ g | = g (d ⊗ id + (−1) id ⊗ e )
n−1 n C ⊗K n−1 i K C n−i
i n−i n−i i
i−1
= d ◦ d ⊗ id + (−1) d ⊗ e
i−1 i K i n−i
n−i
i i i
+ (−1) d ⊗ e + (−1) (−1) id ⊗ e ◦ e
i n−i C n−i−1 n−i
i
= 0.
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
Koszul complexes
Let R be a commutative ring, M a left R-module, and x ∈ R. The Koszul
complex of x and M is
x
K (x; M) : 0 → M −→ M → 0
•
↑ ↑
1 0
If x , . . . , x ∈ R, then the Koszul complex K (x , . . . , x ; M) of x , . . . , x
1 n • 1 n 1 n
and M is the total complex of K (x , . . . , x ; M) ⊗ K (x ; R), defined in-
• 1 n−1 • n
ductively.
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz
K (x , x ; M) explicitly: From
• 1 2
(cid:181) (cid:182) (cid:181) (cid:182)
x x
0 → M −→1 M → 0 0 → R −→2 R → 0
⊗
1 0 1 0
we get the total complex
(cid:183) (cid:184)
−x
2
x [x x ]
1 1 2
0 → M ⊗ R −−−→ M ⊗ R ⊕ M ⊗ R −−−−−−→ M ⊗ R → 0
1 1 1 0 0 1 0 0
complex willdo projinj Hom tensorcompl tensorprodcomp Koszul Koszul23 Koszulfunct homTorExt mapcompl fivelemma sesles Koszuldec Koszreg Koszregex HilbSyz